05 spring Lecture 03 - epab.bme.hu

Geometrical Constructios 2 Lecture Notes 2005 Spring Semester #03

Budapest University of Technology and Economics ? Faculty of Architecture ? Department of Architectural Representation

1

Platonic Solids

A regular polyhedron is one whose faces are identical regular polygons.

The solids as drawn in Kepler’s Mysterium Cosmographicum:

tetrahedron octahedron icosahedron cube dodecahedron

(Fire) (Air) (Water) (Earth) (Universe)



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Faces around a vertex

P

P

P

PP

Only five regular solids are possible. Schläfli symbol {p, q} means: the faces are regular p-gons, q surrounding each vertex.

{4, 3}

{5, 3}

{3, 3}{3, 4}

{3, 5}



3

Archimedean Polyhedra

The 13 Archimedean solids are the

convex polyhedra that have a similar

arrangement of nonintersecting regular

convex polygons of two or more

different types arranged in the same

way about each vertex with all sides

the same length (Cromwell 1997,

pp. 91-92).

http://mathworld.wolfram.com/Archim

edeanSolid.html



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Archimedean Polyhedra



5

Fullerains (named after Buckminster Fuller)

A highlight of one of the pentagonal rings

A highlight of one of the hexagonal rings

The Royal Swedish Academy of Sciences has awarded the 1996 Nobel Prize for Chemistry jointly to: •Professor Robert F. Curl, Jr., Rice University, Houston, USA •Professor Sir Harry W. Kroto FRS, University of Sussex, Brighton, UK •Professor Richard E. Smalley, Rice University, Houston, USA For their Discovery of Fullerenes.

Buckminster Fuller's Dome -Expo '67 Montreal

In 1985 one of the greatest new discoveries in science was made when chemists Richard Smalley and Harold Kroto discovered the existence of a third form of carbon. Unlike the two other forms of carbon, diamond and graphite, this amazing 60-atom cage molecule was shaped like a soccer ball.Both Kroto and Smalley felt it most appropriate to name it, "buckminsterfullerene" for its striking resemblance to a geodesic dome. A new family of these molecules have since been found called "fullerenes." (Note: Diamond is a molecular network crystal with each carbon bonded to four others in a tetrahedral configuration. Graphite is formed in flat sheets with each carbon bonded to three others in a hexagonal configuration.)



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Regular Star Polyhedra

Two star polyhedra were discovered by Poinsot in 1809. The others were discovered about 200 years before that by Johannes Kepler (1571-1630), the German astronomer and natural philosopher noted for formulating the three laws of planetary motion, now known as Kepler's laws, including the law that celestial bodies have elliptical, not circular orbits.

Stellation is the process of constructing polyhedron by extending the facial planes past the polyhedron edges of a given polyhedron until they intersect (Wenninger 1989). The set of all possible polyhedron edges of the stellations can be obtained by finding all intersections on the facial planes. The Kepler-Poinsot solids consist of the three dodecahedron stellations and one of the icosahedron stellations, and these are the only stellations of Platonic solids which are uniform polyhedra.



7

Art and Science

JACOPO DE 'BARBERI: Luca Pacioli, c. 1499

This painting shows Fra Luca Pacioli and hisstudent, Guidobaldo, Duke of Urbino. In the upperleft is a rhombi-cuboctahedron, and on the table is a dodecahedron on top of a copy of Euclid'sElements.

Leonardo's Illustrations for Luca's book.Da Divina Proportione

Luca Pacioli wrote a book called Da DivinaProportione (1509) which contained a section onthe Platonic Solids and other solids, which has 60 plates of solids by none other than his studentLeonardo da Vinci.



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M. C. ESCHER (1902-1972)

Stars, 1948 Note the similarity betweenthis polyhedron and Leonardo'sillustrations for Pacioli's book

Escher made a set of nested Platonic Solids. When he moved to a new studio he have awaymost of his belongings but took his belovedmodel.



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Models



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Links

http://mathworld.wolfram.com/ArchimedeanSolid.htmlhttp://www.math.bme.hu/~prok/RegPoly/index.htmlhttp://www.korthalsaltes.comhttp://www.math.dartmouth.edu/~matc/math5.geometry/

Documents

05 spring Lecture 03 - epab.bme.hu