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Notes and References
Notes and References for Chapter 2
Page 14 limit yourself to using only one ortwo different shapes . . . These types of startingpoints were suggested by Adrian Pine. For manymore suggestions for polyhedra-making activi-ties, see Pinel’s 24-page booklet MathematicalActivity Tiles Handbook, Association of Teachersof Mathematics, Unit 7 Prime Industrial Park,Shaftesbury Street, Derby DEE23 8YB, UnitedKingdom website; www.atm.org.uk/.
Page 17 “What-If-Not Strategy” . . . S. I.Brown and M. I. Walter, The Art of ProblemPosing, Hillsdale, N.J.; Lawrence Erlbaum, 1983.
Page 18 easy to visualize. . . . MarionWalter, “On Constructing Deltahedra,” WiskobasBulletin, Jaargang 5/6, I.O.W.O., Utrecht,Aug. 1976.
Page 19 may be ordered . . . from the As-sociation of Teachers of Mathematics (addressabove).
Page 28 hexagonal kaleidocycle . . .D. Schattschneider and W. Walker, M. C. EscherKaleidocycles, Pomegranate Communications,Petaluma, CA; Taschen, Cologne, Germany.
Page 33 we suggest you try it . . . Ifyou get stuck, see Jim Morey, YouTube:theAmatour’s channel, http://www.youtube.com/user/theAmatour, instructional videos ofballoon polyhedra since November 6, 2007. Seealso http://moria.wesleyancollege.edu/faculty/morey/.
Page 34 a practical construction . . .see Morey above, and also Jeremy Shafer,“Icosahedron balloon ball,” Bay Area RapidFolders Newsletter, Summer/Fall 2007, andRolf Eckhardt, Asif Karim, and MarcusRehbein, “Balloon molecules,” http://www.balloonmolecules.com/, teaching molecularmodels from balloons since 2000. First describedin Nachrichten aus der Chemie 48:1541–1542,December 2000. The German website, http://www.ballonmolekuele.de/, has more informationincluding actual constructions.
Page 37 family of computationallyintractable “NP-complete” problems . . .Michael R. Garey and David S. Johnson,Computers and Intractability: A Guide to theTheory of NP-Completeness, W. H. Freeman &Co., 1979.
Page 37 the graph’s bloon number . . .Erik D. Demaine, Martin L. Demaine, andVi Hart. “Computational balloon twisting: Thetheory of balloon polyhedra,” in Proceedingsof the 20th Canadian Conference on Compu-tational Geometry, Montreal, Canada, August2008.
Page 37 tetrahedron requires only two bal-loons . . . see Eckhardt et al., above.
Page 37 a snub dodecahedron from thirtyballoons . . . see Mishel Sabbah, “Polyhe-drons: Snub dodecahedron and snub truncatedicosahedron,” Balloon Animal Forum posting,
291
292
March 7, 2008, http://www.balloon-animals.com/forum/index.php?topic=931.0. See also theYouTube video http://www.youtube.com/watch?v=Hf4tfoZC L4.
Page 37 Preserves all or most of the sym-metry of the polyhedron . . . see Demaine et al.,above.
Page 37 family of difficult “NP-complete”problems . . . see Demaine et al., above.
Page 37 regular polylinks . . . Alan Holden,Orderly Tangles: Cloverleafs, Gordian Knots,and Regular Polylinks, Columbia UniversityPress, 1983.
Page 39 the six-pentagon tangle and thefour-triangle tangle . . . see George W. Hart.“Orderly tangles revisited,” in MathematicalWizardry for a Gardner, pages 187–210, 2009,and http://www.georgehart.com/orderly-tangles-revisited/tangles.htm.
Page 37 computer software to find the rightthicknesses of the pieces . . . see Hart, above.
Page 39 polypolyhedra . . . Robert J.Lang. “Polypolyhedra in origami,” in Origami3:Proceedings of the 3rd International Meeting ofOrigami Science, Math, and Education, pages153–168, 2002, and http://www.langorigami.com/science/polypolyhedra/polypolyhedra.php4.
Page 39 balloon twisting . . . Carl D.Worth. “Balloon twisting,” http://www.cworth.org/balloon twisting/, June 5, 2007, and Demaineet al., above.
Page 39 video instructions . . . see, in par-ticular, Morey (above) and Eckhardt (above).
Notes and References for Chapter 3
Page 42 a very nice book on the history ofthese things . . . B. L. van der Waerden ScientificAwakening, rev. ed (Leiden: Noordhoff Interna-tional Publishing; New York, Oxford UniversityPress, 1974).
Page 42 Federico gives the details of thisstory. . . . P. J. Federico, Descartes on Polyhedra,(New York, Springer-Verlag, 1982).
Page 44 very nicely described in TheSleepwalkers by Arthur Koestler . . . A. KoestlerThe Watershed [from Koestler’s larger book, TheSleepwalkers] (Garden City, NY, Anchor Books,1960).
Page 45 In his little book Symmetry, . . .H. Weyl, Symmetry Princeton, NJ, PrincetonUniversity Press, 1952.
Page 45 Felix Klein did a hundred yearsago . . . F. Klein, Lectures on the Icosahedron,English trans., 2nd ed., New York, DoverPublications, 1956.
Page 51 may be any number except one.. . . Grunbaum and T. S. Motzkin “The Numberof Hexagons and the Simplicity of Geodesicson Certain Polyhedra,” Canadian Journal ofMathematics 15, (1963): 744–51.
Notes and References for Chapter 4
Page 53 two papyri . . . Gillings, R. J.,Mathematics in the Time of the Pharaohs. Cam-bridge, Mass.: MIT Press, 1972.
Page 54 volume of the frustrum of a pyramid. . . Gillings, R. J., “The Volume of a TruncatedPyramid in Ancient Egyptian Papyri,” The Math-ematics Teacher 57 (1964):552–55.
Page 55 singling them out for study . . .William C. Waterhouse, “The Discovery of theRegular Solids,” Archive for History of ExactSciences 9 (1972):212–21.
Page 55 By the time we get to Euclid . . .Heath, T. L. The Thirteen Books of Euclid’s Ele-ments. Cambridge, England: Cambridge Univer-sity Press, 1925; New York: Dover Publications,1956.
Page 56 “semiregular” solids discoveredby Archimedes . . . Pappus. La CollectionMathematique. P. Ver Eecke, trans. Paris: DeBrouwer/Blanchard, 1933. See also Como, S.,Pappus of Alexandria and the Mathematics ofLate Antiquity, New York: Cambridge U. Press,2000.
Notes and References 293
Page 56 They may date from theRoman period . . . Charriere, “NouvellesHypotheses sur les dodecaedres Gallo-Romains.”Revue Archeologique de l’Est et du Centre-Est16 (1965):143–59. See also Artmann, Benno,“Antike Darstellungen des Ikosaeders [Antiquerepresentations of the icosahedron],” Mitt. Dtsch.Math.-Ver. 13 (2005): 46–50.
Page 56 that have been cited by scholars . . .Conze, Westdeutsche Zeitschrift fur Geschichteund Kunst 11 (1892):204–10; Thevenot, E., “Lamystique des nombres chez les Gallo-Romains,dodecaedres boulets et taureauxtricomes,” Re-vue Archeolgique de l’Est et du Centre-Est 6(1955):291–95.
Page 56 they were used as surveying instru-ments . . . Friedrich Kurzweil, “Das Pentagon-dodekaeder des Museum Carnuntinun und seineZweckbestimmung,” Carnuntum Jahrbuch 1956(1957):23–29.
Page 56 the best guess is that they werecandle holders . . . F. H. Thompson, “Dodecahe-drons Again,” The Antiquaries Journal 50, pt. I(1970):93–96.
Page 56 A large collection of suchpolyhedral objects . . . F. Lindemann, “ZurGeschichte der Polyeder und der Zahlze-ichen,” Sitzungsberichte der Mathematisch-Physikalische Klasse der Koeniglich Akademieder Wissenschaften [Munich] 26 (1890):625–783and 9 plates.
Page 57 situation during the Renaissance isextremely complicated . . . Schreiber, Peter andGisela Fischer, Maria Luise Sternath, “New lighton the rediscovery of the Archimedean solidsduring the Renaissance.” Archive for History ofExact Sciences 62 2008:457–467.
Page 57 Pacioli actually made polyhedronmodels . . . Pacioli, L. Divina Proportione,Milan: 1509; Milan: Fontes Ambrosioni, 1956.
Page 57 related to the emergent study ofperspective . . . Descarques, P. Perspective, NewYork: Van Nostrand Reinhold, 1982.
Page 57 Professor Coxeter discusses . . .Coxeter, H. S. M., “Kepler and Mathematics,”Chapter 11.3 in Arthur Beer and Peter Beer, eds.,Kepler: Four Hundred Years, Vistas in Astronomy,vol. 18 (1974), pp. 661–70.
Page 57 Durer made nets for the dodec-ahedron . . . Durer, Albrecht, Unterweysungder Messung mit dem Zyrkel und Rychtscheyd,Nurnberg: 1525.
Page 58 two “new” “regular” (star) poly-hedra . . . Field J. V., “Kepler’s Star Polyhedra,”Vistas in Astronomy 23 (1979):109–41.
Page 58 Kepler’s work is constantly cited. . . Kepler, Johannes, Harmonices Mundi,Linz, 1619; Opera Omnia, vol. 5, pp. 75–334,Frankfort: Heyden und Zimmer, 1864; M. Caspar,ed., Johannes Kepler Gesammelte Werke, vol. 6,Munich: Beck, 1938.
Page 58 the papers of Lebesgue . . .Lebesgue, H., “Remarques sur les deux premieresdemonstrations du theoreme d’Euler, relatif auxpolyedres,” Bulletin de la Societe Mathematiquede France 52 (1924):315–36.
Page 58 the same statement in Polya . . .George Polya, Mathematical Discovery (NewYork: John Wiley and Sons, 1981), combineded., vol. 2, p. 154.
Page 58 nice paper by Peter Hiltonand Jean Pedersen . . . P. Hilton and J.Pedersen, “Descartes, Euler, Poincare, Polyaand Polyhedra,” L’Enseignment Mathematique27 (1981):327–43.
Page 58 an extensive summary of this debate. . . P. J. Federico, Descartes on Polyhedra. NewYork: Springer-Verlag, 1982.
Page 58 evident as they were, I had notperceived them . . . Jacques Hadamard, An Essayon the Psychology of Invention in the Mathemat-ical Field (Princeton, N.J.: Princeton UniversityPress, 1945), 51.
Page 59 elongated square gyrobicupola(Johnson solid J37) . . . Norman W. Johnson,“Convex Solids with Regular Faces”, CanadianJournal of Mathematics, 18, 1966, pages169–200.
Page 59 we can continue to honor his mem-ory . . . Grunbaum, Branko, “An Enduring Error,”Elemente der Mathematik 64 (2009):89–101.
Page 59 Euler, although he found hisformula, was not successful in proving it. . . Euler, Leonhard, “Elementa DoctrinaeSolidorum,” Novi Commentarii AcademiaeScientiarum Petropolitanae 4 (1752–53):
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109–40 (Opera Mathematica, vol. 26, pp. 71–93); Euler, Leonhard, “Demonstratio Non-nullarum Insignium Proprietatus Quibus SolidaHedris Planis Inclusa Sunt Praedita,” NoviCommentarii Academiae Scientiarum Petropoli-tanae 4 (1752–53):140–60 (Opera Mathematica,vol. 26, pp. 94–108).
Page 59 The first proof was providedby Legendre . . . Legendre, A., Elements degeometrie. 1st ed. Paris: Firmin Didot, 1794;or to him: there are claims that Meyer Hirschgave a correct proof (Hirsch, M. SammlungGeometrischer Aufgaben, part 2, Berlin, 1807)prior to Legendre (Malkevitch, Joseph, “TheFirst Proof of Euler’s Theorem,” Mitteilungen,Mathematisches Seminar, Universitat Giessen165 (1986):77–82.) I believe that this is an errorthat came about due to a misreading (Steinitz,Ernst, and H. Rademacher, Vorlesungen uber dieTheorie des Polyeder, Berlin: Springer-Verlag,1934) of something in Max Bruckner’s book,Bruckner, M, Vielecke und Vielflache, Leipzig:Teubner, 1900.
Page 59 providing different proofs . . .Lakatos, I., Proofs and Refutations. New York:Cambridge University Press, 1976; see alsoRicheson, D. S., Euler’s Gem, Princeton:Princeton U. Press, 2008.
Page 60 Poinsot’s work on the starpolyhedra . . . Poinsot, L., “Memoire sur lespolygones et les polyedres,” Journal de l’EcolePolytechnique 10 (1810):16–48; Poinsot, L,“Note sur la theorie des polyedres,” ComptesRendus 46 (1858):65–79.
Page 60 Other contributors . . . Cauchy,A. L, “Sur les polygones et les polyedres,”J. de l’Ecole Polytechnique 9 (1813):87–98; J. Bertrand (1858), and Cayley, A., “OnPoinsot’s Four New Regular Solids,” The London,Edinburgh, and Dublin Philosophical Magazineand Journal of Science, ser. 4, 17 (1859):123–28.
Page 60 Cauchy made major contributions. . . Cauchy, A. L, “Recherches sur lespolyedres,” Journal de l’Ecole Imp. Polytech-nique 9 (1813):68–86; Oeuvres Completes, ser. 2,vol. 1, pp. 1–25, Paris: 1905. See also Cauchy’spaper cited above.
Page 58 duals of the Archimedeanpolyhedra . . . Catalan, M. E., “Memoire surla theorie des polyedres,” Journal de l’Ecole Imp.Polytechnique 41 (1865):1–71.
Page 60 Max Bruckner published . . . seeBruckner, above.
Page 60 a paper in 1905 of D. M. Y.Sommerville . . . Sommerville, D. M. Y., “Semi-regular Networks of the Plane in AbsoluteGeometry,” Transactions of the Royal Society[Edinburgh] 41 (1905):725–47 and plates I–XII.
Page 61 challenges to mathematics inthe future . . . Hilbert, David, Mathematicalproblems, Lecture delivered at the Intern.Congress of Mathematicians at Paris in 1900,Bull. Amer. Math. Soc. (N.S) 37(2000), no. 4,407–436, reprinted from Bull. Amer. Math. Soc.8 (1902), 437–479.
Page 61 cutting a polyhedron into piecesand assembling them into another polyhedron. . . Boltianskii, V., Hilbert’s Third Problem, NewYork: John Wiley and Sons, 1975.
Page 61 these pieces can be assembled . . .Boltyanskii, V., Equivalent and Equidecompos-able Figs., Boston: D.C. Heath, 1963.
Page 61 extended Dehn’s work . . . see Had-wiger, Hugo and Paul Glur, “Zerlegungsgleich-heit ebener Polygone,” Elemente der Mathematik6 1951:97106.
Page 61 the theory of equidecomposability. . . Rajwade, A. R., Convex Polyhedra with Reg-ularity Conditions and Hilbert’s Third Problem,New Delhi: Hindustan Book Agency, 2001.
Page 61 can not be decomposed, using ex-isting vertices, into tetrahedra . . . Schoenhardt,E., “Uber die Zerlegung von Dreieckspolyed-ern in Tetraeder,” Mathematische Annalen, 981928:309–312; Toussaint, Godfried T. and ClarkVerbrugge, Cao An Wang, Binhai Zhu, “Tetra-hedralization of Simple and Non-Simple Polyhe-dra,” Proceedings of the Fifth Canadian Confer-ence on Computational Geometry, 1993:24–29.
Page 61 Lennes polyhedra . . . Lennes, N.J.,“On the simple finite polygon and polyhedron,”Amer. J. Math. 33 (1911): 37–62.
Page 61 the most important early twentieth-century contributor to the theory of polyhedra . . .Steinitz, Ernst, “Polyeder und Raumeinteilun-
Notes and References 295
gen,” in W. F. Meyerand and H. Mohrmann,eds., Encyklopadie der mathematischen Wis-senschaften, vol. 3. Leipzig: B. G. Teubner,1914–31.
Page 61 Lennes polyhedra . . . seeRademacher, above.
Page 62 if one starts with a 1 � 1 square. . . see Alexander, Dyson, and O’Rourke, “Theconvex polyhedra foldable from a square,” Proc.2002 Japan Conference on Discrete Computa-tional Geometry. Volume 2866, Lecture Notes inComputer Science, pp. 38–50, Berlin: Springer,(2003). Good sources of relatively recent workon a wide variety of aspects of polyhedra (con-vex and non-convex) from a combinatorial pointof view (e.g rigidity, Bellow’s conjecture, nets(folding to polyhedra), and unfolding algorithms,Steinitz’s Theorem (including the circle packingapproach to proving this seminal theorem, as wellas the dramatic extension due to the late OdedSchramm)) are Geometric Folding Algorithms:Linkages, Origami, and Polyhedra by Eric De-maine and Joseph O’Rourke (Demaine, E. and J.O’Rourke, Geometric Folding Algorithms, NewYork: Cambridge, U. Press. 2007) and Pak, I.Lectures on Discrete and Polyhedral Geometry(to appear).
Page 62 the stellated icosahedra . . .Coxeter, H. S. M., P. du Val, H. T. Flather, and J.F. Petrie, The Fifty-nine Icosahedra,. New York:Springer-Verlag, 1982 (reprint of 1938 edition.)
Page 62 famous work on uniform polyhedra. . . Coxeter, H. S. M., M. S. Longuet-Higgins,and J. C. P. Miller, Uniform Polyhedra, Philo-sophical Transactions of the Royal Society [Lon-don], sec. A, 246 (1953/56):401–50; Skilling,J., “The Complete Set of Uniform Polyhedra,”Philosophical Transactions of the Royal Society[London], ser. A, 278, (1975):111–35.
Page 62 many important results in severalbranches of mathematics . . . Coxeter, H. S. M.Regular Polytopes. 3rd ed. New York: DoverPublications, 1973; Coxeter, H. S. M., RegularComplex Polytopes. London: Cambridge Univer-sity Press, 1974.
Page 62 O. Rausenberger shows . . .Rausenberger, O., “Konvex pseudoregularePolyeder.” Zeitschr. fur math. u. maturwiss. Utr-
erricht 1915:135–142. Probably independentlydiscovered by Hans Freudenthal and B. L. vander Waerden, “Over een bewering van Euclides,”Simon Stevin 25 (1946/47) 115–121.
Page 62 Johnson, Grunbaum, V. A.Zalgaller et al. prove Johnson’s conjecture.. . . Grunbaum, Branko, and N. W. Johnson.“The Faces of a Regular-faced Polyhedron,”Journal of the London Mathematical Society40 (1965):577–86; Johnson, N. W., “ConvexPolyhedra with Regular Faces.” CanadianJournal of Mathematics 18 (1966):169–200;Zalgaller, V. A., Convex Polyhedra with RegularFaces, New York: Consultants Bureau, 1969.See also Cromwell, P., Polyhedra, New York:Cambridge U. Press, 2008.
Page 63 his beautiful book . . . Grunbaum,Branko. Convex Polytopes. New York: John Wi-ley and Sons, 1967. Second Edition, New York:Springer, 2003.
Page 63 he published an article in 1977. . . Grunbaum, Branko, “Regular Polyhedra–Old and New.” Aequationes Mathematicae 16(1977):1–20.
Page 63 list of regular polyhedra thatGrunbaum gave is complete . . . Dress, AndreasW. M., “A Combinatorial Theory of Grunbaum’sNew Regular Polyhedra. Part I. Grunbaum’sNew Regular Polyhedra and Their Automor-phism Groups,” Aequationes Mathematicae 23(1981):252–65; Dress, Andreas W. M., “A Com-binatorial Theory of Grunbaum’s New RegularPolyhedra, Part II. Complete Enumeration,”Aequationes Math. 29 (1985) 222–243.
Notes and References for Chapter 5
Page 67 Dividing Equation 5.1 by 2E andthen substituting . . . The division of both sidesof Equation 5.1 by the summation is based on theassumption that rav and nav remain finite even inthe limit of infinite number of vertices and edges.The detailed justification of this assumption isbeyond the scope of this paper, but it is limitedto surfaces in which the density of vertices andedges (averaged over a region whose area is
296
large compared to that of a face) is reasonablyuniform throughout the structure. Obviously thisassumption is valid for periodically repeatingstructures.
Page 68 the number of hexagons can be anypositive integer except 1 . . . B. Grunbaum andT.S. Motzkin, “The Number of Hexagons and theSimplicity of Geodesics on Certain Polyhedra,”Canadian Journal of Mathematics 15 (1963):744–51.
Page 68 Equation 5.6 still yields anenumerable set of solutions . . . Arthur L.Loeb, Space Structures: Their Harmony andCounterpoint, Reading, Mass. Addison-Wesley,Advanced Book Program, 1976.
Notes and References for Chapter 6
Page 77 masterwork on geometry . . .Albrecht Durer. The painter’s manual: A manualof measurement of lines, areas, and solids bymeans of compass and ruler assembled byAlbrecht Durer for the use of all lovers of artwith appropriate illustrations arranged to beprinted in the year MDXXV, New York: AbarisBooks, 1977, 1525, English translation by WalterL. Strauss of ‘Unterweysung der Messung mitdem Zirkel un Richtscheyt in Linien Ebnen uhndGantzen Corporen’.
Page 79 sharp mathematical question . . .Geoffrey C. Shephard, “Convex polytopes withconvex nets,” Math. Proc. Camb. Phil. Soc.,78:389–403, 1975.
Page 79 researchers independently dis-covered . . . Alexey S. Tarasov, “Polyhe-dra with no natural unfoldings,” RussianMath. Surv., 54(3):656–657, 1999; Mar-shall Bern, Erik D. Demaine, David Epp-stein, Eric Kuo, Andrea Mantler, and JackSnoeyink, “Ununfoldable polyhedra withconvex faces,” Comput. Geom. Theory Appl.,24(2):51–62, 2003; Branko Grunbaum, “No-net polyhedra,” Geombinatorics, 12:111–114,2002.
Page 83 number is exponential in the squareroot of the number of faces . . . Erik D. De-maine and Joseph O’Rourke, Geometric FoldingAlgorithms: Linkages, Origami, Polyhedra, Cam-bridge University Press, July 2007, http://www.gfalop.org.
Page 83 every prismoid has a net . . . JosephO’Rourke, Unfolding prismoids without overlap,Unpublished manuscript, May 2001.
Page 84 three different proofs . . . Demaineand O’Rourke, above; Alex Benton and JosephO’Rourke, Unfolding polyhedra via cut-treetruncation, In Proc. 19th Canad. Conf. Comput.Geom., pages 77–80, 2007; Val Pincu, “On thefewest nets problem for convex polyhedra,”In Proc. 19th Canad. Conf. Comput. Geom.,pages 21–24, 2007.
Page 85 we describe here . . . (see Demaineand O’Rourke, above, for another).
Page 85 This concept was introducedby Alexandrov . . . Aleksandr DanilovichAlexandrov, Vupyklue Mnogogranniki. Gosy-darstvennoe Izdatelstvo Tehno-TeoreticheskoiLiteraturu, 1950. In Russian; Aleksandr D.Alexandrov, Convex Polyhedra, Springer-Verlag,Berlin, 2005, Monographs in Mathematics.Translation of the 1950 Russian edition byN. S. Dairbekov, S. S. Kutateladze, andA. B. Sossinsky.
Page 85 Alexandrov unfolding . . . EzraMiller and Igor Pak, “Metric combinatorics ofconvex polyhedra: Cut loci and nonoverlappingunfoldings, Discrete Comput. Geom.,” 39:339–388, 2008.
Page 85 proved to be non-overlapping. . . Boris Aronov and Joseph O’Rourke,“Nonoverlap of the star unfolding,” DiscreteComput. Geom., 8:219–250, 1992.
Page 86 when Q shrinks down to x . . .Jin-ichi Itoh, Joseph O’Rourke, and CostinVılcu, “Star unfolding convex polyhedra viaquasigeodesic loops,” Discrete Comput. Geom.,44, 2010, 35–54.
Page 86 This enticing problem remains un-solved today . . . See Demaine and O’Rourke,above.
Notes and References 297
Page 86 unfolds any orthogonal polyhedron. . . Mirela Damian, Robin Flatland, andJoseph O’Rourke, “Epsilon-unfolding orthog-onal polyhedra,” Graphs and Combinatorics,23[Suppl]:179–194, 2007. Akiyama-ChvatalFestschrift.
Page 86 Unsolved problems abound in thistopic . . . see Demaine and O’Rourke, Chapter25, above; and Joseph O’Rourke, “Folding poly-gons to convex polyhedra,” In Timothy V. Craineand Rheta Rubenstein, editors, UnderstandingGeometry for a Changing World: National Coun-cil of Teachers of Mathematics, 71st Yearbook,pages 77–87. National Council of Teachers ofMathematics, 2009.
Notes and References for Chapter 8
Page 123 variations on a theme is reallythe crux of creativity . . . Douglas R. Hofstadter,“Metamagical Themas: Variations on a ThemeAs the Essence of Imagination,” Scientific Amer-ican 247, no. 4 (October 1982): 20–29.
Page 123 You see things, and you say ‘why?’. . . George Bernard Shaw, Back to Methuselath,London: Constable (1921).
Notes and References for Chapter 9
Page 126 forms that are simultaneouslymathematical and organic . . . George W. Hart,website, http://www.georgehart.com
Page 126 black and white version becameofficial . . . http://www.soccerballworld.com/History.htm accessed, June 2009.
Page 128 particle detectors sphericallyarranged with the structure of GP(2,1) . . .N.G. Nicolis, “Polyhedral Designs of DetectionSystems for Nuclear Physics Studies”, MeetingAlhambra, Proceedings of Bridges 2003,J. Barrallo et al. eds., University of Granada,Granada, Spain, pp. 219–228.
Page 129 spherical allotropes of carbon . . .Aldersey-Williams, Hugh, The Most BeautifulMolecule: The Discovery of the Buckyball. Wiley,1995; P. W. Fowler, D. E. Manolopoulos, An Atlasof Fullerenes, Oxford University Press, 1995.
Page 129 truncated icosahedron was al-ready a well-known mathematical structure . . .see Aldersey-Williams, above.
Page 129 If we want to have a structure,we have to have triangles . . . R. BuckminsterFuller, Synergetics: Explorations in the Ge-ometry of Thinking, MacMillan, 1975, Section610.12
Page 132 the basic geometric ideas . . .see Clinton, J.D., 1971, Advanced StructuralGeometry Studies Part I - Polyhedral SubdivisionConcepts for Structural Applications, Wash-ington, D.C.: NASA Tech. Report CR-1734.,1971; H.S.M. Coxeter, “Virus Macromoleculesand Geodesic Domes,” in A Spectrum ofMathematics, J.C. Butcher (editor), Aukland,1971; J. Francois Gabriel, Beyond the Cube:The Architecture of Space Frames and Polyhedra,Wiley, 1997; Hugh Kenner, Geodesic Math andHow to Use It, University of California Press,1976; Magnus J. Wenninger, Spherical Models,Cambridge, 1979, (Dover reprint 1999).
Page 133 In the optimal solution . . .Michael Goldberg, “The Isoperimetric Problemfor Polyhedra,” Tohoku Mathematics Journal, 40,1934/5, pp. 226–236.
Page 133 Details of one algorithm for thegeneral (a,b) case . . . George W. Hart, “Retic-ulated Geodesic Constructions” Computers andGraphics, Vol 24, Dec, 2000, pp. 907–910.
Page 133 the center of gravity of thesetangency points is the origin . . . Gunter M.Ziegler, Lectures on Polytopes, Springer-Verlag,1995, p. 118.
Page 133 a simple fixed-point iteration . . .George W. Hart, “Calculating Canonical Polyhe-dra,” Mathematica in Education and Research,Vol 6 No. 3, Summer 1997, pp. 5–10.
Page 134 any desired number of differentforms . . . Michael Goldberg, “A Class of Multi-Symmetric Polyhedra,” Tohoku MathematicsJournal, 43, 1937, pp. 104–108.
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Page 134 Paper models . . . See MagnusJ. Wenninger, “Artistic Tessellation Patterns onthe Spherical Surface,” International Journal ofSpace Structures, Vol. 5, 1990, pp. 247–253.
Page 135 models far too tedious to cutwith scissors . . . George W. Hart, “ModularKirigami,” Proceedings of Bridges Donostia, SanSebastian, Spain, 2007, pp. 1–8.
Page 135 an equal-area projection . . .Snyder, John P. (1992), “An equal-area mapprojection for polyhedral globes,” Cartographica29: 10–21.
Page 137 models of two-layer spheres. . . George W. Hart, Two-layer dome instruc-tions, http://www.georgehart.com/MathCamp-2008/dome-two-layer.html
Notes and References for Chapter 10
Page 139 nonperiodic arrangements existin the solid state . . . Concerning crystalswith 5-fold symmetry, see D. Shechtman, I.Gratias, and J. W. Cahn [“Metallic Phasewith Long-range Orientational Order and NoTranslational Symmetry,” Physical ReviewLetters 53 (1984): 1951–53]. Their paper haschanged our views of the solid state. For anoverview of the rapidly-developing mathematicaltheory—including a role for polyhedra—seeM. Senechal, Quasicrystals and Geometry,Cambridge University Press, corrected paperbackedition, 1996; reprinted in 2009.
Page 142 the methods are used for the de-scription of crystal structures . . . see, for ex-ample, A. L. Loeb, “A Systematic Survey ofCubic Crystal Structures,” Journal of Solid StateChemistry I (1970):237–67.
Page 142 a different kind of basic unit . . .G. Lejeune Dirichlet, “Uber die Reduction derpositiven quadratischen Formen mit drei unbes-timmten ganzen Zahlen,” Journal fur die reineund angewandte Mathematik 40 (1850):209–27.
Page 145 all cubic crystal structures . . .For details of the classification see Chung Chieh,“The Archimedean Truncated Octahedron,and Packing of Geometric Units in Cubic
Crystal Structures,” Acta Crystallographica,sec. A, 35 (1979):946–52; “The ArchimedeanTruncated Octahedron II. Crystal Structureswith Geometric Units of Symmetry 43m,” ActaCrystallographica, sec. A, 36(1980):819–26.“The Archimedean Truncated Octahedron III.Crystal Structures with Geometric Units ofSymmetry m3m,” Acta Crystallographica, sec.A, 38 (1982):346–49. Chung Chieh, HansBurzlaff, and Helmuth Zimmerman, “Commentson the Relationship between ‘The ArchimedeanTruncated Octahedron, and Packing of GeometricUnits in Cubic Crystal Structures’ and ‘Onthe Choice of Origins in the Description ofSpace Groups’,” Acta Crystallographica, sec.A, 38(1982):746–47.
Page 145 crystal structure of a � brass. . . M. H. Booth, J. K. Brandon, R.Y. Brizard,C. Chieh, and W. B. Pearson, “� Brasses withF Cells,” Acta Crystallographica, sec. B, 33(1977):30–30 and references therein.
Page 146 Dirichlet domains of tetragonal,rhombohedral, and hexagonal lattices . . .Chung Chieh, “Geometric Units in Hexagonaland Rhombohedral Space Groups,” ActaCrystallographica, sec. A, 40(1984):567–71.
Page 148 geometric plan of the tetrago-nal space groups . . . Chung Chieh, “Geomet-ric Units in Tetragonal Crystal Structures,” ActaCrystallographica, sec. A, 39 (1983):415–21.
Page 148 a series of organometallic com-pounds . . . Chung Chieh, “Crystal Chemistryof Tetraphenyl Derivatives of Group IVB Ele-ments,” Journal of the Chemical Society [Lon-don] Dalton Transactions (1972):1207–1208.
Notes and References for Chapter 11
Page 153 Coxeter has said . . . H.S.M Cox-eter, “Preface to the First Edition,” in RegularPolytopes 3rd ed., New York: Dover Publications,1973.
Page 153 termed polyhedral chemistry. . . .A more extensive discussion of the shapes andsymmetries of molecules and crystals can be
Notes and References 299
found in M. Hargittai and I. Hargittai, Symmetrythrough the Eyes of a Chemist, Third Edition,Springer, hardcover, 2009; softcover, 2010.
Page 153 Muetterties movingly describedhis attraction . . . Earl L. Muetterties, ed., BoronHydride Chemistry, New York: Academic Press,1975.
Page 154 Two independent studies . . . V.PSpiridonov and G. I. Mamaeva, “Issledovaniemolekuli bordigrida tsirkoniya metodom gazovoielektronografii,” Zhurnal Strukturnoi Khimii,10 (1969), 133–35. Vernon Plato and KennethHedberg, “An Electron-Diffraction Investigationof Zirconium Tetraborohydride, Zr.BH4/4,”Inorganic Chemistry 10 (1970), 590–94. Inone interpretation (Spiridonov and Mamaeva,“Issledovanie molekuli bordigrida tsirkoniyametodom gazovoi elektronografii”) there arefour Zr � B bonds; according to the other : : :there is no Zr � B bond (Plato and Hedberg, “AnElectron-Diffraction Investigation of ZirconiumTetraborohydride, Zr.BH4/4.”)
Page 154 sites are taken by carbon atoms. . . see Muetterties, above.
Page 154 removing one or more polyhedralsites . . . Robert E. Williams, “Carboranes andBoranes; Polyhedra and Polyhedral Fragments,”Inorganic Chemistry 10 (1971), 210–14; RalphW. Rudolph, “Boranes and Heteroboranes: AParadigm for the Electron Requirements of Clus-ters?” Accounts of Chemical Research 9 (1976),446–52.
Page 155 energetically most advantageous. . . A. Greenberg and J.F. Liebman, StrainedOrganic Molecules, New York: Academic Press,1978.
Page 155 an attractive and challengingplayground . . . See the previous reference andLloyd N. Ferguson, “Alicyclic Chemistry: ThePlayground for Organic Chemists,” Journal ofChemical Education 46, (1969), 404–12.
Page 155 amazingly stable . . . GuntherMaier, Stephan Pfriem, Ulrich Shafer, andRudolph Matush, “Tetra-tert-butyltetrahedrane,”Angewandte Chemie, International Edition inEnglish 17 (1978), 520–21.
Page 155 known for some time . . . PhilipE. Eaton and Thomas J. Cole, “Cubane,” Journal
of the American Chemical Society 86 (1964),3157–58.
Page 155 prepared more recently . . . RobertJ. Ternansky, Douglas W. Balogh, and Leo A.Paquette, “Dodecahedrane,” Journal of the Amer-ican Chemical Society 104 (1982), 4503–04.
Page 155 was predicted . . . H.P. Schultz,“Topological Organic Chemistry. Polyhedranesand Prismanes,” Journal of Organic Chemistry 30(1965), 1361–64
Page 157 Triprismane . . . Thomas J. Katzand Nancy Acton, “Synthesis of Prismane,” Jour-nal of the American Chemical Society 95 (1973),2738–39; Nicholas J. Turro, V. Ramamurthy, andThomas J. Katz, “Energy Storage and Release.Direct and Sensitized Photoreactions of DewarBenzene and Prismane,” Nouveau Journal deChimie1 (1977), 363–65.
Page 157 and pentaprismane . . . Philip E.Eaton, Yat Sun Or, and Stephen J. Branca, “Pen-taprismane,” Journal of the American ChemicalSociety 103 (1981), 2134–36.
Page 157 only a few are stable . . . DanFarcasiu, Erik Wiskott, Eiji Osawa, Wilfried Thi-elecke, Edward M. Engler, Joel Slutsky, Paul v.R. Schleyer, and Gerald J. Kent, “Ethanoadaman-tane. The Most Stable C12H18 Isomer,” Jour-nal of the American Chemical Society 96(1974),4669–71.
Page 157 The name iceane was proposedby Fieser . . . Louis F. Fieser, “Extensions in theUse of Plastic Tetrahedral Models,” Journal ofChemical Education 42 (1965), 408–12.
Page 158 The challenge was met. . . . ChrisA. Cupas and Leonard Hodakowski, “Iceane,”Journal of the American Chemical Society 96(1974), 4668–69.
Page 158 Diamond has even been called. . . Chris Cupas, P. von R. Schleyer, and DavidJ. Trecker, “Congressane,” Journal of the Ameri-can Chemical Society 87 (1965) 917–18. TamaraM. Gund, Eiji Osawa, Van Zandt Williams, andP. v R. Schleyer, “Diamantane. I. Preparationof Diamantane. Physical and Spectral Proper-ties.” Journal of Organic Chemistry 39 (1974)2979–87. The high symmetry of adamantane isemphasized when its structure is described byfour imaginary cubes: Here it is assumed that the
300
two different kinds of C – H bonds in adamantanehave equal length. See I. Hargittai and K. Hed-berg, in S.J Cyvin, ed., Molecular Structures andVibrations (Amsterdam: Elsevier, 1972).
Page 159 Joined tetrahedra are even moreobvious . . . A. F. Wells, Structural InorganicChemistry 5th Edition (New York: Oxford Uni-versity Press, 1984).
Page 159 similar consequences in molec-ular shape and symmetry . . . M. Hargittai andI. Hargittai, The Molecular Geometries of Coor-dination Compounds in the Vapour Phase (Bu-dapest: Akademiai Kiado; Amsterdam: Elsevier,1977).
Page 159 The H3N � AlCl3 donor–acceptorcomplex . . . Magdolna Hargittai, Istvan Hargit-tai, Victor P. Spiridonov, Michel Pelissier, andJean F. Labarre, “Electron Diffraction Study andCNDO/2 Calculations on the Complex of Alu-minum Trichloride with Ammonia,” Journal ofMolecular Structure 24 (1975), 27–39.
Page 160 the model with two halogenbridges . . . E. Vajda, I. Hargittai, and J.Tremmel, “Electron Diffraction Investigationof the Vapour Phase Molecular Structure ofPotassium Tetrafluoro Aluminate,” InorganicChimica Acta 25 (1977), L143–L145.
Page 160 the barrier to rotation and free-energy difference . . . A. Haaland and J.E. Nils-son, “The Determination of Barriers to InternalRotation by Means of Electron Diffraction. Fer-rocene and Ruthenocene,” Acta Chemica Scandi-navica 22 (1968), 2653–70.
Page 160 structures with beautiful andhighly symmetric polyhedral shapes . . . F.A. Cotton and R. A. Walton, Multiple Bondsbetween Metal Atoms, New York: Wiley-Interscience, 1982.
Page 160 One example is . . . F. A. Cottonand C. B. Harris, “The Crystal and MolecularStructure of Dipotassium Octachlorodirhen-ate(III) Dihydrate, K2ŒRe2Cl8� � 2H2O,” In-organic Chemistry 4 (1965), 330–33; V.G.Kuznetsov and P. A. Koz’min, “A Study of theStructure of .PyH/HReCl4,” Zhurnal StrukturnoiKhimii 4 (1963), 55–62.
Page 160 the paddlelike structure ofdimolybdenum tetra-acetate . . . M. H. Kelley
and M. Fink, “The Molecular Structure ofDimolybdenum Tetra-acetate,” Journal ofChemical Physics 76 (1982), 1407–16.
Page 160 hydrocarbons called paddlanes. . . E. H. Hahn, H. Bohm, and D. Ginsburg, “TheSynthesis of Paddlanes: Compounds in WhichQuaternary Bridgehead Carbons Are Joinedby Four Chains,” Tetrahedron Letters (1973),507–10.
Page 160 in which interactions betweenbridgehead carbons have been interpreted . . .Kenneth B. Wiberg and Frederick H. Walker,“[1.1.1] Propellane,” Journal of the AmericanChemical Society 104 (1982), 5239–40; James E.Jackson and Leland C. Allen, “The C1–C3 Bondin [1.1.1] Propellane,” Journal of the AmericanChemical Society 106 (1984), 591–99.
Page 160 distances are equal withinexperimental error . . . Vernon Plato, WilliamD. Hartford, and Kenneth Hedberg, “Electron-Diffraction Investigation of the MolecularStructure of Trifluoramine Oxide, F3NO,”Journal of Chemical Physics 53 (1970), 3488–94.
Page 160 nonbonded distances . . . IstvanHargittai, “On the Size of the Tetra-fluoro 1,3-dithietane Molecule,” Journal of MolecularStructure 54 (1979), 287–88.
Page 161 remarkably constant at 2.48 A. . . Istvan Hargittai, “Group Electronegativities:as Empirically Estimated from Geometrical andVibrational Data on Sulphones,” Zeitschrift furNaturforschung, part A 34 (1979), 755–60.
Page 161 depending on the X and Y ligands. . . I. Hargittai, The Structure of Volatile SulphurCompounds (Budapest: Akademiai Kiado; Dor-drecht: Reidel, 1985).
Page 162 the SO bond distances differ by0.15 A . . . Robert L. Kuczkowski, R. D. Suen-ram, and Frank J. Lovas “Microwave Spectrum,Structure and Dipole Moment of Sulfuric Acid,”Journal of the American Chemical Society 103(1981), 2561–66.
Page 162 such molecules are nearly regulartetrahedral . . . K. P. Petrov, V. V. Ugarov,and N. G. Rambidi, “Elektronograficheskoeissledovanie stroeniya molekuli Tl2SO4,”Zhurnal Strukturnoi Khimii 21 (1980), 159–61.
Notes and References 301
Page 162 simple and successful model . . .R. J. Gillespie, Molecular Geometry, New York:Van Nostrand Reinhold, 1972.
Page 163 nicely simulated by balloons . . .H. R. Jones and R. B. Bentley, “Electron-Pair Re-pulsions, A Mechanical Analogy,” Proceedingsof the Chemical Society [London]. (1961), 438–440; : : : nut clusters on walnut trees see GavrilNiac and Cornel Florea, “Walnut Models of Sim-ple Molecules,” Journal of Chemical Education57 (1980), 4334–35.
Page 165 attention to angles of lone pairs. . . I. Hargittai, “Trigonal-Bipyramidal Molecu-lar Structures and the VSEPR Model,” InorganicChemistry 21 (1982), 4334–35.
Page 165 results from quantum chemicalcalculations . . . Ann Schmiedekamp, D. W. J.Cruickshank, Steen Skaarup, Peter Pulay, IstvanHargittai, and James E. Boggs, “Investigationof the Basis of the Valence Shell Electron PairRepulsion Model by Ab Initio Calculation ofGeometry Variations in a Series of Tetrahedraland Related Molecules,” Journal of the Ameri-can Chemical Society 101 (1979), 2002–10; P.Scharfenberg, L. Harsanyi, and I. Hargittai, un-published calculations, 1984.
Page 165 equivalent to a rotation . . . R.S. Berry, “The New Experimental Challengesto Theorists,” in R. G. Woolley, ed., QuantumDynamics of Molecules, New York: PlenumPress, 1980.
Page 165 lifetime of a configuration andthe time scale of the investigating technique . . .E. L Muetterties, “Stereochemically NonrigidStructures,” Inorganic Chemistry 4 (1965),769–71.
Page 165 experimentally determined struc-tures occur . . . Zoltan Varga, Maria Kolonits,Magdolna Hargittai, “Gas-Phase Structures ofIron Trihalides: A Computational Study of allIron Trihalides and an Electron Diffraction Studyof Iron Trichloride.” Inorganic Chemistry 43(2010):1039–45; Zoltan Varga, Maria Kolonits,Magdolna Hargittai, “Iron dihalides: structuresand thermodynamic properties from computationand an electron diffraction study of irondiiodide” Structural Chemistry, 22 (2011) 327–36; Magdolna Hargittai, “Metal Halide Molecular
Structures” Chemical Reviews 100 (2000):2233–301.
Page 165 different kinds of carbon positions. . . W. v. E. Doering and W. R. Roth, “A RapidlyReversible Degenerate Cope Rearrangement.Bicyclo[5.1.0]octa-2, 5-diene,” Tetrahedron 19(1963), 715–737; G. Schroder, “Preparationand Properties of Tricyclo[3, 3, 2, 04;6]deca-2, 7, 9-triene (Bullvalene),” Angewandte Chemie,International Edition in English 2 (1963), 481–82; Martin Saunders, “Measurement of the Rateof Rearrangement of Bullvalene,” TetrahedronLetters (1963), 1699–1702.
Page 165 hydrocarbon whose trivial namewas chosen . . . J. S. McKennis, Lazaro Brener, J.S. Ward, and R. Pettit, “The Degenerate Cope Re-arrangement in Hypostrophene, a Novel C10H10
Hydrocarbon,” Journal of the American Chemi-cal Society 93 (1971), 4957–58.
Page 166 discovered by Berry . . . R.Stephen Berry, “Correlation of Rates ofIntramolecular Tunneling Processes, withApplication to Some Group V Compounds,”Journal of Chemical Physics 32 (1960), 933–38.
Page 167 similar pathway was established. . . George M. Whitesides and H. LeeMitchell, “Pseudorotation in .CH3/2NPF4,” Jour-nal of the American Chemical Society 91 (1969),5384–86.
Page 167 permutation of nuclei in five-atom polyhedral boranes . . . Earl L. Muettertiesand Walter H. Knoth, Polyhedral Boranes,New York: Marcel Dekker, 1968. In onemechanism (W. N. Lipscomb, “FrameworkRearrangement in Boranes and Carboranes,”Science 153 (1966), 373–78); illustrated byinterconversion (R. K. Bohn and M. D Bohn,“The Molecular Structure of 1,2-,1,7-, and 1,12-Dicarba-closo-dodecaborane(12), B10C2H12,”Inorganic Chemistry 10 (1971), 350–55.). Asimilar model has been proposed (Brian F. G.Johnson and Robert E. Benfield, “Structuresof Binary Carbonyls and Related Compounds.Part 1. A New Approach to Fluxional Behaviour,”Journal of the Chemical Society [London] DaltonTransactions (1978), 1554–68.)
Page 168 even in the solid state . . . Brian E.Hanson, Mark J. Sullivan, and Robert J. Davis,
302
“Direct Evidence for Bridge-Terminal CarbonylExchange in Solid Dicobalt Octacarbonyl byVariable Temperature Magic Angle Spinning13C NMR Spectroscopy,” Journal of theAmerican Chemical Society 106 (1984),251–53.
Page 168 polyhedron whose vertices areoccupied by carbonyl oxygens . . . RobertE. Benfield and Brian F. G. Johnson, “TheStructures and Fluxional Behaviour of the BinaryCarbonyls; A New Approach. Part 2. ClusterCarbonyls Mm.CO/n (n = 12, 13, 14, 15, or 16),”Journal of the Chemical Society [London] DaltonTransactions (1980), 1743–67.
Page 168 Kepler’s planetary system . . .Johannes Kepler, Mysterium Cosmographicum,1595.
Page 168 observed in mass spectra . . .E. A. Rohlfing, D. M. Cox, and A. Kaldor,“Production and Characterization of SupersonicCarbon Cluster Beams,” Journal of ChemicalPhysics 81 (1984), 3322–3330; the structurewas assigned to it (H. W. Kroto, J. R. Heath,S. C. O’Brien, R. F. Curl, and R. E. Smalley,“C60:Buckminsterfullerene,” Nature 318 (1985),162–163); produced for the first time (W.Kratschmer, L. D. Lamb, K. Fostiropoulos,and D. R. Huffman, “Solid C60: a new formof carbon,” Nature 347 (1990), 354–358).
Notes and References for Chapter 12
Page 172 Lucretius (99–50 B.C.E.) arguedthat motion is impossible . . . Titus LucretiusCarus, De Rerum Natura: book I, lines 330–575;book II, lines 85-308; see Cyril Bailey, trans.,Lucretius on the Nature of Things (Oxford:Oxford University Press, 1910), pp. 38–45,68–75.
Page 173 William Rankine (1820–1872)proposed a theory of molecular vortices . . .William John Macquorn Rankine, “On theHypothesis of Molecular Vortices, or CentrifugalTheory of Elasticity, and Its Connexion with theTheory of Heat,” Philosophical Magazine, ser. 4,10 (1855):411.
Page 173 Hermann von Helmholtz (1821–1894) derived mathematical expressions . . . H.Helmholtz, “On Integrals of the HydrodynamicalEquations, Which Express Vortex-Motion,”Philosophical Magazine, ser. 4, 33 suppl.(1867):485; “from Crelle’s Journal, LV (1858),kindly communicated by Professor Tait.”
Page 173 Lord Kelvin . . . Sir WilliamThomson, “On Vortex Atoms,” PhilosophicalMagazine, ser. 4, 34 (1867):15.
Page 173 a lecture demonstration of smokerings by Tait . . . Cargill Gilston Knott, Lifeand Scientific Work of Peter Guthrie Tait, Cam-bridge, England: Cambridge University Press,1911, p. 68.
Page 173 Tait described an apparatus toproduce smoke rings . . . P. G. Tait, Lectureson Some Recent Advances in Physical Science,London, 1876, p. 291.
Page 173 investigations on the analyticgeometry of knots . . . Peter Guthrie Tait,Scientific Papers, 2 vols., Cambridge UniversityPress, 1898, I, pp. 270–347; papers originallypublished 1867–85.
Page 173 his search after the true interpre-tation of the phenomena . . . J. C. Maxwell, “OnPhysical Lines of Force. Part II. —The Theoryof Molecular Vortices applied to Electric Cur-rents.” Philosophical Magazine, ser. 4, 21 (1861):281–91.
Page 174 demonstrating that there isno aether . . . Loyd S. Swenson, Jr., “TheMichelson-Morley-Miller Experiments beforeand after 1905,” The Journal for the History ofAstronomy 1 (1970):56–78.
Page 174 introduced into the mainstream ofEuropean chemistry . . . Joseph Le Bel, Bulletinde la Societe Chimique [Paris], 22 (November1874), 337–47; J. H. van ’t Hoff, The Arrange-ment of Atoms in Space, 2nd ed., London: Long-mans, Green, 1898.
Page 175 Alfred Werner (1866–1919) usedoctahedra to model metal complexes . . . A.Werner, “Beitrag zur Konstitution anorganischerVerbindungen,” Zeitschrift fur anorganischeChemie 3 (1893):267–330.
Page 175 useful in teaching general chem-istry . . . G. N. Lewis, Valence and The Structure
Notes and References 303
of Atoms and Molecules, New York: The Chemi-cal Catalog Company, 1923, p. 29.
Page 175 developed the cubic model in amore formal manner . . . G. N. Lewis, “TheAtom and the Molecule,” Journal of the Ameri-can Chemical Society 38, (1916): 762–85.
Page 175 60-carbon cluster molecule . . . H.W. Kroto, J. R. Heath, S. C. O’Brien, R. F. Curl,and R. E. Smalley, “C60: Buckminsterfullerene,”Nature 318 (1985): 162–63.
Page 176 Francis Crick and James Watsonsuggested . . . F. H. C. Crick and J. D. Wat-son, “Structure of Small Viruses,” Nature 177(1956):473–75.
Page 176 the design is embodied in thespecific bonding properties of the parts . . . D.L. D. Caspar and A. Klug. “Physical Principlesin the Construction of Regular Viruses.” ColdSpring Harbor Symposia on Quantitative Biology27 (1962):1–24.
Page 177 a radical departure from the ideaof quasi-equivalence . . . I. Rayment, T. S. Baker,D. L. D. Caspar, and W. T. Murakami, “PolyomaVirus Capsid Structure at 22.5 PA Resolution,”Nature 295 (1982):110–15.
Page 178 collection of plasticene balls . . .See J. D. Bernal, “A Geometrical Approach to theStructure of Liquids,” Nature 183 (1959):141–47; “The Structure of Liquids,” Scientific Amer-ican 203 (1960):124–34. See also J. L. Finney,“Random Packings and the Structure of SimpleLiquids. I. The Geometry of Random Close Pack-ing,” Proceedings of the Royal Society [London],sec. A, 319 (1970):479–93.
Page 178 Stephen Hales (1677–1761) usedthis strategy . . . Stephen Hales, Vegetable Stat-icks, London: W. & J. Innys, 1727. Reprint (Can-ton, Mass.: Watson, Neale, 1969).
Page 178 Extending experimental methodsused by Plateau . . . Joseph A. F. Plateau,Statique experimentale et theorique des liquidessoumis aux seules forces moleculaires (Paris:Gand, 1873).
Page 179 William Barlow (1845–1934)observed . . . W. Barlow, “Probable Nature ofthe Internal Symmetry of Crystals,” Nature 29(1883–84):186–88, 205–07, 404.
Page 179 close-packing spheres with ori-ented binding sites . . . Sir William Thomson,“Molecular Constitution of Matter,” Proceedingsof the Royal Society of Edinburgh 16 (1888–89):693–724.
Page 179 Pauling developed in detail themodel of atoms-as-spheres . . . Linus Pauling,“Interatomic Distances and Their Relation to theStructure of Molecules and Crystals,” ch. 5 inThe Nature of the Chemical Bond (Ithaca, N.Y.:Cornell University Press, 1939).
Page 179 polyhera as models for plant cells. . . Ralph O. Erickson, “Polyhedral Cell Shapes,”in Grzegorz Rozenberg and Arto Salomaa, eds,The Book of L (Berlin: Spring-Verlag, 1986),pp. 111–124.
Page 179 Erickson described experiments. . . James W. Marvin. “Cell Shape Studies inthe Pith of Eupatorium purpureum,” AmericanJournal of Botany 26 (1939):487–504. Asurvey of other similiar work is given inEdwin B. Matzke and Regina M. Duffy, “TheThree-Dimensional Shape of Interphrase Cellswithin the Apical Meristem of Anacharisdensa,” American Journal of Botany 42 (1955):937–45.
Page 179 Previous investigators had takenKelvin’s tetrakaidecahedra . . . Sir WilliamThomson. “On the Division of Space withMinimum Partitional Area.” The London,Edinburgh, and Dublin Philosophical Magazineand Journal of Science, ser. 5, 24 (1887):503–14.
Page 180 In a series of Kepler-type experi-ments, Matzke and Marvin . . . Edwin B. Matzke,“Volume-Shape Relationships in Lead Shot andTheir Bearing on Cell Shapes,” American Journalof Botany 26 (1939):288–95; James W. Marvin,“The Shape of Compressed Lead Shot and itsRelation to Cell Shape,” American Journal ofBotany 26 (1939):280–88.
Page 180 Matzke also used another model. . . E. B. Matzke, “The Three-dimensional Shapeof Bubbles in Foam—An Analysis of the Roleof Surface Forces in Three-dimensional CellShape Determination,” American Journal ofBotany 33.
304
Page 184 Dynamic computational geometry. . . Godfried T. Toussaint. “On Translating a Setof Polyhedra.” McGill University School of Com-puter Science Technical Report No. SOCS-84.6,1984. See also Godfried T. Toussaint, “MovableSeparability of Sets,” in G. T. Toussaint, ed.,Computational Geometry, (Amsterdam: North-Holland, 1985), pp. 335–75.
Page 185 translated without collisions. . . Godfried T. Toussaint, “Shortest PathSolves Translation Separability of Polygons,”McGill University School of Computer ScienceTechnical Report No. SOCS-85.27, 1985.
Page 185 The answer is yes. . . . BinayK. Bhattacharya and Godfried T. Toussaint, “ALinear Algorithm for Determining TranslationSeparability of Two Simple Polygons,” McGillUniversity School of Computer Science Techni-cal Report No. SOCS-86.1, 1986.
Page 185 for any set of circles of arbitrarysizes . . . G. T. Toussaint, “On Translating a Set ofSpheres,” Technical Report SOCS-84.4, Schoolof Computer Science, McGill University, March1984.
Page 185 This problem can be generalized. . . G. T. Toussaint, “The Complexity of Move-ment,” IEEE International Symposium on Infor-mation Theory, St. Jovite, Canada, September1983; J.-R. Sack and G. T. Toussaint, “Movabilityof Objects,” IEEE International Symposium onInformation Theory, St. Jovite, Canada, Septem-ber 1983. G. T. Toussaint and J.-R. Sack, “SomeNew Results on Moving Polygons in the Plane,”Proceedings of the Robotic Intelligence and Pro-ductivity Conference, Detroit: Wayne State Uni-versity (1983) pp. 158–63.
Page 186 no translation ordering existsfor some directions. . . . L. J. Guibas and Y.F. Yao, “On Translating a Set of Rectangles.”Proceedings of the 12th Annual ACM Symposiumon Theory of Computing, Association forComputing Machinery Symposium on Theory ofComputing, Conference Proceedings, 12 (1980),pp. 154–160.
Page 186 translated in any direction with-out disturbing the others. . . . Robert Dawson,“On Removing a Ball Without Distrubing the
Others,” Mathematics Magazine 57, no. 1 (1984):27–30.
Page 186 no one can be moved withoutdisturbing the others. . . . K. A. Post, “SixInterlocking Cylinders with Respect to AllDirections,” unpublished paper, University ofEindhoven, The Netherlands, December 1983.
Page 187 can be separated with a singletranslation . . . Toussaint, “On Translating a Setof Spheres.”
Page 187 star-shaped polyhedra in 3-space . . . Toussaint, “On Translating a Set ofSpheres.”
Page 187 separated with a single translation. . . Godfried T. Toussaint and Hossam A. ElGindy, “Separation of Two Monotone Polygonsin Linear Time,” Robotica 2 (1984):215–20.
Page 187 known in Japanese carpentry asthe ari-kake joint . . . Kiyosi Seike, The Art ofJapanese Joinery, New York: John Weatherhill,1977.
Page 189 examined (by numerical calcula-tions) . . . H. M. Princen and P. Levinson, “TheSurface Area of Kelvin’s Minimal Tetrakaideca-hedron: The Ideal Foam Cell,” Journal of Colloidand Interface Science, vol. 120, no. 1, pp. 172–175 (1978).
Page 189 alternative unit cell . . . D. Weaireand R. Phelan, “A Counter-Example to Kelvin’sConjecture on Minimal Surfaces,” PhilosophicalMagazine Letters, vol. 69, no. 2, pp. 107–110(1994).
Notes and References for Chapter 13
Page 193 Proofs and Refutations . . . ImreLakatos, Proofs and Refutations, New York:Cambridge University Press, 1976.
Page 194 “Branko Grunbaum’s definition of“polyhedron” . . . B. Grunbaum, “Regular Poly-hedra, Old and New,” Aequationes Mathematicae16 (1977):1–20.
Page 197 the theory is still evolving. . . .B. Grunbaum and G. C. Shephard, Tilings andPatterns, San Francisco: W. H. Freeman, 1986.
Notes and References 305
Page 198 The numbers of combinatorialtypes . . . P. Engel, “On the Enumeration of Poly-hedra,” Discrete Mathematics 41 (1982):215–18.
Page 198 surprisingly low bounds for thenumber of combinatorially distinct polytopes. . . J. E. Goodman and R. Pollack, “ThereAre Asymptotically Far Fewer Polytopes ThanWe Thought,” Bulletin (New Series) AmericanMathematical Society 14, no. 1 (January1986):127–29.
Notes and References for Chapter 14
Page 202 SolidWorks . . . Solidworks: 3dmechanical design and 3d cad software, http://www.solidworks.com/
Page 202 Cauchy’s famous theorem (1813). . . A. L. Cauchy, “Sur le polygones et lespolyedres, seconde memoire,” Journal de l’EcolePolytechnique, XVIe Cahier (Tome IX):87–90,1813, and Oevres completes, IIme Serie, Vol. I,Paris, 1905, 26–38. For modern presentations,see P. R. Cromwell, Polyhedra, CambridgeUniversity Press, 1997, and M. Aigner and G. M.Ziegler, Proofs From the Book, Springer-Verlag,Berlin, 1998.
Page 202 theorems and implications due toDehn, Weyl and A.D. Alexandrov . . . Dehn, Weyland Alexandrov obsrved that Cauchy’s proof canbe adapted to yield an infinitesimal rigidity theo-rem; see M. Dehn, “Uber die Starrheit konvexerPolyeder,” Mathematische Annalen, 77:466–473,1916; H. Weyl, “Uber die Starrheit der Eiflachenund konvexer Polyeder,” In Gesammelte Abhand-lungen, Springer Verlag, Berlin, 1917; and A. D.Alexandrov, Convex Polyhedra, Springer Mono-graphs in Mathematics. Springer Verlag, BerlinHeidelberg, 2005, English Translation of Rus-sian edition, Gosudarstv. Izdat.Tekhn.-Teor. Lit.,Moscow-Leningrad, 1950.
Page 202 characterized by Steinitz’s theo-rem . . . E. Steinitz, Polyeder und Raumeinteilun-gen, volume 3 (Geometrie) of Enzyklopadie derMathematischen Wissenschaften, 1–139, 1922.
Page 203 3-connected planar graphs. . . .We refer to them as skeleta of convex polyhedra.
Page 203 a later theorem due to Gluck . . .H. Gluck, Almost all simply connected closedsurfaces are rigid, Lecture Notes in Mathematics,438:225–239, 1975.
Page 204 the Tay graph . . . T.-S. Tay,“Rigidity of multigraphs I: linking rigid bodiesin n-space,” Journal of Combinatorial Theory,Series B, 26:95–112, 1984; T.-S. Tay, “Linking.n � 2/-dimensional panels in n-space II:.n � 2; 2/-frameworks and body and hingestructures,” Graphs and Combinatorics, 5:245–273, 1989.
Page 204 Tay and Whiteley proved . . .T.-S. Tay and W. Whiteley, “Recent advancesin the generic rigidity of structures,” StructuralTopology, 9:31–38, 1984.
Page 204 Their guess was proved true . . .N. Katoh and S. Tanigawa, “A proof of themolecular conjecture,” in Proc. 25th Symp. onComputational Geometry (SoCG’09), pages 296–305, 2009, http://arxiv.org/abs/0902.0236.
Page 205 the structure theorem for .k; k/-sparse graphs . . . A. Lee and I. Streinu, “Pebblegame algorithms and sparse graphs,” DiscreteMathematics, 308(8):1425–1437, April 2008.
Page 206 the resulting graph G0 is sparse. . . For further structural properties of sparsegraphs, especially the .6; 6/ case, see A. Lee andI. Streinu, cited above.
Page 206 It relies on the pebble game algo-rithm . . . A. Lee and I. Streinu, cited above.
Page 208 a single-vertex origami . . . I.Streinu and W. Whiteley, “Single-vertex origamiand spherical expansive motions,” in J. Akiyamaand M. Kano, editors, Proc. Japan Conf. Discreteand Computational Geometry (JCDCG 2004),volume 3742 of Lecture Notes in Computer Sci-ence, pages 161–173, Tokai University, Tokyo,2005. Springer Verlag.
Notes and References for Chapter 15
Page 212 the eighteenth-century works ofEuler and Meister . . . see L. Euler, “Elementadoctrinae solidorum,” Novi Comm. Acad. Sci.Imp. Petropol. 4 (1752–53):109–40; see also
306
in Opera Mathematica, vol. 26, pp. 71–93.P. J. Federico, Descartes on Polyhedra, NewYork: Springer-Verlag, 1982, pp. 65–69. A. L. F.Meister, “Commentatio de solidis geometricis,”Commentationes soc. reg. scient. Gottingensis,cl. math. 7 (1785). M. Bruckner, Vielecke undVielflache (Leipzig: Teubner, 1900), p. 74.
Page 212 each face of one is the polar . . .see H. S. M. Coxeter, Regular Polytopes, 3rd ed.,New York: Dover Publications, 1973, p. 126.
Page 213 are duals of each other . . . seeB. Jessen, “Orthogonal Icosahedron,” NordiskMatematisk Tidskrift 15 (1967):90–96; J. Oun-sted, “An Unfamiliar Dodecahedron,” Mathemat-ics Teaching, 83 (1978):45–47. B. M. Stewart,Adventures among the Toroids, 2nd ed. (Okemos,Mich.: B. M. Stewart, 1980), p. 254, and B.Grunbaum and G. C. Shephard, “Polyhedra withTransitivity Properties,” Mathematical Reportsof the Academy of Science [Canada] 6, (1984):61–66.
Page 213 Csaszar polyhedron . . . see,for example, “A Polyhedron without Diag-onals,” Acta Scientiarum Mathematicarum13 (1949):140–42. B. Grunbaum, ConvexPolytopes (London Interscience, 1967), p. 253.M. Gardner, “On the Remarkable CsaszarPolyhedron and its Applications in ProblemSolving,” Scientific American 232, no. 5 (May1975):102–107. Stewart, Adventures among theToroids, p. 244.
Page 213 Szilassi polyhedron . . . see L.Szilassi, “A Polyhedron in Which Any Two FacesAre Contiguous” [In Hungarian, with Russiansummary], A Juhasz Gyula Tanarkepzo FoiskolaTudomanyos Kozlemenyei, 1977, Szeged. M.Gardner, “Mathematical Games, in Which aMathematical Aesthetic Is Applied to ModernMinimal Art,” Scientific American 239, no. 5(November 1973):22–30. Stewart, Adventuresamong the Toroids, p. 244. P. Gritzmann,“Polyedrische Realisierungen geschlossener 2-dimensionaler Mannigfaltigkeiten im R3,” Ph.D.thesis, Universitat Siegen, 1980.
Page 213 this goal may be unattainable . . .B. Grunbaum and G. C. Shephard, “Polyhedrawith Transitivity Properties.”
Page 216 this generalization of theconcept of a polyhedron . . . See, for example,A. F. Mobius, “Uber die Bestimmung desInhaltes eines Polyeders” (1865), in GesammelteWerke, vol. 2 (1886), pp. 473–512. E. Hess,Einleitung in die Lehre von der Kugelteilung(Leipzig: Teubner, 1883). Bruckner, Vieleckeund Vielflache, p. 48. H. S. M. Coxeter,Regular Polytopes. H. S. M. Coxeter, M. S.Longuet-Higgins, and J. C. P. MIller, “UniformPolyhedra,” Philosophical Transactions of theRoyal Society [London], sec. A. 246 (1953–54):401–450. J. Skilling, “The Complete Set ofUniform Polyhedra,” Philosophical Transactionsof the Royal Society [London], sec. A. 278(1975):111–135. M. J. Wenninger, PolyhedronModels (London: Cambridge University Press,1971). M. Norgate, “Non-Convex Pentahedra,”Mathematical Gazette 54 (1970): 115–24.
Notes and References for Chapter 16
Page 218 combinatorial analogue of thetiling problem for higher dimensions . . . seeL. Danzer, B. Grunbaum, and G. C. Shephard,“Does Every Type of Polyhedron Tile Three-space?” Structural Topology 8 (1983):3–14, and D. G Larman and C. A. Rogers,“Durham Symosium on the Relations betweenInfinite-dimensional and Finitely-dimensionalConvexity,” Bulletin of the London MathematicalSociety 8 (1976):1–33.
Page 220 we can prove this generalization. . . E. Schulte, “Tiling Three-space by Com-binatorially Equivalent Convex Polytopes,” Pro-ceedings of the London Mathematical Society 49,no. 3, (1984):128–40.
Page 220 operation, due to Steinitz . . .Branko Grunbaum, Convex Polytopes, New York:John Wiley and Sons, 1967.
Page 221 all simplicial polytopes give lo-cally finite face-to-face tilings . . . B. Grunbaum,P. Mani-Levitska, and G. C. Shephard, “TilingThree-dimensional Space with Polyhedral Tilesof a Given Isomorphism Type,” Journal of
Notes and References 307
the London Mathematical Society 29, no. 2(1984):181–91.
Page 221 the reverse is true . . . see E.Schulte, “Tiling Three-space by CombinatoriallyEquivalent Convex Polytopes,” Proceedings ofthe London Mathematical Society 49, no. 3,(1984):128–40; E. Schulte, “The Existence ofNontiles and Nonfacets in Three Dimensions,”Journal of Combinatorial Theory, ser. A,38 (1985):75–81; E. Schulte, “Nontiles andNonfacets for the Euclidean Space, SphericalComplexes and Convex Polytopes.” Journal furdie Reine und Angewandte Mathematik, 352(1984):161–83.
Page 221 the fundamental region for thesymmetry group of the regular tessellation of E3
by cubes is a 3-simplex . . . H. S. M. Coxeter,Regular Polytopes, 3rd ed., New York: DoverPublications, 1973
Page 222 examples of space-filling toroids. . . D. Wheeler and D. Sklar, “A Space-fillingTorus,” The Two-Year College Mathematics Jour-nal 12, no. 4 (1981):246–48.
Page 222 Polyhedra with this propertyhave recently been studied . . . P. McMullen, C.Schulz, and J. M. Wills, “Polyhedral Manifolds inE3 with Unusually Large Genus,” Israel Journalof Mathematics 46 (1983):127–44.
Page 222 monotypic tilings of M bytopological polytopes or tiles of anotherhomeomorphism type . . . E. Schulte, “Reg-ular Incidence-polytopes with Euclidean orToroidal Faces and Vertex-figures,” Journalof Combinatorial Theory, ser. A, 40 (1985):305–30.
Notes and References for Chapter 17
Page 223 infinitely many equivelar mani-folds . . . B. Grunbaum and G. Shephard, “Poly-hedra with Transitivity Properties,” MathematicalReports of the Academy of Science [Canada] 6(1984):61–66; P. McMullen, C. Schulz, and J.M. Wills, “Polyhedral 2-Manifolds in E3 withUnusually Large Genus,” Israel Journal of Math-ematics 46 (1983):124–44.
Page 224 there exist no face-transitive poly-hedra with g > 0 . . . see Grunbaum and Shep-hard, above.
Page 224 first vertex-transitive polyhedronwith g > 0 . . . U. Brehm and W. Kuhnel,“Smooth Approximation of Polyhedral SurfacesRegarding Curvatures,” Geometriae Dedicata 12(1982):438.
Page 224 two more polyhedra with vertex-transitivity under symmetries . . . see Gunbaumand Shephard, above.
Page 225 an impression of four Platonohe-dra . . . J. M. Wills, “Semi-Platonic Manifolds”in P. Gruber and J. M. Wills, eds., Convexityand Its Applications (Basel: Birkhauser, 1983); J.M. Wills,“On Polyhedra with Transitivity Prop-erties,” Discrete and Computational Geometry 1(1986):195–99.
Page 225 a figure of f3; 8I 5g from whicha three-dimensional construction is possible . . .see Grunbaum and Shephard, above.
Page 225 a precise construction of f3; 8I 3g.. . . E. Schulte and J. M. Wills, “Geometric Real-izations for Dyck’s Regular Map on a Surface ofGenus 3,” Discrete and Computational Geometry1 (1986):141–53.
Page 226 the last four can be realizedin E4 . . . H. S. M. Coxeter, “Regular SkewPolyhedra in Three and Four Dimensions, andTheir Topological Analogues,” Proceedings ofthe London Mathematical Society, ser. 2, 43(1937):33–62.
Page 226 The geometric construction tracesback to Alicia Boole-Stott . . . A. Boole-Stott, Ge-ometrical Reduction of Semiregular from RegularPolytopes and Space Fillings, Amsterdam: Ver.d. K. Atkademie van Wetenschappen, 1910; P.McMullen, C. Schulz, and J. M. Wills, “EquivelarPolyhedral Manifolds in E3,” Israel Journal ofMathematics 41 (1982):331–46. They are projec-tions of Coxeter’s regular skew polyhedra (seeSchulte and Wills, above).
Page 226 polyhedral realization of FelixKlein’s famous quartic . . . E. Schulte and J. M.Wills, “A Polyhedral Realization of Felix Klein’sMap f3; 7g8 on a Riemann Surface of Genus 3,”Journal of the London Mathematical Society 32(1985):539–47.
308
Page 228 Do equivelar manifolds exist withp � 5 and q � 5? . . . see McMullen, Schultz,and Wills, 1982, above; see also P. McMullen, C.Schulz, and J. M. Wills, “Two Remarks on Equiv-elar Manifolds,” Israel Journal of Mathematics52 (1985):28–32.
Page 228 for q D 4 no such manifold exists. . . see McMullen, Schultz, and Wills, 1985,above.
Page 229 a nice and interesting proof. . . Gunter Ziegler and Michael Joswig “Poly-hedral surfaces of high genus,” Oberwolfachseminars, vol. 38 (2009), Discrete Differen-tial Geometry (Bobenko, Sullivan, Schrder,Ziegler).
Page 229 This polyhedron was found byDavid McCooey in 2009. . . . David McCooey, Anon-selfintersecting polyhedral realization of theall-heptagon Klein map, Symmetry, vol. 20, no. 1-(2009), 247–268.
Notes and References for Chapter 18
Page 231 to determine when they were rigid. . . The technical term is infinitesimally rigid. Abar-and-joint framework is infinitesimally rigid ifevery set of velocity vectors which preserves thelengths of the bars represents a Euclidean motionof the whole space.
Page 231 the reciprocal figure . . . J. C.Maxwell, “On Reciprocal Diagrams and Dia-grams of Forces,” Philosophical Magazine, 4,27(1864):250–61, J.C. Maxwell, “On Recipro-cal Diagrams, Frames and Diagrams of Forces.”Transactions of the Royal Society of Edinburgh26 (1869–72):1–40.
Page 231 the field of graphical statics . . .L. Cremona, Graphical Statics, English trans.,London: Oxford University Press, 1890.
Page 231 grows from these geometric roots. . . H. Crapo and W. Whiteley, “Statics ofFrameworks and Motions of Panel Structures:A projective Geometric Introduction,” StructuralTopology 6, (1982):43–82, H. Crapo andW. Whiteley, “Plane Stresses and ProjectedPolyhedra I, the basic pattern” Structural
Topology 20 (1993), 55–68. http://www-iri.upc.es/people/ros/StructuralTopology/, W. Whiteley,“Motions and Stresses of Projected Polyhedra,”Structural Topology 7 (1982):13–38.
Page 232 Several workers independently ob-served . . . D. Huffman, “A Duality Concept forthe analysis of Polyhedral Scenes,” in E. W. El-cock and D. Michie (eds.), Machine Intelligence8 [Ellis Horwood, England] (1977):475–92.A. K.Mackworth, “Interpreting Pictures of PolyhedralScenes,” Artifical Intelligence 4 (1973):121–37.
Page 232 necessary and sufficient condi-tion for correct pictures . . . H. Crapo and W.Whiteley, “Plane Stresses and Projected Polyhe-dra I, the basic pattern” Structural Topology 20(1993), 55–68. http://www-iri.upc.es/people/ros/StructuralTopology/.
Page 232 projected from the point of tan-gency of one face . . . K. Q. Brown, “VoronoiDiagrams from Convex Hulls,” Information Pro-cessing Letters 9 (1979):223–38.
Page 232 the diagram of centers formsa classical reciprocal figure . . . P. Ash andE. Bolker, “Recognizing Dirichlet Tessellations,”Geometriae Dedicata 19 (1985):175–206.
Page 232 last link in the proof . . . P. Ash andE. Bolker, “Generalized Dirichlet Tessellations,”Geometriae Dedicata 20 (1986):209–43.
Page 233 an explicit construction of apolyhedron. . . H. Edelsbrunner and R. Seidel,“Voronoi Diagrams and Arrangements,” Discreteand Computational Geometry I (1986):25–44.
Page 235 Delauney triangulation. . . In aDirichlet tessellation the centers of the cellswhich share a vertex are equidistant from thatvertex. If the centers are chosen at random, then(with probability 1) no four lie on a circle. So theresulting Dirichlet tessellation has only 3-valentvertices.
Page 236 a proof can be found. . . P. Ash andE. Bolker, “Recognizing Dirichlet Tessellations,”Geometriae Dedicata 19 (1985):175–206.
Page 237 Ash and Bolker proved. . . P.Ash and E. Bolker, “Generalized DirichletTessellations,” Geometriae Dedicata 20 (1986):209–43.
Page 237 model a simple biologicalphenomenon. . . H. Edelsbrunner and R. Seidel,
Notes and References 309
“Voronoi Diagrams and Arrangements,” Discreteand Computational Geometry I (1986):25–44.
Page 237 this need not always betrue. . . Other examples and a more completebibliography are given in P. Ash and E.Bolker, “Recognizing Dirichlet Tessellations,”Geometriae Dedicata 19 (1985):175–206.
Page 238 they are rigid in the plane. . .Other examples and a more complete bibliographyare given in P. Ash and E. Bolker, “RecognizingDirichlet Tessellations,” Geometriae Dedicata 19(1985):175–206.
Page 238 the appearance of a spider websignals that it is shaky. . . R. Connelly, “Rigid-ity and Energy,” Inventiones Mathematicae 66(1982):11–33.
Page 238 cannot be made denser. . . H.Crapo and W. Whiteley, “Statics of Frameworksand Motions of Panel Strucutres: A projectiveGeometric Introduction,” Structural Topology6, (1982):43–82, W. Whiteley, “Motions andStresses of Projected Polyhedra,” StructuralTopology 7 (1982):13–38.
Page 238 we have constructed a convexreciprocal. . . R. Connelly, “Rigid Circle andSphere Packings I. Finite Packings,” StructuralTopology, 14 (1988) 43–60. R. Connelly, “RigidCircle and Sphere Packings II. Infinite Packingswith Finite Motion,” Structural Topology, 16(1990) 57–76.
Page 238 Thus we have proved. . . J.C. Maxwell, “On Reciprocal Diagrams andDiagrams of Forces,” Philosophical Magazine, 4,27(1864):250–61, H. Crapo and W. Whiteley,“Plane Stresses and Projected Polyhedra I,the basic pattern” Structural Topology 20(1993), 55–68. http://www-iri.upc.es/people/ros/StructuralTopology/.
Page 240 a projective polarity aboutthe Maxwell paraboloid. . . H. Crapo andW. Whiteley, “Plane Stresses and ProjectedPolyhedra I, the basic pattern” StructuralTopology 20 (1993), 55–68. http://www-iri.upc.es/people/ros/StructuralTopology/.
Page 240 the positions of the remainingplanes can be deduced. . . J. C. Maxwell, “OnReciprocal Diagrams and Diagrams of Forces,”Philosophical Magazine, 4, 27(1864):250–61, H.
Crapo and W. Whiteley, “Plane Stresses and Pro-jected Polyhedra I, the basic pattern” StructuralTopology 20 (1993), 55–68. http://www-iri.upc.es/people/ros/StructuralTopology/.
Page 241 Choose the centers to bethe points. . . H. Crapo and W. Whiteley,“Plane Stresses and Projected Polyhedra I,the basic pattern” Structural Topology 20(1993), 55–68. http://www-iri.upc.es/people/ros/StructuralTopology/, Section 4.
Page 242 in the cell of the sectional Dirich-let tessellation. . . H. Edelsbrunner and R. Seidel,“Voronoi Diagrams and Arrangements,” Dis-crete and Computational Geometry I (1986):25–44.
Page 242 This completes the converse. . . H.Crapo and W. Whiteley, “Plane Stresses and Pro-jected Polyhedra I, the basic pattern” StructuralTopology 20 (1993), 55–68. http://www-iri.upc.es/people/ros/StructuralTopology/.
Page 243 furthest-point Dirichlet tessella-tion of centers. . . P. Ash and E. Bolker, “Gener-alized Dirichlet Tessellations,” Geometriae Ded-icata 20 (1986):209–43.
Page 243 the picture of some convex polyhe-dron. . . H. Edelsbrunner and R. Seidel, “VoronoiDiagrams and Arrangements,” Discrete and Co-mutational Geometry I (1986):25–44.
Page 244 This argument and its converseprove. . . see B. Roth and W. Whiteley, “Tenseg-rity Frameworks,” Transactions of the Ameri-can Mathematical Society 265 (1981):419–45,Whiteley, “Motions and Stresses of ProjectedPolyhedra,” Structural Topology 7 (1982):13–38,and R. Connelly, “Rigidity and Energy,” Inven-tiones Mathematicae 66 (1982):11–33.
Page 244 This gives. . . R. Connelly, “Rigid-ity and Energy,” Inventiones Mathematicae 66(1982):11–33.
Page 245 positions of the vertices and the di-rections of the infinite edges. . . Brown, “VoronoiDiagrams from Convex Hulls,” above.
Page 246 any triply connected planargraph can be realized as a convex polyhedron. . .Roth and Whiteley, “Tensegrity Framworks,”above.
Page 246 conditions that must be satis-fied if there is to be a stress on all members. . .
310
B. Grunbaum, Convex Polytopes, New York: In-terscience, 1967.
Page 246 one for each finite cell in thegraph. . . N. White and W. Whiteley, “The Al-gebraic Geometry of Stresses in Frameworks,”S.I.A.M. Journal of Algebraic and Discrete Meth-ods 4 (1983):481–511.
Page 246 study of White and Whiteley. . . C.Davis, “The Set of Non-Linearity of a ConvexPiece-wise Linear Function,” Scripta Mathemat-ica 24 (1959):219–28.
Page 246 the dual bowl over this reciprocalwill be a zonohedral cap. . . White and Whiteley,“The Algebraic Geometry of Stresses in Frame-works,” above.
Page 247 edge graphs of convex polyhedrawith inspheres. . . H. S. M Coxeter, “The Clas-sification of Zonohedra by Means of ProjectiveDiagrams,” Journal de Mathematiques Pures etAppliquees, ser. 9, 42(1962):137–56.
Page 247 projection of a polyhedronwhich cannot have an insphere. . . E. Schulte,“Analogues of Steinitz’s Theorem about Non-Inscribable Polytopes,” Colloquia Mathe-matica Societatis Janos Bolyai 48 (1985),503–516.
Page 247 a geometric characterization. . .Grunbaum, Convex Polytopes, above.
Page 248 the one first studied in graphicalstatics. . . P. Ash and E. Bolker, “RecognizingDirichlet Tessellations,” Geometriae Dedicata 19(1985):175–206.
Page 248 both truly belong to projectivegeometry. . . J. C. Maxwell, “On ReciprocalDiagrams and Diagrams of Forces,” Philosoph-ical Magazine, 4, 27(1864):250–61, Cremona,Graphical Statics (above).
Page 248 Connelly discovered. . . J.C.Maxwell, “On Reciprocal Diagrams, Framesand Diagrams of Forces.” Transactions of theRoyal Society of Edinburgh 26 (1869-72):1–40, W. Whiteley, “The Projective Geometry ofRigid Frameworks,” in C. Baker and L. Batten,eds, Proceedings of the Winnipeg Conference onFinite Geometries, New York: Marcel Dekker,1985, pp. 353–70.
Page 249 remains true. . . Connelly, “RigidCircle and Sphere Packings: I and II,” above.
Page 250 this algebraic definition allows. . .H. Edelsbrunner and R. Seidel, “Voronoi Dia-grams and Arrangements,” Discrete and Comu-tational Geometry I (1986):25–44.
Page 250 in the spatial Dirichlet tessellatonit sections. . . H. Crapo and W. Whiteley,“Plane Stresses and Projected Polyhedra I,the basic pattern” Structural Topology 20(1993), 55–68. http://www-iri.upc.es/people/ros/StructuralTopology/.
Page 250 called a pentagrid. . . N. G.de Brujin, “Algebraic Theory of Penrose’sNon-periodic Tilings, I, II,” Akademie vanWetenschappen [Amsterdam], Proceedings, ser.A, 43 (1981):32–52, 53–66.
Page 251 publications on Voronoi Di-agrams. . . see F. Aurenhammer, “Voronoidiagrams: a survey of a fundamental geometricdata structure,” ACM Computing Surveys (CSUR)Volume 23 (1991), 345–405; A. Okabe, B. Boots,K. Sugihara, Spatial tessellations : conceptsand applications of voronoi diagrams, Wiley& Sons, 1992; S. Fortune, “Voronoi diagramsand Delaunay triangulations,” in Computing inEuclidean Geometry, Ding-zhu Du, Frank Hwang(eds), Lecture notes series on computing; vol. 1World Scientific (1992), 193–234.
Page 251 continuing work on weightedtessellations. . . D. Letscher “Vector WeightedVoronoi Diagrams and Delaunay Triangulations,”Canadian Conference on Computational Ge-ometry 2007, Ottawa, Ontario, August 20–22,2007.
Page 251 advances on reciprocal diagramsin the plane. . . H. Crapo and W. Whiteley.“Spaces of stresses, projections, and paralleldrawings for spherical polyhedra”, Beitraegezur Algebra und Geometrie / Contribu-tions to Algebra and Geometry 35 (1994),259–281.
Page 251 zonohedral tilings (cubic partialcubes). . . D. Eppstein, “Cubic partial cubes fromsimplicial arrangements,” The Electronic Journalof Combinatorics. 13 (2006); R79.
Page 251 projections of higher dimensionalcell complexes. . . K. Rybnikov, “Stresses andliftings of cell-complexes,” Discrete Comp.Geom. 21, No. 4, (1999), 481–517; R. M. Erdahl,
Notes and References 311
K. A. Rybnikov, and S. S. Ryshkov, “On traces ofd -stresses in the skeletons of lower dimensionsof homology d -manifolds,” Europ. J. Combin.22, No. 6, (2001), 801–820; W. Whiteley, “SomeMatroids from Discrete Geometry”, in MatroidTheory, J. Bonin, J. Oxley and B. Servatius(eds.), AMS Contemporary Mathematics, 1996,171–313.
Notes and References for Chapter 19
Page 253 the coordinates are given by sys-tematic choices . . . H. S. M. Coxeter, RegularPolytopes, New York: Dover Publications, 1973,pp. 50–53, 156–162.
Page 254 see Hardy and Wright’s . . . G. H.Hardy and E. M. Wright, An Introduction to theTheory of Numbers, 4th ed., London: OxfordUniversity Press, 1960, pp. 221–222.
Notes and References for Chapter 20
Page 257 the way various three-dimensionalfaces fit together in 4-space . . . to form reg-ular polytopes; see for example D.M.Y. Som-merville’s description of the regular polytopesin 4-space, Geometry of n Dimensions, London:Methuen, 1929, or H.S.M. Coxeter’s treatment,Regular Polytopes, 3rd ed., New York: DoverPublications, 1973.
Page 258 This polytope is complicatedenough . . . Similar treatments of the 120-cell andthe 600-cell are implicit in the work of severalmathematicians, notable Coxeter whose RegularComplex Polytopes, Cambridge: CambridgeUniversity Press, 1974, is the primary sourcefor all material of this sort.
Page 262 In 4-space the sphere of pointsat unit distance . . . One direct analogy with theusual coordinate system on the 2-sphere in 3-space would be to use
.x; y; u; v/D.cos.�/ sin.'/ sin. /; sin.�/ sin.'/
sin. /; cos.'/ sin. /; cos.'//: (22.3)
which would suggest the same sort ofdecomposition of the 3-sphere into “parallel2-spheres of latitude.” Such a decomposition hasbeen carried out by several authors (includingD.M.Y. Sommerville), and an early computergenerated film by George Olshevsky usessuch an approach to display the slices ofregular polyhedra in 4-space by sequences ofhyperplanes perpendicular to various coordinateaxes.
Page 264 the 16 vertices of the hypercube onthe unit hypersphere . . . may be given by 1
2Œ˙1˙
i;˙1 ˙ i �: Compare p. 37 of Coxeter, RegularComplex Polytopes.
Page 266 mapped to the vertices of a reg-ular tetrahedron inscribed in the unit 2-sphere. . . The coordinates for the 24-cell obtained inthis way are very similar to those which ap-pear in Coxeter’s discussion of the 24-cell inTwisted Honeycombs (Regional Conference Se-ries in Mathematics, no. 4, American Mathemat-ical Society, Providence 1970), although he doesnot explicitly use the Hopf mapping in any ofhis constructions. Professor Coxeter pointed outthat these coordinates also appear in a slightlydifferent form in the 1951 dissertation of G.S.Shephard.
Notes and References for Chapter 21
Page 267 The key idea in that problem . . .See R. Connelly, “Expansive motions. Surveys ondiscrete and computational geometry,” Contemp.Math., (2008) 453, 213–229 Amer. Math. Soc.,Providence, RI.
Page 267 the theory of tensegrities can beapplied . . . For more about these ideas, seeAleksandar Donev; Salvatore Torquato; Frank H.Stillinger and Robert Connelly, “A linear pro-gramming algorithm to test for jamming in hard-sphere packings,” J. Comput. Phys., 197, (2004),no. 1, 139–166.
Page 269 stress associated to a tensegrity. . . Note that in the paper by B. Roth and W.Whiteley, “Tensegrity frameworks,” Trans. Amer.Math. Soc., 265, (1981), no. 2, 419–446, a proper
312
stress is called what is strict and proper here,whereas in my definition proper stresses need notbe strict. And don’t confuse this notion of stresswith that used in structure analysis, physics orengineering. In those fields, stress is defined as aforce per cross-sectional area. There are no cross-sections in my definition; the scalar !ij is betterinterpreted as a force per unit length.
Page 269 are congruences . . . this is alsocalled local rigidity as in S. Gortler, A. Healy,and D. Thurston, “Characterizing generic globalrigidity,” arXiv:0710.0907 v1, 2007.
Page 269 a good body of work . . . seethe surveys in Andras Recski, “Combinatorialconditions for the rigidity of tensegrityframeworks. Horizons of combinatorics,” 163–177, Bolyai Soc. Math. Stud, 17, Springer, Berlin,2008; Walter Whiteley, “Infinitesimally rigidpolyhedra. I. Statics of frameworks.” Trans.Amer. Math. Soc., 285, (1984), no. 2, 431–465; Walter Whiteley, “Infinitesimally rigidpolyhedra. II. Weaving lines and tensegrityframeworks.” Geom. Dedicata, 30 (1989), no. 3,255–279.
Page 270 when all directional derivativesgiven by p0 D .p0
1; : : : ; p0n/ starting at p are 0
. . . We perform the following calculation startingfrom (21.1) for 0 � t � 1:
E!.p C tp0/ DX
i<j
!ij ..pi � pj /2
C 2t.pi � pj /.p0i � p0
j /C t2.p0i � p0
j /2/:
Taking derivatives and evaluating at t D 0, weget:
d
dtE!.pCtp0/jtD0 D 2
X
i<j
!ij .pi �pj /.p0i�p0
j /:
At a critical configuration p, this equation musthold for all directions p0.
Page 270 so is any affine transformation . . .This is seen by the following calculation:
X
j
!ij .qj � qi /DX
j
!ij .Apj C b � Api � b/
D AX
j
!ij .pj � pi / D 0:
Page 271 quadric at infinity . . . The reasonfor this terminology is that real projective spaceRpd�1 can be regarded as the set of lines throughthe origin in Ed , and Equation (21.5) is thedefinition of a quadric in Rpd�1.
Page 271 We can prove . . . Converselysuppose that the member directions of a bartensegrity G.p/ lie on a quadric at infinity inEd given by a non-zero symmetric matrixQ. Bythe spectral theorem for symmetric matrices, weknow that there is an orthogonal d -by-d matrixX D .XT /�1 such that:
XTQX D
0BBBBB@
�1 0 0 � � � 00 �2 0 � � � 00 0 �3 � � � 0:::::::::: : :
:::
0 0 0 � � � �d
1CCCCCA:
Let �� be the smallest �i , and let �C be thelargest �i . Note 1 � 1=�� < 1=�C � 1, ��is non-positive, and �C is non-negative when Qdefines a non-empty quadric and when 1=�� �t � 1=�C, 1 � t�i � 0 for all i D 1; : : : ; d .Working Equation (21.4) backwards for 1=�� �t � 1=�C we define:
At D XT
0
BBBBBB@
p1 � t�1 0 0 � � � 0
0p
1 � t�2 0 � � � 0
0 0p
1 � t�3 � � � 0
:::
:::
:::
: : ::::
0 0 0 � � � p1 � t�d
1
CCCCCCAX:
(22.4)
Substituting this expression for At into Equation(21.4), we see that it provides a non-trivial affineflex of G.p/. If the configuration is contained ina lower dimensional hyperplane, we should reallyrestrict to that hyperplane since there are non-orthogonal affine transformations that are rigidwhen restricted to the configuration itself.
Page 271 lie on same line on either ruling. . . Consider the diagonal matrix Q with diag-onal entries �1 D �2 D 1; �3 D �1. Whenone node of each bar is translated to a singlepoint, they all lie on a circle at infinity given byQ. The flex given by Formula (22.4) flexes theconfiguration until the nodes lie on a line when
Notes and References 313
t D 1=�C D 1 because two of the eigenvaluesfor Q vanish for that value of t , and in the otherdirection, when t D 1=�� D �1, the nodes lie ina plane. This structure is easy to build with dowelrods and rubber bands securing the joints wherethe rulings intersect.
The space of d -by-d symmetric matrices isof dimension d C .d 2 � d/=2 D d.d C 1/=2.So if the vector directions of a tensegrity are lessthan d.d C 1/=2, then it is possible to find anon-zero d -by-d symmetric matrix that satisfiesEquation (21.5), and then flex it into a lowerdimensional subspace.
Page 271 Barvinok proved . . . Barvi-nok, A. I., “Problems of distance geometryand convex properties of quadratic maps.”Discrete Comput. Geom., 13, (1995), no. 2,189–202.
Page 272 Maria Belk and I showed . . .Maria Belk and Robert Connelly, “Realizabilityof graphs,” Discrete Comput. Geom., 37, (2007),no. 2, 125–137; Maria Belk, “Realizability ofgraphs in three dimensions,” Discrete Comput.Geom., 37, (2007), no. 2, 139–162.
Page 272 Tensegrity techniques are used ina significant way. . . . see Belk, cited above.
Page 273 when the configuration p D.p1; : : : ; pn/ in Ed is universal . . . It isnot difficult to prove that if p is a universalconfiguration for !, any other configuration qwhich is in equilibrium with respect to ! is anaffine image of p.
Page 274 I showed that a tensegrity . . .Robert Connelly, “Rigidity and energy”, Invent.Math., 66, (1982), no. 1, 11–33. These resultsanswered some questions Grunbaum posed in“Lectures on Lost Mathematics” (1975); notesdigitized and reissued at Structural Topology Re-visited Conference (2006), http://hdl.handle.net/1773/15700.
Page 275 two examples in the plane andin three-space . . . this tensegrity is describedin Karoly Bezdek and Robert Connelly, Two-distance preserving functions from Euclideanspace. Discrete geometry and rigidity (Budapest,1999). Period. Math. Hungar., 39, (1999),no. 1–3, 185–200.
Page 275 L. Lovasz showed . . . LaszloLovasz, Steinitz representations of polyhedra andthe Colin de Verdiere number. J. Combin. TheorySer. B, 82, (2001), no. 2, 223–236.
Page 275 such a tensegrity is super stable. . . this is explained in Karoly Bezdek and RobertConnelly, Stress Matrices and M Matrices, Ober-wolfach Reports Vol. 3, No. 1 (2006), 678–680; itanswers a question of K. Bezdek.
Page 275 P(n,k) is super stable . . . R.Connelly; M. Terrell: Tensegrites symetriquesglobalement rigides. [Globally rigid symmetrictensegrities] Dual French-English text. StructuralTopology, No. 21 (1995), 59–78.
Page 276 a website where one can view . . .this is available at http://www.math.cornell.edu/�tens/.
Page 277 resulting tensegrity is super stable. . . J. Y. Zhang: Simon D. Guest; Makoto Ohsaki;Robert Connelly, “Dihedral ‘Star’ TensegrityStructures,” Int. J. Solids Struct. (2009).
Page 278 This follows from the proofof the carpenter’s rule property . . . RobertConnelly, Erik D. Demaine, and Gunter Rote,Straightening polygonal arcs and convexifyingpolygonal cycles. U.S.-Hungarian Workshops onDiscrete Geometry and Convexity (Budapest,1999/Auburn, AL, 2000). Discrete Comput.Geom., 30 (2003), no. 2, 205–239.
Page 278 There’s more – much more – tosay . . . Let’s begin with Generic global rigidity.The configurations in previous sections must beconstructed carefully. What about a bar frame-work where the configuration is more general? Itturns out that the problem of determining whena bar framework is globally rigid is equivalent toa long list of problems known to be hard. (See,for example, James B. Saxe, Embeddability ofweighted graphs in k-space is strongly NP-hard.Technical report, Computer Science Department,Carnegie Mellon University, 1979.) The prob-lem of whether a cyclic chain of edges in theline has another realization with the same barlengths is equivalent to the uniqueness of a so-lution of the knapsack problem. This is one ofthe many problems on the list of NP completeproblems.
314
One way to avoid this difficulty is to assumethat the configuration’s coordinates are generic.This means that the coordinates of p in Ed
are algebraically independent over the rationalnumbers, which means that there is no non-zeropolynomial with rational coordinates satisfied bythe coordinates of p. This implies, among otherthings, that no d C 2 nodes lie in a hyperplane,for example, and a lot more. I proved (“Genericglobal rigidity,” Discrete Comp. Geometry 33(2005), pp 549–563) that:
Theorem. If p D .p1; : : : ; pn/ in Ed is genericand G.p/ is a rigid bar tensegrity in Ed with anon-zero stress matrix˝ of rank n� d � 1, thenG.p/ is globally rigid in Ed .
Notice that the hypothesis includes Conditions 2and 3 of Theorem 23 in the text. The idea of theproof is to show that since the configuration pis generic, if G.q/ has the same bar lengths asG.p/, then they should have the same stresses.Then Proposition 21.1 applies.
Thurston proved the converse (see Thurston,cited above):
Theorem. If p D .p1; : : : ; pn/ in Ed is genericandG.p/ is a globally rigid bar tensegrity in Ed ,then eitherG.p/ is a bar simplex or there is stressmatrix ˝ for G.p/ with rank n � d � 1.
The idea here, very roughly, is to show thata map from an appropriate quotient of an appro-priate portion of the space of all configurationshas even topological degree when mapped intothe space of edge lengths.
As Dylan Thurston pointed out, using theseresults it is possible to find a polynomial timenumerical (probablistic) algorithm that calculateswhether a given graph is generically globallyrigid in Ed , and that the property of being glob-ally rigid is a generic property. In other words, ifG.p/ is globally rigid in Ed at one generic con-figuration p, it is globally rigid at all generic con-figurations. Interestingly, he also showned that ifp is generic in Ed , and G.q/ has the same barlengths in G.p/ in Ed , then G.p/ can be flexed
to G.q/ in EdC1, similar to the discussion ofcompound tensegrities.
A bar graph G is generically redundantlyrigid in Ed if G.p/ is rigid at a generic config-uration p and remains rigid after the removal ofany bar. A graph is vertex k-connected if it takesthe removal of at least k vertices to disconnect therest of the vertices of G. The following theoremof Hendrickson (B. Hendrickson, Conditions forunique graph realizations, SIAM J. Comput 21(1992), pp 65–84) provides two necessary con-ditions for generic global rigidity.
Theorem. If p is a generic configuration in Ed ,and the bar tensegrity G.p/ is globally rigid inEd , then
1. G is vertex .d C 1/-connected, and2. G.p/ is redundantly rigid in Ed .
Condition 1 on vertex connectivity is clearsince otherwise it is possible to reflect onecomponent ofG about the hyperplane determinedby some d or fewer vertices. Condition 2 onredundant rigidity is natural since if, after abar fpi ; pj g is removed, G.p/ is flexible, onewatches as the distance between pi and pj
changes during the flex, and waits until thedistance comes back to its original length. If p isgeneric to start with, the new configuration willbe not congruent to the original configuration.
Hendrickson conjectured that Conditions 1and 2 were also sufficient for generic globalrigidity, but it turns out that the complete bipartitegraph K5;5 in E3 is a counterexample (R. Con-nelly, On generic global rigidity, in Applied Ge-ometry and Discrete Mathematics, DIMACS Ser.Discrete Math, Theoret. Comput. Scie 4, AMS,1991, pp 147–155). This is easy to see as follows.
Similar to the analysis of Radon tenseg-rities, for each of the nodes for the twopartitions of K5;5 consider the affine lineardependency
P5iD1 �ipi D 0;
P5iD1 �i D 0
andP10
iD6 �ipi D 0;P10
iD6 �i D 0, where.p1; : : : ; p5/ and .p6; : : : ; p10/ are the twopartitions of K5;5. When the configurationp D .p1; : : : ; p10/ is generic in E3, then, up to ascaling factor, the stress matrix for K5;5.p/ is
Notes and References 315
˝ D
0
BBBBBBBB@
0
0
B@�6
:::
�10
1
CA��1 � � � �5
�
0
B@�1
:::
�5
1
CA��6 � � � �10
�0
1
CCCCCCCCA
:
See Bolker, E. D. and Roth, B., “When isa bipartite graph a rigid framework?” Pacific J.Math. 90 (1980), no. 1, 27–44. But the rank of� is 2 < 10 � 3 � 1 D 6, while rank 6 isneeded for generic global rigidity in this case byTheorem 25.
In E3, K5;5 is the only counterexample toHendrickson’s conjecture that I know of. On theother hand, a graphG is generically globally rigidinEd if and only if the cone overG is genericallyglobally rigid in EdC1 (see R. Connelly and W.Whiteley, “Global Rigidity: The effect of con-ing,” submitted). This gives more examples in di-mensions greater than 3, and there are some otherbipartite graphs as well in higher dimensions byan argument similar to the one here.
Meanwhile, the situation in the plane isbetter. SupposeG is a graph and fi; j g is an edgeof G, determined by nodes i and j . Remove thisedge, add another node k and join k to i , j , andd � 1 distinct other nodes not i or j . This iscalled a Henneberg operation or sometimes edgesplitting.
It is not hard to show that edge splittingpreserves generic global rigidity in Ed . Whenthe added node lies in the relative interior of theline segment of the bar that is being split, thereis a natural stress for the new bar tensegrity, andthe subdivided tensegrity is also universal withrespect to the new stress. If the original config-uration is generically rigid, a small perturbationof the new configuration to a generic one will notchange the rank of the stress matrix. Thus genericglobal rigidity is preserved under edge splitting.A. Berg and T. Jordan and later B. Jackson(A. Berg and T. Jordan, A proof of Connelly’sconjecture on 3-connected circuits of the rigiditymatroid. J. Combinatorial Theory Ser. B., 88,2003: pp 77–97) and T. Jordan (B. Jackson, and T.Jordan, Connected rigidity matroids and unique
realization graphs, J. Combinatorial Theory B 942005, pp 1–29) solved a conjecture of mine:
Theorem. If a graph G is vertex 3-connected(Condition 23) for d D 2 and is genericallyredundantly rigid in the plane (Condtion 23) ford D 2 then G can be obtained from the graphK4, by a sequence of edge splits and insertionsof additional bars.
Thus Hendrickson’s conjecture, thatConditions 1 and 2 are sufficient as well asnecessary for generic global rigidity in theplane, is correct. This also gives an efficientnon-probablistic polynomial-time algorithm fordetermining generic global rigidity in the plane.
Notes and References for Chapter 22
Notes on Problem 1
Albrecht Durer’s Underweysung der Messung mitdem Zirkel und Richtscheydt, Nurnberg, 1525(English translation with commentary by W. L.Strauss: “The Painter’s Manual: Instructions forMeasuring with Compass and Ruler”, New York1977), is an exciting piece of art and science. Theoriginal source for the unfolding polytopes prob-lem is Geoffrey C. Shephard, “Convex polytopeswith convex nets,” Math. Proceedings CambridgeMath. Soc., 1975, 78, 389–403.
The example of an overlapping unfoldingof a tetrahedron is reported by Komei Fukudain “Strange Unfoldings of Convex Polytopes,”http://www.ifor.math.ethz.ch/�fukuda/unfoldhome/unfold open.html, March/June 1997.
The first figure in our presentation is takenfrom Wolfram Schlickenrieder’s Master’s Thesis,“Nets of polyhedra,” TU Berlin, 1997, with kindpermission of the author.
For more detailed treatments of nets andunfolding and for rich sources of related material,see Joe O’Rourke’s chapter in this volume, andErik D. Demaine and Joseph O’Rourke, Geo-metric Folding Algorithms: Linkages, Origami,Polyhedra, Cambridge University Press, 2008.
316
See also Igor Pak, Lectures on Discrete andPolyhedral Geometry (in preparation), http://www.math.ucla.edu/�pak/book.htm, where thesource and star unfoldings are presented anddiscussed.
Notes on Problem 2
The neighborly triangulation of the torus with 7vertices (and
�72
� D 21 edges) was described byMobius in 1861, but the first polytopal realizationwithout self-intersections was provided by AkosCsaszar in 1948, in his article “A polyhedronwithout diagonals,” Acta Sci. Math. (Szeged),1949/50, 13, 140–142; see also Frank H.Lutz, “Csaszar’s Torus,” April 2002, ElectronicGeometry Model No. 2001.02.069, http://www.eg-models.de/models/ClassicalModels/2001.02.069.
Neighborly triangulations of orientable sur-faces for all possible parameters were providedby Ringel et al. as part of the Map Color The-orem: see Gerhard Ringel, Map Color Theorem,Springer-Verlag, New York, 1974. Beyond n D 4
(the boundary of a tetrahedron) and n D 7 (theCsaszar torus) the next possible value is n D12: But Jurgen Bokowski and Antonio Guedesde Oliveira, in “On the generation of orientedmatroids,” Discrete Comput. Geom., 2000, 24,197–208, and Lars Schewe in “Nonrealizableminimal vertex triangulations of surfaces: Show-ing nonrealizability using oriented matroids andsatisfiability solvers”, Discrete Comput. Geome-try, 2010, 43, 289–302, showed that there is norealization of any of the 59 combinatorial types ofa neighborly surface with 12 vertices and
�122
� D66 edges (of genus 6) without self-intersectionsin R3.
The McMullen–Schulz–Wills surfaces “withunusually large genus” were constructed by PeterMcMullen, Christoph Schulz and Jorg M. Willsin “Polyhedral 2-manifolds in E3 with unusuallylarge genus,” Israel J. Math., 1983, 46, 127–144; see also Gunter M. Ziegler, “Polyhedral sur-faces of high genus” in Discrete Differential Ge-ometry, Oberwolfach Seminars, 38, Birkhauser,Basel 2008, 191–213. The question about almost
disjoint triangles was posed by Gil Kalai; seeGyula Karolyi and Jozsef Solymosi, “Almostdisjoint triangles in 3-space,” Discrete Comput.Geometry. 2002, 28, 577–583, for the problemand for the lower bound of n3=2.
Notes on Problem 3
Steinitz’ theorem is a fundamental result: seeErnst Steinitz, “Polyeder und Raumeinteilun-gen” in Encyklopadie der mathematischenWissenschaften, Dritter Band: Geometrie,III.1.2., Heft 9, Kapitel III A B 12,1–139, 1922,B. G. Teubner, Leipzig, and Ernst Steinitzand Hans Rademacher, Vorlesungen uber dieTheorie der Polyeder, Springer-Verlag, Berlin,For modern treatments, see Branko Grunbaum,Convex Polytopes, Springer-Verlag, 2003 andGunter M. Ziegler, Lectures on Polytopes, Secondedition, Springer, 1995, revised edition, 1998;seventh updated printing 2007.
Steinitz’ proofs imply that a realization withinteger vertex coordinates exists for every combi-natorial type. Furthermore, there are only finitelymany different combinatorial types for each n, sof .n/ exists and is finite. The first explicit upperbounds on f .n/ were derived by Shmuel Onn andBernd Sturmfels in “A quantitative Steinitz’ theo-rem,” Beitrage zur Algebra und Geometrie, 1994,35, 125–129, from the rubber band realizationmethod of Tutte (see William T. Tutte, “Convexrepresentations of graphs,” Proceedings LondonMath. Soc., 1960, 10, 304–320).
Since Jurgen Richter-Gebert’s expositionin Realization Spaces of Polytopes, Springer-Verlag, Berlin Heidelberg, 1996, there hasbeen a great deal of research to improve theupper bounds; see in particular the Ph.D.thesis of Ares Ribo Mor, “Realization andCounting Problems for Planar Structures: Treesand Linkages, Polytopes and Polyominoes,”FU Berlin, http://www.diss.fuberlin.de/diss/receive/FUDISS thesis 000000002075, and AresRibo Mor, Gunter Rote and Andre Schulz,“Embedding 3-polytopes on a small grid,” Proc.23rd Annual Symposium on ComputationalGeometry (Gyeongju, South Korea, June 6–8,
Notes and References 317
2007), Association for Computing Machinery,New York, 112–118.
The upper bound f .n/ < 148n can be found inKevin Buchin and Andre Schulz, “On the numberof spanning trees a planar graph can have,” (2010)arXiv:0912:0712v2
The result about stacked polytopes wasachieved by Erik Demaine and A. Schulz,“Embedding stacked polytopes on a polynomial-size grid,” in: Proc. 22nd ACM-SIAM Sym-posium on Discrete Algorithms (SODA), SanFrancisco, 2011, ACM Press, 1177–1187.
A lower bound of type f .n/ � n3=2 followsfrom the fact that grids of such a size are neededto realize a convex n-gon; compare TorstenThiele, “Extremalprobleme fur Punktmengen,”Master’s Thesis, Freie Universitat Berlin, 1991;the minimal n�n-grid on which a convexm-gon
can be embedded has size n D 2��
m12
�3=2 CO.m logm/.
The theorem about edge-tangent realizationsof polytopes via circle packings is detailed inGunter M. Ziegler,“Convex Polytopes: Extremalconstructions and f -vector shapes,” in Geomet-ric Combinatorics, Proc. Park City MathematicalInstitute (PCMI) 2004, American Math. Society,2007; we refer to that exposition also for furtherreferences.
Notes on Problem 4
A survey of the theory of tilings can be found inEgon Schulte, “Tilings,” in Handbook of ConvexGeometry, v. B, North-Holland, 1993, 899–932.For tilings with congruent polytopes, we refer tothe survey by Branko Grunbaum and Geoffrey C.Shephard, “Tiling with congruent tiles,” BulletinAmer. Math. Soc., 3, 951–973. Their book, Tilingsand Patterns, Freeman, New York, 1987, is a richsource of information on planar tilings.
For the problem about the maximal number offaces, see also Peter Brass, William O. J. Moserand Janos Pach, Research Problems in DiscreteGeometry, Springer, New York, 2005.
Peter Engel presented his tilings by congruentpolytopes with up to 38 faces in “UberWirkungsbereichsteilungen von kubischer
Symmetrie,” Zeitschrift f. Kristallographie, 1981,154, 199–215, and Geometric Crystallography,D. Reidel, 1986.
Notes on Problem 5
The fascinating history of the original work byDescartes—lost, reconstructed and rediscoveredseveral times—is discussed in Chapters 3 and 4of this volume.
The paper by Ernst Steinitz describing thef -vectors .f0; f1; f2/ of 3-polytopes completelyis “Uber die Eulerschen Polyederrelationen,”Archiv fur Mathematik und Physik, 1906, 11,86–88.
The fatness parameter first appears (with aslightly different definition) in Gunter M. Ziegler,“Face Numbers of 4-Polytopes and 3-Spheres,”in Proceedings of the International Congress ofMathematicians (ICM 2002, Beijing), 625–634,Higher Education Press, Beijing; see also GunterM. Ziegler, “Convex Polytopes: Extremal con-structions and f -vector shapes,” cited above forProblem 3.
The 720-cell was apparently first found andpresented by Gabor Gevay, “Kepler hypersolids,”in Intuitive Geometry (Szeged, 1991), North-Holland, 1994, 119–129. The “projecteddeformed products of polygons” were introducedin Gunter M. Ziegler, “Projected Products ofPolygons,” Electronic Research AnnouncementsAMS, 2004,10,122. For a complete combinatorialanalysis, see Raman Sanyal and Gunter M.Ziegler, “Construction and analysis of projecteddeformed products,” Discrete Comput. Geom.,2010, 43, 412–435.
Notes on Problem 6
The Hirsch conjecture appears in GeorgeDantzig’s classic book, Linear Programming andExtensions, Princeton University Press, 1963. Forsurveys see Chapter 16 of Branko Grunbaum,Convex Polytopes, cited above; Victor Klee andPeter Kleinschmidt, “The d -step conjecture andits relatives,” Math. Operations Research, 1987,
318
12, 718–755; Gunter M. Ziegler, Lectures onPolytopes, cited above, and most recently EdwardD. Kim and Francisco Santos, “An update on theHirsch conjecture,” Jahresbericht DMV, 2010,112, 73–98. The Kim–Santos paper in particularexplains very nicely many bad examples for theHirsch conjecture. Santos’ long-awaited counter-example to the Hirsch conjecture appears in “Acounterexample to the Hirsch conjecture,” Annalsof Math., 2012, 176, 383–412.
The Kalai–Kleitman quasipolynomial upperbound (Gil, Kalai, and Daniel J., Kleitman, “Aquasi-polynomial bound for the diameter ofgraphs of polyhedra,” Bulletin Amer. Math. Soc.,1992, 26, 315–416, can also be found in Lectureson Polytopes. David Larman published his resultin “Paths on polytopes,” Proc. London Math. Soc,1970, 20, 161–178.
For the connection to Linear Programmingwe refer to Lectures on Polytopes; JirıMatousek, Micha Sharir and Emo Welzl, “Asubexponential bound for linear program-ming,” Proc. Eighth Annual ACM Symp.Computational Geometry (Berlin 1992), ACMPress, 1992; Gil Kalai, “Linear program-ming, the simplex algorithm and simplepolytopes,” Math. Programming, Ser. B,1997, 79, 217–233; and Volker Kaibel,Rafael Mechtel, Micha Sharir, and Gunter M.Ziegler, “The simplex algorithm in dimen-sion three,” SIAM J. Computing, 2005, 34,475–497.
Notes on Problem 7
The upper-bound theorem was proved by PeterMcMullen in “The maximum numbers of facesof a convex polytope,” Mathematika, 1970, 17,179–184. Very high-dimensional simplicial poly-topes with non-unimodal f -vectors were appar-ently first constructed by Ludwig Danzer in the1960s. The results quoted about non-unimodald -polytopes, d � 8 and about non-unimodalsimplicial d -polytopes, d � 20, are due toAnders Bjorner (“The unimodality conjecture forconvex polytopes,” Bulletin Amer. Math. Soc.,1981, 4, 187–188 and “Face numbers of com-
plexes and polytopes,” Proceedings of the Inter-national Congress of Mathematicians (BerkeleyCA, 1986), 1408–1418; Carl W. Lee, “Bound-ing the numbers of faces of polytope pairs andsimple polyhedra,” Convexity and Graph Theory(Jerusalem, 1981), North-Holland, 1984, 215–232; and Jurgen Eckhoff, “Combinatorial prop-erties of f -vectors of convex polytopes,” 1985,unpublished, and “Combinatorial properties off -vectors of convex polytopes,” Normat, 2006,146–159.
For a current survey concerning general (non-simplicial) polytopes see Axel Werner, “Linearconstraints on face numbers of polytopes,” Ph.D. Thesis, TU Berlin, 2009, http://opus.kobv.de/tuberlin/volltexte/2009/2263/. A detailed dis-cussion of cyclic polytopes and their propertiescan be found in Lectures on Polytopes, Sec.8.6. The example of a non-unimodal 8-polytopeglued from a cyclic polytope and its dual alsoappears there, as Example 841. The proof of theunimodality conjecture for cyclic polytopes wasachieved only recently by Laszlo Major, Log-concavity of face vectors of cyclic and ordinarypolytopes, http://arxiv.org/abs/1112.1713.
Notes on Problem 8
The relation C v � T was proved by Adam Blissand Francis Su in “Lower Bounds for SimplicialCovers and Triangulations of Cubes,” DiscreteComput. Geom., 2005, 33, 4, 669–686. The re-sults by Smith can be found in Warren D. Smith,“A lower bound for the simplexity of the n-cubevia hyperbolic volumes,” Eur. J. of Comb., 2000,21, 131–138, and those proved by Glazyrin inAlexey Glazyrin, “On Simplicial Partitions ofPolytopes,” Mathematical Notes, 2009, (6) 85,799–806. A lot more on triangulations in generalcan be found in the book by Jesus De Loera,Jorg Rambau and Francisco Santos, Triangula-tions, Springer-Verlag, 2010, while ChuanmingZong’s book The Cube—A window to Convex andDiscrete Geometry, Cambridge Tracts in Mathe-matics, Cambridge University Press, 2006 is anin-depth look at Id .
Notes and References 319
Notes on Problem 9
Zong posed his conjecture at a geometry meetingin 1994 in Vienna and published it in “Whatis known about unit cubes?”, Bulletin of theAmer. Math. Soc, 2005, 42, 181–211. LaszloFejes Toth’s Lagerungen in der Ebene, auf derKugel und im Raum, Springer-Verlag, 1972, con-tains the solutions for the case d D 3. See LeoniDalla, David Larman, Peter Mani-Levitska, andChuanming Zong, “The blocking numbers ofconvex bodies,” Discrete Comput. Geom., 2000,24, 267–277, for the proof of Zong’s conjecturein dimension 4.
Notes on Problem 10
The f -vector problem for 3-polytopes wassolved by Ernst Steinitz in “Uber die EulerschenPolyederrelationen”. The case d D 4 is surveyedin Gunter M. Ziegler, “Face Numbers of 4-Polytopes and 3-Spheres,” cited above.
For the g-Theorem see Section 8.6 ofLectures on Polytopes. Kalai’s 3d conjectureis “Conjecture A” in Gil Kalai, “The number offaces of centrally-symmetric polytopes (ResearchProblem),” Graphs and Combinatorics, 1989,5, 389–391; the conjecture is verified ford � 4 in Raman Sanyal, Axel Werner andGunter M. Ziegler, “On Kalai’s conjecturesabout centrally symmetric polytopes,” DiscreteComput. Geometry, 2009, 41, 183–198.
The Hanner polytopes were introduced byOlof Hanner in “Intersections of translatesof convex bodies,” Math. Scand., 1956, 4,67–89; The Mahler conjecture goes back to1939—see Mahler’s original paper, K. Mahler,“Ein Ubertragungsprinzip fur konvexe Korper,”Casopis Pest. Mat. Fys., 1939, 68, 184–189;a detailed current discussion of the Mahlerconjecture with recent related references appearsin Terry Tao’s blog, “Open question: the Mahlerconjecture on convex bodies,” http://terrytao.wordpress.com/2007/03/08/open-problem-the-mahler-conjecture-on-convex-bodies/.
Sources and Acknowledgments
Sources
We are grateful to all the authors, artists, mu-seums, publishers, and copyright holders whosework appears in this volume for permission toprint it.
Preface
All photographs by Stan Sherer.
Chapter 1
Figures 1.1 and 1.2. Photographs by Stan Sherer.Figures 1.4, 1.13, and 1.18. From Henry MartynCundy and A. P. Rollett, Mathematical Models,2nd edition, Oxford University Press, 1981.Figures 1.5 and 1.6. c�M. C. Escher Heirs c/oCordon Art-Baarn-Holland. Figure 1.6 is fromBruno Ernst, The Magic Mirror of M. C. Escher,New York: Ballantine Books, 1976.Figure 1.7. From Peter S. Stevens, Patterns inNature. (Boston: Little, Brown and Company,1974). Reprinted by permission.Figures 1.9 and 1.20. From Paolo Portoghesi, TheRome of Borromini: Architecture as Language,New York: George Braziller, 1968. Photographscourtesy of Electa, Milano.Figure 1.11. From Cedric Rogers, Rock andMinerals. London, Triune Books, 1973.
Figure 1.10. From Linus Pauling and RogerHayward, The Architecture of Molecules, SanFrancisco, W. H. Freeman, 1964.Figure 1.8. Photograph in the Sophia Smith Col-lection, Smith College.Figure 1.16. Wenzel Jamnitzer, Perspectiva Cor-porum Regularium, 1568; facsimile reproduction,Akademische Druck- u. Verlagsanstalt, 1973,Graz, Austria.Figure 1.19. Courtesy of Scienza e Tecnica 76,Mondadori.Figure 1.21. From Bruce L. Chilton and GeorgeOlshevsky, How to Build a Yog-Sothoth (GeorgeOlshevsky, P. O. Box 11021, San Diego, Califor-nia, 92111–0010, 1986).
Chapter 2
The photographs in Figures 2.1–2.3 and2.15 are by Ken R. O’Connell, and thephotographs in Figures 2.2 and 2.4–2.14are by Marion Walter. The photographs inFigures 2.18, 2.19, 2.25, 2.29, 2.33, and 2.37are by Stan Sherer.
Chapter 3
Figure 3.2. Drawing from Johannes Kepler,Harmonices Mundi, 1619.Figure 3.6. Drawing from Johannes Kepler,Mysterium Cosmographicum, 1595.
321
322 Sources and Acknowledgments
Figures 3.7 and 3.17–3.20. Drawn by PatrickDuVal for his book Homographies, Quaternions,and Rotations, London: Oxford University Press,1964.Figures 3.25–3.28. Photographs by Stan Sherer.
Chapter 4
Figure 4.1. The Metropolitan Museum of Art,New York. 27.122.5, Fletcher Fund, 1927.Figure 4.2. The Metropolitan Museum of Art,New York. 37.11.3, Museum purchase, 1937.Figure 4.3. Photograph reproduced by courtesy ofthe Society of Antiquaries of London.
Chapter 5
Figures 5.1–5.3. c�M.C Escher Heirs c/o CordonArt – Baarn – Holland.Figures 5.4, 5.10, 5.21 and 5.22. From theTeaching Collection in the Carpenter Center forthe Visual Arts, Harvard University. Reproducedwith permission of the Curator. The egg inFigure 5.4 was painted by Beth Saidel; the modelin Figure 5.10 was created by Brett Tomlinson.The models in Figures 5.21 and 5.22 weredesigned and constructed by Jonathan Lessersonand photographed by C. Todd Stuart.Table 5.1 and Figures 5.9, 5.11, 5.12, and 5.14are from Arthur L. Loeb, Space Structures: TheirHarmony and Counterpoint, Reading, Mass.Addison-Wesley, Advanced Book Program,1976.
Chapter 6
Figures 6.1–6.3, 6.5–6.7, 6.10b, 6.12 arereprinted, with the permission of Cambridge Uni-versity Press, from Eric D. Demaine and JosephO’Rourke, Geometric Folding Algorithms: Link-ages, Origami, Polyhedra, Cambridge UniversityPress, July 2007. In that volume they are, respec-tively, Figures ??, 21.3, 22.7, 22.9, 22.8, 22.17,22.32, and 22.2.
Chapter 7
Figure 7.1. Photograph by Hirmer Verlag,Munich.Figure 7.2. U.S. Air Force Photo by EddieMcCrossan.Figures 7.3, 7.4, 7.37, and 7.38. Photographs byIrwin Hauer.Figure 7.5. Photograph courtesy of the Buckmin-ster Fuller Institute, Los Angeles.Figures 7.6, 7.9, 7.13, 7.56, Photographs byWendy Klemyk.Figures 7.7 and 7.53. From Domebook 2(Bolinas, Calif.: Shelter Publications, 1971).Reprinted by permission of Steve Baer.Figure 7.8. Photograph by Steve Long. Universityof Massachusetts Photocenter.Figures 7.10 and 7.11. Courtesy of Zvi Hecker,architect.Figures 7.12, 7.24, 7.33 (top), 7.34, 7.39, 7.45,7.50, 7.52, and 7.54. Photographs by Stan Sherer.Figure 7.24 is used by permission of the Trusteesof the Smith College.Figure 7.14. From Wolf Strache, Forms and Pat-terns in Nature, New York: Pantheon Books, aDivision of Random House, Inc., 1956.Figure 7.15. From Vincenzo de Michele, Min-erali (Milan: Istituto Geografico de Agostini-Novara, 1971).Figure 7.16. From Earl H. Pemberton, Mineralsof California, New York: Van Nostrand ReinholdCompany Inc., 1983.Figures 7.17 and 7.43. From Viktor Goldschmidt,Atlas der Krystallformen, Heidelberg: CarlWinters, 1913–1923.Figure 7.18. Photograph by Carl Roessler.Figure 7.19. E. Haeckel, The Voyage of H.M.S.Challenger (Berlin: Georg Reimer, 1887), plates12, 20, and 63.Figure 7.20. Photograph by Lawrence Conner,Ph.D., entomologist.Figure 7.21. Karl Blossfeldt, Wundergarten derNatur, Berlin: Verlag fur Kunstwissenschaft,1932.Figure 7.22. Museo e Gallerie Nazionali diCapodimonte, Naples. Illustration by permissionof Soprintendenza ai B.A.S. di Napoli.
Sources and Acknowledgments 323
Figure 7.23. Alinari/Art Resource, New York.Figure 7.25. National Gallery of Art, Washington.Chester Dale Collection.Figure 7.26. The Salvador Dali Foundation, Inc.,St. Petersburg, Fla.Figure 7.27. Photograph supplied by artist, MaryBauermeister.Figure 7.28. Reproduced by permission of Mrs.Anni Albers and the Josef Albers Foundation,Inc.Figures 7.29, 7.30, and 7.51. c�M. C. EscherHeirs c/o Cordon Art-Baarn-Holland. Figure 7.51is reproduced from Bruno Ernst, The Magic Mir-ror of M. C. Escher, New York: Ballantine Books,1976.Figure 7.31. Photograph by permission of theMarine Midland Bank, Corporate Communica-tions Group.Figure 7.32. Photograph by Jeremiah O.Bragstad.Figure 7.33 (bottom). Photograph reproducedfrom Arthur L. Loeb, Space Structures: TheirHarmony and Counterpoint, Reading, Mass.:Addison-Wesley, Advanced Book Program,1976.Figure 7.35. Sculpture by Hugo F. Verheyen.Figure 7.36. From Gyorgy Kepes, The New Land-scape in Art and Science, Chicago: Paul Theobaldand Company, 1956.Figure 7.40. Photograph by Bob Thayer. c�1984by the Providence Journal Company.Figure 7.41. Photograph supplied by the sculptor,Robinson Fredenthal.Figure 7.42. From William Blackwell, Geometryin Architecture, p. 155. c�Copyright 1984 byJohn Wiley & Sons, Inc. Reprinted by permissionof John Wiley & Sons, Inc.Figure 7.46. From Peter S. Stevens, Patterns inNature, Boston: Little, Brown, and Company,1974.Figure 7.47. Photographs by R. W. G. Wyckoff.Figure 7.49. Cardboard models by Lucio Saffaro.Figure 7.55. Lampshade by Bahman Negahban,architect, and Ezat O. Negahban, calligrapher.Figure 7.57. Reprinted by permission of Dick A.Termes.
Figure 7.58. Reprinted from Better Homes andGardens Christmas Ideas. Copyright MeredithCorportation, 1957. All rights reserved.
Chapter 8
Figure 8.2. Photograph by Stan Sherer.
Chapter 9
Figures 9.1 and 9.3 are Leonardo da Vinci’sdrawings for Luca Pacioli’s Divina Proportione,Milan, 1509.
Chapter 10
Figures 10.8 and 10.11. c�M.C. Escher Heirs c/oCordon Art–Baarn–Holland.
Chapter 11
Figure 11.4. From Ralph W. Rudolph, “Boranesand Heteroboranes: A Paradigm for the ElectronRequirements of Clusters?” Accounts of Chemi-cal Research, 9 (1976):446–52. Copyright 1976American Chemical Society.Figure 11.21. Drawing by the artist FerencLantos.
Chapter 12
Figure 12.1. After A. W. Hofmann, “On theCombining Power of Atoms,” Proceedings ofthe Royal Institution of Great Britian 4 (1865):401–30.Figure 12.2. Reprinted from PhilosophicalMagazine, series 4, 21, (1861): Plate V, Figure 2.Figures 12.3–12.5. From J. H. van ’t Hoff,The Arrangement of Atoms in Space, 2nd ed.(London: Longmans, Green, 1898).
324 Sources and Acknowledgments
Figure 12.6. From Linus Pauling and RogerHayward, The Architecture of Molecules. W.H. Freeman and Company. Copyright c�1964.Figure 12.7. Reproduced by permission ofPergamon Press.Figure 12.8. Reprinted from Linus Pauling, TheNature of the Chemical Bond and the Structure ofMolecules and Crystals: An Introduction to Mod-ern Structural Chemistry, 3rd edition. Copyright1960 by Cornell University. Used by permissionof Cornell University Press.Figure 12.9. Used by permission of John C.Spurlino and Florentine A. Quiocho, Departmentof Biochemistry, Rice University.Figures 12.10 and 12.11. Courtesy of D. L. D.Caspar.Figure 12.12. From E. B. Matzke, “The Three-Dimensional Shape of Bubbles in Foam —An Analysis of the Role of Surface Forces inThree-Dimensional Cell Shape Determination,”American Journal of Botany 33Figures 12.15–12.16. Models made by CharlesIngersoll, Sr. Photographs by Fred Clow.Figure 12.23. Model made by Charles Ingersoll,Sr. Photogarph by William Saunders.
Chapter 21
Figure 21.1, Hirshhorn Museum and SculptureGarden, Washington, D.C.
All illustrations NOT listed above werecreated for Shaping Space by (or for) the authorsof the chapters in which they appear.
Acknowledgments
I am grateful to Ann Kostant, long-timeMathematics Editor of Springer, New York,for encouraging me to prepare this volume andfor making it possible. Smith College studentsAmy Wesolowski and Marissa Neal typed theLaTeX code and made helpful comments andsuggestions.
Stan Sherer, many of whose photographs ap-pear in this volume, scanned and enhanced theillustrations and helped in countless ways.
My Smith College colleague George Fleck’ssound advice and eagle eye for detail have been,once again, invaluable.
Marjorie Senechal
Chapter 2
Jean Pedersen thanks Les Lange, Editor of Cal-ifornia Mathematics, for giving permission touse in this article some of the ideas that wereoriginally part of “Pop-up Polyhedra,” CaliforniaMathematics (April 1983):37–41.
Chapter 4
Joe Malkevitch thanks D. M. Bailey of theBritish Museum and Dr. Maxwell Andersonof the Metropolitan Museum of Art fortheir cooperation in obtaining access to theinformation about early man-made polyhedra.J. Wills (Siegen) provided him with a copy ofLindemann’s paper and Robert Machalow (YorkCollege Library) helped obtain copies of manyarticles, obscure and otherwise.
Chapter 10
Chung Chieh thanks Nancy McLean for redoingsome of the drawings.
Chapter 11
This chapter was written when the authors wereat the University of Connecticut (Istvan Hargittaias Visiting Professor of Physics (1983–84) andVisiting Professor of Chemistry (1984–85) andMagdolna Hargittai as Visiting Scientist at the In-stitute of Materials Science), both on leave fromthe Hungarian Academy of Sciences, Budapest.They express their appreciation to the University
Sources and Acknowledgments 325
of Connecticut and their colleagues there forhospitality, and to Professor Arthur Greenberg ofthe New Jersey Institute of Technology for manyuseful references on polycyclic hydrocarbons andfor his comments on the manuscript.
Chapter 15
Research for this chapter was supportedby the National Science Foundation GrantMCS8301971.
Chapter 16
Egon Schulte thanks Professors Ludwig Danzerand Branko Grunbaum for many helpful sugges-tions.
Chapter 18
Research for this chapter was supported in partby a grant to Henry Crapo from National Scienceand Engineering Research Council of Canada andin part by grants to Walter Whiteley from Fondspour la Formation de Chercheurs et l’Aide a laRecherche Quebec and NSERC, Canada. Thischapter also grows out of previous joint workby Crapo and Whiteley, Ash and Bolker, andBolker and Whiteley. The unification which isits theme developed while they were writing it.
Because time was short, the pleasure of shapingthe chapter and working out the details fell tothe final author, Henry Crapo. They all share theresponsibility for any errors in the text.
Chapter 20
Tom Banchoff acknowledges with gratitudethe help he received from correspondence andconversations with Professor Coxeter during thecourse of preparation of this paper. Computerimages in this chapter were generated incollaboration with David Laidlaw and DavidMargolis, and with the cooperation of the entiregraphics group at Brown University. Severalof the illustrations are taken from the filmThe Hypersphere: Foliation and Projection byHuseyin Kocak, David Laidlaw, David Margolis,and the author.
Chapter 22
Moritz Schmitt thanks the DFG ResearchCenter MATHEON, Institut fur Mathematik,Freie Universitat Berlin, Arnimallee 2, 14195Berlin, Germany, for support. Gunter M.Ziegler’s work was partially supported byDFG, Research Training Group “Methods forDiscrete Structures,” also at the Institut furMathematik.
Index
AAcetylene, 159, 175Adamantane, structure of, 157–159Adjacency, 116, 119Adonis pernalis, 96Aether, 173, 174, 188Affine
hull, 221polyhedra, 120properties, 113, 117
Affleck, 122After the Flood, 97Air, 5, 21, 22, 39, 42, 57, 140, 267Albers, Anni, 323Albers, Josef, 96, 99, 323
structural constellation, 99Alexandrov, Aleksandr D., 62, 85, 202, 296, 305Alum, 140Aluminum silicates, 141Alves, Corraine, 28Ammonia, 162Angeles, Los, 322Angles, 58
dihedral, 32, 35, 78, 129, 135Antiprism
bicapped square, 154, 155definition of, 9ferrocene as, 160, 161hexagonal, 181H3N�AlCl3 as, 160non-convex isogonal, 215octagonal, 215pentagonal, 18, 42, 112semiregular, 9, 57–58, 120square, 69, 154triangular, 159, 160
Antiprismatic polygon, definition of, 195Application as geometrical action, 110Archambault, Louis, 122Archimedean polyhedra
definition of, 59hydrocarbons as, 155
Archimedean solids, 8, 37, 42, 44, 47, 55, 57, 59, 60, 62,77, 80, 227
definition of, 47flag diagram of, 227
Archimedes, 8, 47, 54–59, 126, 171, 197, 199Ari-kake joint, 187Aristotle, 140, 172, 178Ashkinuse, V.G., 59, 60Ash, Peter, 231–251Automorphism of polyhedron, 225, 229Axis of symmetry, 213, 257, 262, 275
BBack, Allen, 276Baer, Steve, 91, 323Balloons
shapes of groups, 163twisting, 33, 37, 39–40
Banchoff, Thomas F., 257–266Baracs, Janos, 109–123, 231Bar-and-joint framework, 201–203, 238Barbaro, Daniel, 57Bar directions of a bar tensegrity, 271Barlow, William, 179Bartholin, Erasmus, 140Bauermeister, Mary, 96, 99
Fall-Out, 99Bean mosaic virus, 105Beer mats, 19Benitoite, 92, 94Berry, R. Stephen, 166, 167Bertrand, J., 60Bezdek, Daniel, 84Bicapped square antiprism
borane as, 155carborane as, 155
Billera–Lee, 288Billiard-ball model, 173, 188Bill, Max, 97, 101
Construction with 30 Equal Elements, 97, 101Bipyramid
pentagonal, 16, 154, 155triangular, 16trigonal, 154, 155, 163, 164, 166
Black and white knights, 143–146Blackwell, William, 323Bloon number, 37Blossfeldt, Karl, 322Body-centered lattice, 144, 145
327
328 Index
Bokowski, Jurgen, 280, 316Bolker, Ethan, 231–251, 315Bolyai, Wolfgang Farkar, 61Bombelli, R., 57Bonding
of molecules, 160in ONF3, 160in sulfates, 163in sulfones, 163in tetrafluoro-1,3-dithietane, 160in zirconium borohydride, 154
Boole-Stott, Alicia, 226Boranes
arachno, 155–157closo, 154–157, 167–168nido, 155–157quasi-closo, 154relationships among, 154
Boron hydrides, 153–155. See also BoranesBorromean rings, 37, 39Borromini, 6, 10Boundary, definition of, 194, 195Bounded
definition of, 194, 195regions, 195
Bowl, 136, 239–251Braced grids, 121Bragstad, Jeremiah O., 323Brass tetrahedron, 101Bravais, A., 143, 144Braziller, George, 321Brehm, U., 224Brisson, Harriet E., truncated 600-cell, 97, 102Brodie, Benjamin Collins, 172Bronze dodecahedron, 56, 57Brooks, A. Taeko, 107Brown, Ezra, 232Brown, K.Q., 247Bruckner, Max, 53, 60, 199Buba, Joy, 98Buckminsterfullerene, 129, 168, 169, 175Bullvalene, nuclear interconversion in, 165,
166Burns, A. Lee, 97, 101, 104
tetrahedron, 97, 101
CCalif, Bolinas, 322Camera lucida drawings, 180Candy box, (Escher), 5, 6Capsid, models of, 177Carbonyl scrambling, 168Carborane, 154Carex grayi, 96Caspar, Donald, 176–178, 182, 184, 189, 324Catalan, Eugene Charles, 60, 199Cauchy, Augustin-Louis, 10, 59, 60, 202, 207Cauchy’s theorem, 202, 207Cayley, A., 60
Cell decomposition, 234–250Centroid, 133, 212Challenger, H.M.S., 93Cheese-slicing algorithm, 75Chefren, pyramid of, 89Cheops, pyramid of, 89Chieh, Chung, 139–151Chilton, Bruce L., 10, 11, 321Chinese postman problem, 36Chirality, definition of, 70Chrome alum, 140Circles
great, 48–49, 264packing of, 141, 233, 234, 238, 248,
281, 317Circumcenter, 212Circumcircle, 41Classification as geometrical action, 110Closed set, definition of, 195Clow, Fred, 324Coca-Cola Building, 90Combinatorial dual, definition of, 212Combinatorial geometry, 113Combinatorial prototiles, 217–222Compact set, 195
definition of, 194Computational geometry, 184, 188, 251, 267Computer
graphics by, 196, 258realization by, 133recognition of polyhedra by, 150
Configuration, 30, 62, 72, 109, 141, 145, 154, 155,158, 160, 163–165, 167, 186, 187, 225, 251,267–275, 277, 282
Connelly, Robert, 250, 267–278Conner, Lawrence, 338Connolly, Helen, 106Construction with 30 Equal Elements, 97, 101Continuity, 116Continuous mapping, 116Convex hull, 83, 84, 133, 241, 247, 254, 276,
279, 385Convexity, 77, 117, 185, 211, 213, 214, 218, 239, 246,
251Convex polyhedron
definition of, 195projection of, 232, 233, 241, 247
Convex reciprocal, figure, 233, 235–241, 243, 246,248–250
Coorlawala, Uttara, 104, 106Cornus kousa, 96Corpus Hypercubicus, 96Cosmic Contemplation, 96, 100Covalent radii, 179Coxeter, H.S.M., 5, 41–54, 62, 153, 199
portrait of, ixCrapo, Henry, 231–251Crawl-through toy, 104, 107Crick, Francis, 176Crosspolytope, 220
Index 329
Crystalarchitecture of, 102, 141benitoite, 92, 94chrome alum, 140gold, 91, 94leucite, 91pyrite, 6, 7, 54quartz, 91, 93relationships, 139structures, 99, 139–151, 159, 179systems, 144, 146, 149, 150vanadinite, 140virus, 109wulfenite, 139
Csaszar polyhedron, 213Csaszar torus, 213, 280Cubane, 155, 158Cube-dual, 259Cubes. See also Snub cube
construction of, 21, 30decomposition of, 258, 286–287diagonal, 115with face, 3within octahedron, 103open packing of, 142as platonic element, 42plexiglas, 103as reciprocal of octahedron, 42, 43relation to rhombic dodecahedron, 30–32rotations of, 45, 46Schlegel diagram of, 70, 71, 258snub, 8, 37, 38, 47–50, 70, 71, 77, 78, 80, 82,
83, 253as space filler, 220tessellation by, 221triangulation in, 286truncated, 47–49, 73–75, 103, 253vertices of, 4, 11
Cube-within-a-cube projection, 258–260Cubical nontiles, 221
existence questioned, 221Cubical vase, 107Cubist paintings, 96Cuboctahedron
definition of, 219not combinatorial prototile, 218–220truncated, 47–49as unit of crystal structure, 145, 146vertices of, 73, 253, 262, 265
Cubus simus, 47Cundy, Henry Martyn, 321Cut edge, 81, 82, 208, 209Cycle, Hopf, 266Cyclic permutations, 50, 72, 254Cyclopentane, pseudorotation in, 165
DDali, Salvador, 96, 98
Corpus Hypercubicus, 96
Cosmic Contemplation, 96, 98The Sacrament of the Last Supper, 96, 98
Dalla, Leoni, 335Dalton, John, 172Dance, 165Dantzig, George, 62, 317Danzer, Ludwig, 218, 285Darwin, Charles, 93da Vinci, Leonardo, 42, 57, 96, 323de Barbari, Jacopo, 96Decomposition
cell, 234–238, 245–251of cube, 258, 259finite, 61, 248Hopf, 264–265of hypercube, 258–261, 264–265of octahedron, 258, 259, 262, 265of polytopes, 257–266
de Foix, Francois, 57Dehn, Max, 54, 61, 199Dehn-Sommerville equations, 60–61de Justice, Palais, 122de la Fresnaye, Roger, 96della Francesca, Piero, 57de Louvre, Musee, 97Deltahedra-regular polyhedra, 105Deltahedron
borane as, 154construction of, 104definition of, 17
Demaine, Eric, 13–40, 295Demaine, Martin, 13–40de Michele, Vincenzo, 322Democritus, 54dem Zirckel, 279de Oliveira, Antonio Guedes, 280Desbarats, 122Descartes, Rene, 42, 58, 140–141Design, 29, 89, 109, 114, 115, 118, 119, 123, 125, 126,
131, 133, 135, 139, 151, 169, 176–178, 184,195, 202, 243
de Stein, 122Determination as geometrical action, 110Diamond, 18, 148, 149, 158, 234
structure of, 149Dibenzene chromium, 160, 161Dicarba-closo-dodecaboranes, 168Dice, icosahedral, 41Diego, San, 321Dieudonne, Jean, 211Dihedra, 197Dihedral angles, 32, 35, 78, 129, 135Dihedral rotation group, 46, 224Dimakopolous, 122Dimolybdenum tetra-acetate, 160, 161Diophantine equation, 70, 253–255Dirichlet domain, 142–148Dirichlet, G. Lejeune, 142Dirichlet tessellation, 196, 231–251Disdodecahedron, 225
330 Index
Distances, 30, 44, 45, 49, 112, 113, 117, 118, 129, 134,141, 145, 160–162, 179, 237, 243, 268, 272,284, 287
Dodecahedral crystal, 5, 6, 54, 56Dodecahedral housing complex, 91Dodecahedral recycling bin, 7Dodecahedrane, 155, 158Dodecahedron
borane as, 155bronze, 56, 57carborane as, 155construction of, 30Etruscan, 42ghost of, 3golden, 23icosahedron as reciprocal of, 42as platonic element, 227
Dome, 50, 68, 84, 89, 91, 93, 104, 106, 121, 129,131–132, 136, 137, 169, 275
articulated ring, 121geodesic, 50, 89, 91, 93, 129, 132,
169–170, 177Doughnut, 27, 65, 66, 96Dress, Andreas, 63Dual
combinatorial, 212existence of, 212
Duality, 58, 60, 65, 100, 129, 196, 211–216Durer, Albrecht, 56, 57, 79, 199, 275
Melencolia I, 77Durer’s Problem, 77–86DuVal, Patrick, 337
EEarth, 5, 42, 44, 45, 69, 135, 140Easter egg, tessellation of, 67Eberhard’s theorem, 198Eclipse, 97, 100Edelsbrunner, Herbert, 233Edges
curved, 68, 116definition of, 3length of, 30, 120, 132, 141, 194, 197, 198, 225, 227,
267, 272, 288, 314number of, 10, 17, 33, 34, 36, 50, 59, 67, 69, 70, 114,
134, 203, 205, 218, 227, 280skeleton, 250
Egyptian pyramids, 89Electron microscope, 102Element, 5, 7, 42, 54, 55, 69, 72, 73, 97, 101, 114, 145,
148, 150, 175, 178–181, 199, 202, 212, 214,249, 276. See also Platonic solids
definition of, 5Eliminating dependencies by edge subdivisions, 206Engel, Peter, 282Epicurus, 172, 173Equivelar manifold, 223, 225, 228Erickson, Ralph O., 122, 179–183, 189Ernst, Bruno, 321, 323
Escher, M.C., 5, 6, 65, 66, 96, 99, 103, 106, 142–146,150, 151, 153, 197, 321–323
Angels and Devils, 66Black and White Knights, 143–146candy box, 5, 6contemplating polyhedron, 104contemplating spheres, 66Fishes and Birds, 143, 144, 146Order and Chaos, 99Reptiles, 6Sphere with Angels and Devils, 66Three Spheres II, 66Waterfall, 98
Eskenazi, 122Ethylene, 159, 175Euclid, 41, 54, 55, 58, 59Euclidean geometry, 241Euclidean 3-space (E3/, 217, 218, 223Eudemus of Rhodes, 54Eudoxus, 54, 55Euler characteristic, 59, 280Eulerian path, 33, 35Euler, Leonhard, 11, 33, 34, 58, 59, 99, 212, 283Euler-Poincare equation, 283Euler’s equation, generalization of, 288Euler’s formula, 51, 58, 126, 130, 193, 198, 219Exhaustion, method of, 54Expandable sculpture, 101Exposed vertex, 23
FFace-centered lattice, 145, 146Faces
of cube, 144curved, 5, 6definition of, 3dihedral angles between, 32, 78infinite helical polygon as, 196number of, 9, 10, 17, 41, 51, 59, 67, 83, 100, 126,
134, 137, 182, 207, 221, 282ordering of, 118pentagonal hole in, 68, 126, 180of polytope, 282self-intersecting, 216skew polygon as, 194, 196square hole in, 20, 30, 49, 69, 75, 141, 142, 167, 183,
189, 258–260, 264, 265tetrahedron as, 11, 265
Facet, 217, 219, 221, 282Face-to-face tiling, 217–222, 283Faıence icosahedron, 56Fall-Out, 99Fatness, 283–284, 288Fatness parameter, 283Federico, P.J., 42, 58Federov, E.S., 102Fejes Toth, L., 53Fejes-Toth, Laszlo, 288Ferrocene, 160, 161
Index 331
Fieser, Louis F., 157Fire, 5, 42, 140Fisher, Ed, 53Fishes and Birds, 143, 144, 146Flag, 223, 225, 289
definition of, 223The flag conjecture, 289Flag diagram, 224, 227, 228Flatness, 117Flat torus, 224, 264Fleck, George, 171–189Flexible, 13, 16, 19, 32, 135–137, 177, 187, 198,
201–208, 268, 270, 277–278, 315Flex of the tensegrity, 268Florence, Sabin, 65, 96, 97Florentine hat, 68. See also MazzocchioFoam, 180Fold-out decomposition, 260–261Folklore, 109, 211, 213, 216
4-simplex, 33, 34Framework octahedron, 100Frameworks, 63, 100, 137, 142, 154, 198, 201–208,
231–233, 238, 246, 267, 270–271Francisco, San, 100, 304, 317Fredenthal, Robinson, 103, 323
Ginger and Fred, 103Freeman, W.H., 321, 324Frost, Robert, 193Froth, 100, 104Fuller, R. Buckminster, 50, 89, 109, 129, 130, 169, 170,
267, 322. See also Buckminsterfullerenegeodesic dome, 89, 129
Furthest-point Dirichlet tessellation, 233, 237, 243, 244,247, 250
f-vector, 197, 198, 283, 285, 288
G��Brass, 145–146General unfolding, 85–86Generically rigid, 203, 204, 315Genus, 86, 196, 198, 223, 225, 227– 229, 316
definition of, 195, 280Geodesic dome
construction of, 91house, 91
Geometrymolecular, 153, 162, 163, 174teaching of, 119
Gergonne, J.D., 59Gerwien, Karl, 61Gerwien, P., 61Gilbert, A.C., 189Ginger and Fred, 103Girl with a Mandolin, 15Girvan, 122Giza, pyramids at, 89Glass polyhedra, 57Globally rigid, 270, 274–277, 314–315Global rigidity, 267–278
Gluing joints, 32Glur, Paul, 61Goldbach, Christian, 59Goldberg, Michael, 125, 133Goldberg Polyhedra, 125–139, 168Gold crystals, 94Golden dodecahedron, definition of, 23Golden ratio, 254Goldschmidt, Viktor, 322GPs and Fullerenes, 129Granche, Pierre, 122Graphical statics, 231, 248, 308Graph, planar, 198, 203, 207–209, 211–212, 231, 233,
234, 243, 245–247, 250, 305, 309, 317Gray, Jack, 31Great stellated dodecahedron, 23–25, 42, 43
construction of, 24Group
point, 145, 148, 155, 158theory, 59, 62, 153
Grunbaum, Branko, 51, 53, 62, 63, 109, 194, 199,211–216, 221, 293, 295, 296, 304, 306, 316,317
Gvay, Gabor, 283
HHadamard, Jacques, 58Hadwiger, Hugo, 61Haeckel, Ernst, 93, 322Hales, Stephen, 178Hampshire College, 90
modular residences, 92Hanner, Olof, 289, 319Hardy, G.H., 254Hargittai, Istvan, 153–170, 175Hargittai, Magdolna, 153–170, 175Hart, George, 125–138Hart, Vi, 13–40Hauer, Erwin, 89, 97, 101, 102, 322
Coca-Cola Building, 90Obelisk, 102Rhombidodeca, 101
Hayward, Roger, 175, 321, 324Hecker, Zvi, 90, 322Hexagon
as a face, 68, 83, 89, 96, 125, 126, 181, 189, 213, 225,227
polytope analogous to, 49, 224Hexagonal kaleidocycle, 28Hexagonal lattice, 147, 148
Dirichlet domain of, 147, 148Hexagonal prism, 147
vanadinite crystal, 140Hexagonal pyramid, tiling of E3 by, 222Hexahedron. See CubesHexamethylenetetramine, 145, 147Hexaprismane, 157, 158Higher-order deltahedra, 93Hilbert, David, 60, 61, 282
332 Index
Hilbert’s 18th problem, 282Hilton, Peter, 58Hinges, 30–32, 201–205, 207–209Hirsch conjecture, 284–285, 317, 318Hirsch, Warren, 284Hirsh, Meyer, 59History of polyhedra, 53–63, 196Hofmann, August Wilhelm, 172, 323Hofstadter, Douglas R., 123Holden, Alan, 37, 97, 102
“Ten Tangled Triangles”, 97, 102Hole, 15, 19, 27, 28, 39, 65, 66, 77, 86, 120, 131, 135,
186, 189, 194, 195, 201Holyer’s problem, 37Honeybee comb, 95Honeycomb, 41, 103Hooke, Robert, 141Hopf decomposition, 264–265Hopf, Heinz, 257Hopf mapping, 257, 263–266Hornby, Frank, 188Hull
affine, 221convex, 83, 84, 133, 241, 247, 254, 276, 279, 285,
308, 309Hungarian hut, 324Hydrocarbons, 157, 160
polycyclic, 155–158Hydrogen fluoride, 163Hypercube
decomposition of, 260–261, 264–265projection of, 258, 266
Hypersphere, 257, 263, 264, 266Hypostrophene, 165, 166
intramolecular rearrangements in, 165
IIce, 157, 159
structure of, 148, 149Iceane, 157
structure of, 159Icosahedral candy box, 5Icosahedral crystal, 6, 7Icosahedral kaleidoscope, 49Icosahedral rotation group, 46Icosahedron
B12H2�
12 as, 154, 155borane as, 167carborane as, 154, 155as a deltahedron, 16dodecahedron as reciprocal of, 42incised, 56isogonal, 213model of, 167, 253plane net of, 4truncated, 47, 77–80, 126, 127, 129, 133, 135, 254vertices of, 41, 133, 225, 254, 280
Icosidodecahedron, 47, 100, 220, 254definition of, 47truncated, 47, 254
Icositetrahedral leucite crystal, 93Identity operation, 271Image, 24, 48, 65, 70, 112, 129, 188, 231, 258, 264, 270,
273creation of, 109, 110
Impossible structure, 96Incenter, 212Incidences, 112, 113, 116, 117, 120, 202,
207, 284Incircle, 41, 65Independence and redundancy, 202Infinitesimal rigidity, 203Ingersoll, Charles, Sr., 324Intramolecular motion, 154, 165–168Ion exchangers, 141IRODO, 101Isoaxis, 27
construction of, 28–29Isogonal icosahedron, 213Isogonalities, 213–215Isogonal polyhedron, definition of, 213, 214Isohedral dodecahedron, 213Isohedral polyhedron, definition of, 213Isomerism, permutational, 166Isomorphic polyhedral, 214
definition of, 213, 218Isomorphism classes, 212Isothetic, 185, 186Isotoxal polyhedron, definition of, 213
JJamnitzer, Wenzel, 8, 9, 57, 321Joe-pye weed, 179Johnson, Norman W., 62Joswig, Michael, 229Jupiter, 44, 45
KKalai, Gil, 281, 284, 289Kalai’s question, 289Kaleidocycle, constructi of, 26–29Kaleidoscope, 48, 49Karolyi, Gyula, 281Katoh, N., 204, 208, 209Katoh–Tanigawa theorem, 208Kelvin, Baron of Largs (Lord Kelvin), 173, 178–180,
182, 189Kepes, Gyorgy, 323Kepler, Johannes, 5, 9, 42–45, 56–60, 112, 126, 140, 168,
172, 199, 317planetary system of, 45, 168
Kirkpatrick, Katherine, 23Klein, Felix, 45, 47, 123, 226, 227, 229Kleitman, Daniel, 284Klemyk, Wendy, 322Klug, A., 176, 177, 183Knots, 21, 22, 173Koestler, Arthur, 44, 123
Index 333
LLabyrinth, 101, 102LaFollete, Curtis, 102Lakatos, Imre, 193Lantos, Ferenc, 163, 323Larman, David, 284Lattice, 111, 132, 134, 143–148, 150, 177, 183, 184, 188,
227, 248Lead shot, 180, 181Le Bel, Joseph, 174Lebensold, 122Lebesgue, H., 58Lee, A., 206Legendre, Adrian-Marie, 59Leibniz, Gottfried Wilhelm, 42, 58Lemma, 208, 209Lennes, N.J., 61Lesser rhombicuboctahedron, 73–75Lesserson, Jonathan, 322Leucite, 91Levinson, P., 189Lewis, Gilbert Newton, 175Lhuilier, S., 59Linear arrangement of electron pairs, 162Line-sweep heuristic, 185Lipscomb, W.N., 167Loeb, Arthur L., 13–40, 65–75, 97, 100, 109, 322, 323
Polyhedral Fancy, 97, 100Lovasz, L., 275Lucretius, 172, 173
MThe Mahler conjecture, 289, 319Mahler, K., 289, 319Malkevitch, Joseph, 53–63, 198Man in the Community building, 121, 122Mani, Peter, 221, 319Man the Explorer building, 121, 122Map coloring, 280, 316Mapping
continuous, 116Hopf, 257, 263–266
Mars, 44, 45Martin, George, 60Marvin, James, 179, 180Massey, 122Mathematics Activity Tiles (MATs), 19MATs. See Mathematics Activity Tiles (MATs)Matzke, Edwin B., 179–182, 324Maxwell bowl, 244, 249, 250Maxwell, James Clerk, 173, 231Maxwell paraboloid, 240, 244Mayers, 122Mazzocchio, 96, 98, 104, 106McCooey, David, 229McCrossan, Eddie, 322McMullen, Peter, 254, 281, 285Meister, A.L.F., 212Melencolia I, 77
Mercury, 44, 45Methane, 7, 153, 162, 163, 189Metric polyhedra, 120Metric properties, 55, 110, 118, 119, 123, 197, 198Michelson, Albert, 174Milestones, 53–63, 77, 99Miller, Dayton, 174Miller, J.C.P., 59Miller’s solid, 60Minimally rigid, 202–204, 208, 209Mirror symmetry, 7, 118, 128Mobius, A.F., 47, 48, 195, 216, 316Model
affine, 111aluminum, 141, 188of As4 molecule, 153, 154of atom, 179ball-and-stick, 172, 173, 179, 201billiard-ball, 173, 188of borane, 155, 167of capsid, 177of carbon-carbon bonds, 174of carborane, 155cardboard, 182, 183, 225, 323of [Co6(CO)14]4�, 168, 169construction of, 18, 184of dibenzene chromium, 160, 161of ethane, 159of ferrocene, 160, 161of gas molecule, 173of �-brass atomic structure, 145–147hard-rubber, 174of H3N � AlCl3, 159, 160of icosahedron, 6, 16, 37, 41, 44, 56, 78, 127, 129,
133, 167, 253of KAlF4, 160, 161Lipscomb rearrangement, 167of macromolecular dynamics, 184mechanical, 173, 182, 184, 188, 189of methane, 153, 189of methane molecule, 7, 162metric, 111of Mo2(O2CCH3/4, 160, 161of [2.2.2.2]-paddlane, 160, 162paper, 128, 134, 181of pentagonal bipyramid, 16, 154, 155of plant growth, 182–183of plant structure, 179–182plastic-tubing, 181projective, 111of [1.1.1]-propellane, 160, 162of [Re2Cl8]2�, 160, 161self-assembly, 182–184, 189, 214skeletal, 181, 182, 187of sodalite, 175, 176of sodium chloride crystal structure, 141spatial, 111–113, 115of sulfate ion, 162of tessellation of sphere, 72of tetrafluoroaluminate ion, 160, 162
334 Index
topological, 111, 115of trigonal bipyramid, 154, 155, 163–166of vortices in aether, 173VSEPR, 162–165, 175wooden, 135of zeolite crystal structure, 141of zirconium borohydride, 154
Modular residences, 92Modules for generating a rhombic dodecahedron, 30–32,
58, 101(hinged) Molecular frameworks, 203–207
Molecular geometries, 153–170, 174Molecular polyhedron, 207Molecular sieves, 141
of acetylene, 159, 175of adamantane, 158, 159of As4, 153of dibenzene chromium, 160, 161of dodecahedrane, 155, 158of ethylene, 159, 175of ferrocene, 160of hexaprismane, 157, 158of H3N�AlCl3, 160of iceane, 157, 159of KAlF4, 160, 161of methane, 7, 162of Mo2(O2CCH3/4, 160of ONF3, 160, 162of [2.2.2.2.]-paddlane, 160, 162of pentaprismane, 157, 158, 165of [1.1.1]-propellane, 160, 162of sulfone, 161, 163of sulfuric acid, 161, 163of tetrahedrane, 155of tetra-tert-butyltet-rahedrane, 155of triprismane, 157of zirconium borohydride, 154
Monotonic figure, 187Monson, Barry, 197, 253–255Monster barring, 193Morley, Edward, 174Moscow papyrus, 53Moser, William O.J., 225Mt. Loffa, 57Muetterties, Earl L., 153–155Mundi, Harmonices, 42, 57, 199, 321Mycerinus, pyramid of, 89
NNamiki, Makoto, 279Negahban, Bahman, 323Negahban, Ezat O., 323Negev Desert, synagogue in, 92Neighborly complex, 280, 285, 316Neighborly polyhedra, definition of, 316Neighbor switching, 183, 187Net
of crawl-through toy, 104, 107of cube, 77, 82
definition, 77of dodecahedron, 23, 57of icosahedron, 77of kaleidocycle, 26–27of octahedron, 33of polyhedron, 57, 77, 81, 82of total photo, 104, 107of triangles, 26, 28
Networks, 23, 33, 65, 69, 81, 91, 197, 198Noguchi, Isamu, Red Rhombohedron, 97, 100Nonconvex polyhedron, 82, 86, 214Nonfacet, 221Nontile
definition of, 218toroidal, 222
Normal tiling, 217, 284
OObelisk, 102Oblique prism, 83, 84Octadecahedron
borane as, 154carborane as, 154
Octahedral rotation group, 48Octahedron
borane as, 154carborane as, 154[Co6(CO)14]4�as, 168, 169combinatorial, 218construction of, 21–22crosspolytope as analog, 220cube as reciprocal of, 42, 43decomposition of, 258as a deltahedron, 16, 18electron pairs arranged as, 162–163as model for proteins, 6molecules shaped as, 202plane net of, 4relation to rhombic dodecahedron, 30, 31as space filler, 31within sphere, 100, 265stellation, 24truncated, 47, 48, 51, 69, 73–75, 144–147, 214as unit of crystal structure, 139, 140, 144–147wulfenite crystal, 139
Octetspaceframe, 121, 122theory of chemical bonding, 175
One-balloon constructions, 331-skeleton, 198, 207–209Onn, Shmuel, 281Order and Chaos, 99Organometallic compounds, 148, 153Orientable, definition of, 195O’Rourke, Joseph, 77–86, 322
PPacioli, Luca, 42, 57, 96, 199
portrait of, 96
Index 335
Packingof atoms, 145of circles, 141, 233, 234, 238, 295, 317of cubes, 142of polygons, 144of polyhedra, 141of semiregular polyhedra, 144of spheres, 141, 172, 178–179of tetrahedra, 149, 150of triangles, 141
Paddlanes, 160, 162, 300Pajeau, Charles H., 189Panel-and-hinge, 202–204, 208Paneled, 201–209Paper. See CardboardPappus, 55–57, 59, 199, 292Papyrus
Moscow, 53, 199Rhind, 53, 199
Parallelism, 114, 116, 117Parenchymal tissues, 179Partition of space, 141, 180Pauling, Linus, 175, 179, 321, 324Paving. See Tessellation; TilingPeas, 178(6, 6)-pebble game, 206Pedagogy, 175Pedersen, Jean, 13–40, 58, 293Pemberton, Earl H., 322Pentagon
building, 89, 90as a face, 68, 126, 182
Pentagonal bipyramidborane as, 154carborane as, 154as a deltahedron, 16model of, 154
Pentagonal dodecahedron, 68, 96Pentagonal hexacositahedron
as dual of snub dodecahedron, 71Schlegel diagram of, 71
Pentagonal Hexecontahedron, 128, 129Pentagonal icositetrahedron
as dual of snub cube, 71Schlegel diagram of, 71
Pentagonal pyramid, construction of, 18Pentagonal tessellation
of plane, 68, 70, 72, 73of sphere, 70, 72
Pentagram, definition of, 8, 9Pentaprismane, 157, 158, 165Perception
spatial, 109–123structural, 109
Permutational isomerism, 166Permutations, equivalent to pseudorotation, 166Perry, Charles O., Eclipse, 97, 100Petal unfolding, 83Petrie, J.F., 42Petrie polygon, 42
Pettit, Robert, 189PF5;pseudorotation in, 167Phelan, R., 189Phipps, Jane B., 19Piaget, Jean, 123Picasso, Pablo, Girl with a Mandolin, 96Pinel, Adrian, 19Planar graph, definition of, 198Plane net (unfolding), 4, 78–85, 107, 279–280Plane-sweep heuristic, 185Planets, 5, 44, 45Plateau, Joseph A.F., 178Plato, 5, 55, 56, 96, 140, 150, 171, 172, 174, 178, 188Platonic solids, 5, 34, 39, 41–45, 47, 51, 54, 55, 59, 68,
77, 155, 197, 223–229, 283analogues of, 223–229
Platonohedronconstruction of, 225definition of, 223
Plato, Vernon, 5, 55, 56, 96, 140, 174, 188Playground polyhedra, 155Plexiglas cube, 100Poinsot, Louis, 43, 60, 199, 227, 228Point group, 145, 148, 155, 158Points-on-a-sphere configurations, 163Polya, George, 58Polya, Poincare, 58Polydron, 135, 201, 202, 207, 208Poly-Form, 119, 120Polygon of forces, 231, 239Polygons
antiprismatic, 195definition of, 194Hopf, 264monotone, 187packing of, 183Petrie, 42prismatic, 195regular, 4, 5, 8, 13, 15, 19, 37, 39, 41, 54, 59, 60, 63,
114, 195, 250split, 247star, 9, 60, 198translation of, 235zigzag, 195, 196
Polyhedraanimals as, 92Archimedean, 8, 10, 58–60, 97, 155classification of, 100construction of, 30coordinated, 175, 176coordinates for, 263and crystal structures, 91design of, 29, 184equivelar, 196four-dimensional, 3, 97half-open, 93history of, 53–63isomorphic, 212, 214, 227isothetic, 185, 186juxtaposition of, 30–31
336 Index
kinship structures involving, 97as models for atoms, 165, 301movement of, 185neighborly, 49, 130, 134, 163networks as, 33, 65packing of, 141, 183Petrie-Coxeter, 196, 228plants as, 92recipes for making, 13–40relationships among, 139, 182, 189with seven vertices, 164study of, 196theory of, 50, 53, 54, 58, 60–63, 103, 194–197transformations of, 187and viral structures, 176
Polyhedral art, 93–97Polyhedral bowl, 239–243, 245, 246, 249, 250Polyhedral Fancy, 97, 100Polyhedral housing project, 91Polyhedral 2-manifold, 222, 316Polyhedral molecular geometries, 153–170Polyhedral monster, 4Polyhedral networks, 197Polyhedral society, 97–103Polyhedral torus, 260, 263Polyhedron
affine, 120Archimedean, 59capped cylinder as, 194cardboard, 19, 174chemical journal, 175colored, 20, 22, 32, 197combinatorial dual of, 211, 212combinatorial structure of, 196convex, 10, 51, 59, 62, 77–79, 81, 83–86, 100, 154,
195, 207, 212, 231–233, 241, 243, 244, 246,247
Csaszar, 213definition of, 85, 194, 195deltahedra-regular, 105digonal, 68dual of, 211elements of, 5, 6flag of, 227holes in, 19, 77, 201hollow, 174infinite, 194isogonal, 213, 214isohedral, 213, 215isotoxal, 213Klein, 226, 227, 229metric, 55, 120model of, 110, 174molecule described as, 154nonconvex, 82, 86, 214perforated, 7picture of, 42, 60, 231, 232, 243pop-up, 20–22regular, 5, 121, 125, 155, 164, 225–227, 257, 258, 262regular skew, 226
rigidity of, 198, 202–204, 208, 209self-intersection of, 203, 223semiregular, 8, 121, 258, 265simplicial, 221, 283, 285, 288, 318with six faces, 10, 104, 282skeletal, 137, 183with skew faces, 62, 194, 196, 225as a solid, 4–10, 34, 41–47, 50, 51, 54, 55space-filling, 120, 178, 180, 183, 187, 222spherical, 114, 248star, 8, 9, 43, 58, 60, 93, 103, 194, 196, 199as a surface, 65–75Szilassi, 213, 306tessellation of, 41, 65tetrahedral twins as, 194toroidal, 194, 196, 225toroidal isogonal, 214triangulated, 129, 132, 202uniform, 9, 60, 62, 212, 253–255wooden, 73, 103
Polyhedron Kingdom, 3–11, 89–107, 171, 193–199Poly-Kit, 119–121Polypolyhedra, 39Polytopes
combinatorially equivalent, 217, 218, 220coordinates for, 263definition of, 41equifaceted, 221isomorphic, 217regular, 62, 257–266, 283simplicial, 221, 283, 285, 288
Pop-up polyhedra, 20–22Post, K.A., 186Pretzel, 66Primal substances, 140. See also Platonic solidsPrimitive lattice, 144, 145, 150Princen, H.M., 189Prism
definition of, 89dibenzene chromium as, 161ferrocene as, 161hexagonal, 140, 147octagonal, 214pentagonal, 89, 90semiregular, 8square, 161, 163with terminating facets, 91triangular, 89, 113, 114tricapped trigonal, 154, 155
Prismanes, 160, 299Prismatic building, 89Prismatic polygon, definition of, 195Prismatoid, 84Prismoid, 83–84, 296Projective equivalence classes, 212Projective properties, 116Project Synergy, 199Propellane, 160, 162Proper stress, 269Proteins, 6, 105, 176, 183, 272
Index 337
Prototiles, 217, 221, 222Pseudo-rhombicuboctahedron, 57, 59Pseudorotation, 165, 167Pufferfish, 93, 94Pumpkin, as digonal polyhedron, 68Puzzles, wooden, 103, 105Pyramid, 5, 15, 18, 24, 30, 31, 42, 50, 53–55, 83, 84, 89,
112, 164, 166, 214, 222, 258, 282ammonia molecule as, 162as construction module, 30Egyptian, 89pentagonal, 14, 18, 42, 50, 112square, 18, 30, 31, 42, 53, 163, 258truncated, 53, 54, 258
Pyrite, 6, 7, 54, 103Pythagoras, 42, 54, 171Pythagoreans, 42, 54, 133
QQuadrangle, polytope analogous to, 219Quartz, 91, 93Quasi-equivalence, 177
RRademacher, H., 61Radiolaria, 93, 95Radio telescope, 90, 92, 93, 95Radon’s Theorem, 274Ramot, Israel, housing complex, 91, 92Rankine, William, 173Rausenberger, O., 62Rearrangement
icosahedron/cuboctahedron/icosahedron, 167in polyhedral boranes, 167
Reciprocal diagram, 232, 243, 248, 251Reciprocal figure, 231–243, 245, 247, 249, 250Reciprocation, 42, 58, 211–214Rectangles, isothetic, 185, 186Recycling bins, 7Red Rhombohedron, 97, 100Reed, Dorothy Mott, 98Regularity, 5, 53–55, 57, 62, 184, 196, 211, 220Regular polygon, definition of, 8, 41, 59, 195Regular polyhedron, definition of, 125Regular polylinks, 37, 39Regular polytope, 62, 257–265, 283Regular solid, definition of, 41, 196Regular star polyhedra, 9, 58, 60Reimer, Georg, 322Reinhardt, Karl, 282Reinhold, Van Nostrand, 322Reptiles, 5Rhind papyrus, 53Rhombic dodecahedron
construction of, 30relation to cube, 30, 31relation to octahedron, 30, 31as space filler, 31
Rhombicosidodecahedronfused triple, 91vertices of, 254
Rhombic tiling, 251Rhombic triacontahedron, 56, 58, 127Rhombicuboctahedron, lesser, 73–75Rhombidodeca, 101Rhombohedral lattices, Dirichlet domains of, 146Rhombohedron, Red, 97, 100Right prism, 83, 84Rigidity, 24, 120, 121, 128, 198, 201–209, 267–277Rigidity of paneled polyhedra, 207Rigid structure, 201, 203, 207Ring
of polyhedra, 207of tetrahedra, 27, 28
Ringel, Gerhard, 229, 280Robotics, 135, 171, 184–188, 198Roessler, Carl, 322Rogers, Cedric, 321Rollett, A.P., 321Rome de Lisle, J.B.L., 99Rotational symmetry, 7, 72, 125, 139, 213, 254, 275Rotation group, 45, 47, 48, 224Rout of San Romano, The, 97Rubik, Erno, 53, 115Rumi, 106
SSabin, Florence, 98Sachs, Eva, 55Sacrament of the Last Supper, The, 96, 98Saffaro, Lucio, 323Saidel, Beth, 66, 322Salt, 141San Marco, Basilica of, 9Santissimi Apostoli, Church of, 6Santos, Francisco, 281, 284Sanyal, Raman, 289Saturn, 44, 45Saunders, William, 324Schattschneider, Doris, 13–40, 70Schewe, Lars, 280Schlafli, Ludwig, 41, 43, 196Schlafli symbols, definition of, 41Schlegel diagram
of bubbles in foam, 181of cube, 258definition of, 69dual, 69of pentagonal hexacontahedron, 71of pentagonal icositetrahedron, 71of polyhedron, 257, 262of snub cube, 70, 71of snub dodecahedron, 71of square antiprism, 69of stellated icosahedron, 72
Schlegel, V., 218Schlickenrieder, Wolfram, 280
338 Index
Schmitt, Moritz, 279–289Schulte, Egon, 217–222, 225Schulz, Andre, 282Sea urchin, 93Seidel, Raimund, 233Self-assembly model, 183, 184, 189Self-intersection, 203, 215, 223, 229, 233, 267, 317Semiregular antiprism, 8Semiregular polyhedra packing of, 144Semiregular polyhedron, definition of, 8Semiregular prism, 8Senechal, Marjorie, 3–11, 89–107, 193–199Shaw, George Bernard, 123Shearer, Alice, 18Shephard, Geoffrey C., 79, 109, 211–216, 224, 225, 279,
296, 306Sherer, Stan, 322–324Silicates, 141Simplex method, 62Skeleton
of edges and vertices, 7, 198models, 207of polyhedron, 81of radiolarian, 93
Skew face, 62, 194, 196Skew polyhedron, projection of, 226Small stellated dodecahedron, 42–44Smith college campus school, student artwork, 3Smith, Warren D., 287Smoke rings, 173Snelson, Kenneth, 267, 268Snub cube, Schlegel diagram, 70, 71Snub tessellation, 71Snyder, John, 135Soap bubble, 101, 104, 179, 180Soap film, 6, 101, 178, 180, 181Sodalite, model of, 175, 176Sodium chloride, 141Solid
Archimedean, 8, 37, 42, 44, 47, 55, 57, 59, 60, 62, 77,80, 227
Miller’s, 60Platonic, 5, 34, 39, 41–45, 47, 51, 54, 55, 59, 68, 77,
155, 197, 223–229, 283polyhedron as, 57, 65–75, 174, 212, 257, 260, 263,
266regular, 4–9, 41, 43, 51, 54–63, 140, 172, 174, 196
Solit, Matthew, 104Solymosi, Jozsef, 281Sommerville, D.M.Y., 60–61Space
Euclidean, 211, 219, 222partition of, 141, 180
Space-filling polyhedra, 120, 178, 183Space-filling toroid, 222Spaceframe
prefabricated concrete, 121wood and plastic, 121
Spanning tree, 81–83, 204, 279, 280, 317Sparsity counts for cut polyhedra, 209
Spatial perception, 109–123Sphere
as model for atom, 8packing of, 141, 172, 178, 179points on, 51, 162, 263rotations of, 45
Spherical blackboard, 69Spherical complex, 219, 220, 222Spherical polyhedron, 114, 248Spider web, 196, 231–251Spiked tetrahedron, 81Spiny pufferfish, 94Spurlino, John C., 324Square antiprism, Schlegel diagram, 69Square as a face, 20, 30, 50, 69, 75, 141, 142, 163, 183,
190, 258, 259, 263–266Square pyramid
construction of, 30as space filler, 31
Stanley, 288Star. See Pentagram; Stellated dodecahedron; Stellated
icosahedron; Stellated polyhedraStar polygon, 8, 60, 198Star polyhedron, regular, 9–11, 58, 60Star unfolding, 85, 86, 280Statics, 9, 179, 231–233, 248, 312Steatite icosahedron, 56Steiner, J., 60Steinitz, Ernst, 61–63, 198, 199, 220, 246, 281, 283, 288,
294–295, 319Steinitz’s theorem, 198, 202, 234, 250Stella octangula, 58Stellated dodecahedron, construction of, 58Stellated icosahedron, Schlegel diagram, 70Stellated polyhedra, 58, 60Stellation
of cube, 30, 42, 43of octahedron, 30, 42, 43
Stereoscopic vision, 109Stevens, Peter S., 321, 323Stevin, Simon, 57Stoer, Lorenz, 57Strache, Wolf, 322Streinu, Ileana, 201–209Stress-energy, 269, 273Stress matrix, 267, 273–278, 314, 315Structural Constellation, 99Structural perception, 109Structuration as geometrical action, 110Structure. See also Shape
equilibrium vs. average, 165of tetrafluoro-1,3-dithietane, 160
Stuart, C. Todd, 322Studying polyhedra, 53, 57, 171, 196Sturmfels, Bernd, 281Subgraph isomorphism, 36Subway station, 122Su, Francis, 286Sulfates, 162Sulfones, 161, 163
Index 339
Sulfuric acid, 161, 163Sydler, J., 61Symmetry
color, 144, 197mirror, 7, 118, 128molecular, 165operation, 46, 54, 119, 123, 139, 153of polyhedron, 46, 54, 145, 181, 211, 213, 214,
255–257, 267rotational, 7, 72, 125, 139, 213, 254, 275spherical, 162, 179of uniform polyhedra, 9, 60, 253–255
Synagogue by Zvi Hecker, 90, 92Szilassi, Lajos, 213Szilassi polyhedron, 213, 306
TTait, Peter, 173Tammes, problem of, 175Tanigawa, S., 204, 208, 209Taping joints, 134Tarsia, 9Tay graph, 204–206Tay, T.-S., 204–206Ten Problems in Geometry, 279–289Tensegrities and Global Rigidity, 267–278Tensegrity
centrally symmetric polyhedra, 275compound tensegrities, 276, 277, 314constraints, 269, 270flexible, 270graph, 267, 269, 276highly symmetric tensegrities, 276prismatic tensegrities, 275–276rigid, 268, 270, 277
Tensegrity G.p/; 268–271, 274Ten Tangled Triangles, 97, 102Termes, Dick A., 323Terrell, Robert, 276Tessellation. See also Tiling
colored, 197Dirichlet, 196, 231–251of E3 by cubes, 221, 222of egg, 67of plane, 65, 68, 70, 196of polyhedra, 72, 196, 231–251snub, 71of sphere, 72
Tetrafluoroaluminate ion, 160, 162Tetrafluoro-1,3-dithietane, 160Tetragonal crystal system, 148, 149Tetragonal lattices, dirichlet domains of, 146Tetrahedra
linking of, 121packing of, 150ring of rotating, 28
Tetrahedral rotation group, 46Tetrahedral twins, 194Tetrahedrane, 155, 158
TetrahedronB4Cl4 as, 155brass, 101construction of, 36within cube, 101as a deltahedron, 16, 18electron pairs arranged as, 162–165as face of polyhedron, 33, 37, 41, 42, 45KAIF4 as, 160as model of organic molecule, 145molecule as, 145ONF3 molecule as, 160plane net of, 45as self-dual, 212soap bubble as, 101sulfate ion as, 162tetrafluoroaluminate ion as, 160, 162truncated, 47, 48, 113as unit of crystal structure, 145, 146, 148, 149
Tetrahedronbrass sculpture, 101chemical journal, 172
Tetrakaidecahedra, 178, 179, 189Tetraphenyl compounds, 148Tetra-tert-butyltetrahedrane, 155, 158Thayer, Bob, 323Theatetus, 55Thomson, Sir William. See Kelvin, Baron of LargsThe 3d conjecture, 288–289Three-frequency icosahedron, 129–1303-space, 63, 197, 231–233, 236, 239, 243, 248, 250,
257, 260–262, 264–266, 282, 284,311, 316
Three Spheres II, 66Tiling. See also Tessellation
face-to-face, 217–222, 284locally finite, 217, 218, 220–222monotypic, 218, 220–222normal, 217, 218, 222, 285of plane, 68, 218, 223, 283rhombic, 254
Timaeus, 5, 55, 199Tobacco necrosis virus, 105Tomlinson, Brett, 70, 322Toothpicks as construction materials, 13Topological properties, 257Topology, 59, 109, 116, 123Toroidal polyhedron, 213Toroid, space-filling, 222Torus
coordinates, 7, 261–263decomposition, 257–266flat, 224, 264polyhedral, 62, 260, 263triangulation of, 14, 20, 26
Total photo, 104, 107Toussaint, Godfried, 104, 182, 185, 187Transfiguration as geometrical action, 110, 114Transformations, models of plant growth, 187Tree, dual of, 248
340 Index
Trianglesas faces, 82, 134packing of, 141polytopes analogous to, 218
Triangular antiprism, 159, 160Triangular arrangement of electron pairs, 163Triangular bipyramid as a deltahedron, 16Triangular dipyramid, 34, 36Triangular prism, 89, 113, 115Triangular pyramid, construction of, 30Triangulated polyhedron, 134, 155Triangulation of torus, 213, 316Tricapped trigonal prism
borane as, 154carborane as, 154
Trigonal bipyramidcarborane as, 154dicarborane as, 154electron pairs arranged as, 163model of, 162, 163molecules shaped as, 164pseudorotation in, 165
Trihedral regions, 232Triprismane, 157, 158, 299Trivial flex, 268Truncated 600-Cell, 97, 102Truncated cube, vertices of, 8, 72, 253Truncated cuboctahedron, 47–49Truncated dodecahedron, vertices of, 254Truncated icosahedron, 47, 77–80, 126, 127, 129, 133,
135, 254Truncated icosahedron, vertices of, 254Truncated icosidodecahedron, vertices of, 254Truncated octahedron, vertices of, 54, 72, 253The Truncated Pentagonal Hexecontahedron, 128–129Truncated pyramid, volume of, 53–54Truncated rhombic triacontahedron, 127–128Truncated tetrahedron, 47, 49, 113, 114Truncation, definition of, 97Tunnels in polyhedra, 22724-cell, 258–261, 263, 265–266, 283, 311Twins, tetrahedral, 1942-cell, 194, 195, 2472-cell complex, 1942-manifold
definition of, 215polyhedral, 222, 307
2-sphere, definition of, 194
UUccello, Paolo, 9, 42, 57, 96, 97
After the Flood, 97The Rout of San Romano, 97
Ukrainian Easter egg painting, 67Uniform polyhedron, 10Unit cell, 141–144, 146, 149, 189Universally globally rigid, 270, 274, 275, 277
VValence, 162–165, 175, 219, 220, 222,
227, 303Valence shell electron pair repulsion (VSEPR), 162–173,
175Valence shell electron repulsion, 162, 175Vanadinite, 140van der Waerden, B.L., 42van’t Hoff, Jacobus Henricus, 174Venice, 9Venus, 44Verheyen, Hugo F., 97, 323Verheyen, Hugo F., IRODO, 101Verlag, Hirmer, 322Vertex(ices)
definition of, 41, 57, 80, 193, 217, 249number of, 51, 59, 67, 73, 104, 118, 134, 205,
207, 208, 219, 224, 226, 228, 229, 277,286, 295
permutation of, 51, 72, 73, 152, 261, 290symmetrically equivalent, 10
Vibrations, 154, 165, 166Villains, deltahedral, 18Virus
icosahedral, 176, 177, 184self-assembly of, 183–184southern bean mosaic, 105spherical, 176tobacco necrosis, 105tumor, 177
Void, von Helmholtz, Hermann, 173von Helmholtz, Hermann, 173Von Staudt, 60Voronoi diagram, 232, 237, 250, 251Voronoi domains, 282Voronoi, George, 142Vortex atom, 173, 174VSEPR. See Valence shell electron pair repulsion
(VSEPR)
WWalker, Wallace, 27, 28Wallace, William, 61Walnut clusters, shapes of, 172Walter, Marion, 13–40Water, 5, 42, 140, 157, 163Waterfall, 99Waterhouse, William, 55Watson, James Dewey, 176Weaire, D., 189Weakly neighborly complex, 163Weimer, Diana, 28Wenninger, Magnus, 13–40Werner, Alfred, 175Weyl, Hermann, 45, 202What-If-Not strategy, 17Whirlpools in the aether, 173
Index 341
Whiteley, Walter, 204, 208, 231–251White, N., 246Wiley, John, 323Williams, R.E., 183Wills, Jorg M.,223–229, 281Winters, Carl, 322Wooden
polyhedron, 16, 73puzzles, 103, 105
World FairMontreal 1967, 91New York 1964, 89
WPI, 103Wright, E.M., 254, 311Wrinch, Dorothy, 5, 6Wulfenite, 139Wyckoff, R.W.G., 323
YYao, Y.F., 186Yog-Sothoth, 10, 11
ZZalgaller, V.A., 62Zeolites, 141, 142, 144Ziegler, Gunter M., 229Zigzag polygon, 195, 196Zinc bromide, 148Zirconium borohydride, 154Zodiac, 42Zometool, 135, 201Zonohedra, juxtaposed, 120Zonohedral cap, 246Zonohedron, definition of, 120
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