Ian Stewart*

The catapult that Archimedes built, the gambling-houses that Des- cartes frequented in his dissolute youth, the field where Galois fought his duel, the bridge where Hamilton carved quaternions-- not all of these monuments to mathematical history survive today, but the mathematician on vacation can still find many reminders of our subject's glorious and inglorious past: statues, plaques, graves, the cafd where the famous conjecture was made, the desk where the

famous initials are scratched, birthplaces, houses, memorials. Does your hometown have a mathematical tourist attraction? Have you encountered a mathematical sight on your travels? If so, we invite you to submit to this column a picture, a description of its mathe- matical significance, and either a map or directions so that others may follow in your tracks. Please send all submissions to the Math- ematical Tourist Editor, Ian Stewart.

Pentagonal Decoration in Granada Istv in Hargittai

The uniquely rich geometrical decorations of the Al- hambra in Granada, Spain are widely known [1]. They have inspired artists [2] and have been thoroughly studied for their symmetries [3].

One of the smaller palaces adjacent to the Alhambra is the Generalife. Visitors to Granada usually start with the Alhambra and may or may not get to the General-

* Column Editor's address: Mathematics Institute, University of Warwick, Coventry CV4 7AL England.

ife. During a recent (October 1991) one-day visit, I fol- lowed the usual routine and completed my tour in the farthest tower of the Generalife, where I noticed a wall decoration reproduced in Figure 1. I am aware of no mention, of this particular decoration in the literature. Its special interest lies in the pentagonal/decagonal character of the pattern. Fivefold symmetry has gained prominence recently (see, e.g., [4,5[) due to discover- ies in solid state science (quasicrystals) and in chemis- try (fullerenes).

Figure 1. Photograph of wall decoration in the Generalife (Granada, Spain) taken by the author in October 1991.

Figure 2. Line drawing from David Wade's Pattern in Islamic Art [6], p. 88.

A line drawing analogous to the pattern of Figure 1 is reproduced here (Fig. 2) from David Wade's excel- lent collection Pattern in Islamic Art [6]. Wade gives no reference to the origin or location of this pattern. This is an interlacing version of another noninterlacing pen- tagon/decagon-based pattern analyzed by Wade [6] and shown here in Figure 3.

The Generalife pattern has only local fivefold sym- metries of the pentagons and decagons; the entire dec- oration does not have fivefold symmetry. Due to the interlacing, both the patterns shown in Figures I and 2 are chiral and happen to be heterochiral to each other.

G x

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Acknowledgment: I am grateful to Dr. Emil Makovicky (Copenhagen) for introducing me to David Wade's book.

R e f e r e n c e s

1. M. Vela Torres, ed., La Alhambra, Granada: Grafsur (1987).

2. D. Schattschneider, Visions of Symmetry. Notebooks, Peri- odic Drawings, and Related Work of M. C. Escher, New York: Freeman (1990).

3. B. Grtinbaum, Z. Griinbaum, and G. C. Shepard, Sym- metry in Moorish and other ornaments, Symmetry Uni- fying Human Understanding, (I. Hargittai, ed.), Oxford: Pergamon Press (1986), 641-645.

4. I. Hargittai, ed. Quasicrystals, Networks, and Molecules of Fivefold Symmetry, New York: VCH (1990).

5. I. Hargittai, ed., Fivefold Symmetry, Singapore: World Sci- entific (1992).

6. D. Wade, Pattern in Islamic Art, Woodstock, NY: The Overlook Press (1976).

Technical University Budapest Szt. Gell&t t~r 4 H-1521 Budapest, XI Hungary

l~igure 3. Construction of the noninterlacing version of the pattern of Figure 2 according to David Wade ([6], p. 56). The decagons are surrounded by pentagons. The centers of the sides of the pentagons provide the intersection points for the lines of the secondary pattern.

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2 �9 1993 Springer-Verlag New York 47

The Stadtgottesacker in Halle Manfred Stern

Near downtown Halle is an old graveyard, the so- called Stadtgottesacker (God's acre of the town). Half- way between the main railway station (Hauptbahnhof) and the Marktplatz (the very center of town) is the Leipziger Turm, a tower from the fifteenth century and part of the old fortifications. From there, walk some 100 meters on the Hansering, then turn right. After climbing a few steps you will find the entrance to the graveyard through a gate in the Stadtgottesacker- strasse.

The graveyard has an ample layout built in Renais- sance style (1557-1594) and comparable to the Italian

Camposanto. 1 It is surrounded by a wall of elaborate masonry, and on the interior side the wall opens into niches under strainer arches (Schwibbogen). These niches were the burial places of prominent people: The theologian and pedagogue August Hermann Francke (1636-1727) and the philosopher Christian Thomasius (1655-1728) are buried here. The Stadtgottesacker is also a memorial place for the history of mathematics: Johann Andreas Segner (1704-1777), Eduard Heine (1821-1881), and the lesser known Friedrich Meyer (1842-1898) found here their everlasting rest.

The Hungarian Segner was born in or near Pozsony- Pressburg (today Bratislava, the capital of Slovakia) on October 9, 1704 and died in Halle on October 5, 1777 (the year Carl Friedrich Gauss was born). From 1735 to 1755 he taught mainly physics, mathematics, and

1 For a history of the Stadtgottesacker Halle, see [10].

The Stadtgottesacker (above). Tombstone and portrait of Edward Heine (right).Portrait and tombstone of Johann An- dreas Segner (below).

FriedrichMeyer and details from his tombstone.

chemistry at the University of G6tfingen, where he also founded the observatory. 2 From 1755 until his death, he was a professor of physics, mathematics, and astronomy at the University of Halle. He invented the "Segner-wheel," a precursor of the turbine. Leon- hard Euler, who visited Segner in 1761 in Halle, used Segner's results in his own mechanical investigations. 3 In honor of Segner's astronomical activities a lunar cra- ter was named after him. Segner was buried in a niche under one of the strainer arches (Schwibbogen No. 83). The new headstone of his tomb was sponsored by the Hungarian Segner Society and emplaced in 1977 on the occasion of the 200th anniversary of his death. The text on the headstone is in both Hungarian and German.

Eduard Heine (of Heine-Borel fame) was born on March 16, 1821 in Berlin and died in Halle on October 24, 1881. Heine dedicated his dissertation [3] (Berlin, 1842) to Dirichlet; Kronecker was one of the oppo- nents. The dissertation concludes with five "Theses" clearly proving Heine a staunch proponent of the the- ory of limits. From 1856 until his death, Heine was a full professor (Ordinarius) of the University of Halle. His book [4] on spherical functions has become a clas- sic. Heine also had an influence on Georg Cantor, who came to Halle in 1867. 4 Both Heine and Cantor were actively engaged in the further development of the program initiated by Weierstrass. 5

Friedrich Meyer was born on March 5, 1842 in Mlinsk (Western Prussia) and died in Halle on Decem- ber 5, 1898. He was a teacher of the Stadtgymnasium Halle and remained on close terms with the university as well as with Eduard Heine and Georg Cantor. He already used their ideas from 1880 on, e.g., in his book [8]. 6 It is interesting to note that in 1894 the Honorary

2 See [6]. 3 More information on Segner's life and work is given in [7,12]. 4 For memorial places of Georg Cantor in Halle, see [11]. 5 This is witnessed, for instance, by Heine's paper [5]. Detailed in- formation concerning the mathematical relationship between Heine and Cantor can be found in [1]. 6 Meyer's merits are also mentioned in Fraenkel's biography of Georg Cantor ([2]). More details can be found in [9].

Doctor's Degree of the University of Halle was con- ferred upon Beltrami and Meyer.

Acknowledgments: The photograph of F. Meyer is taken from [9]. All other photographs are from the Archives of the Martin-Luther-Universit/it Halle which is gratefully acknowledged. In particular, I am in- debted to Mrs. M. Heinrichsdorff for taking the pho- tographs at the Stadtgottesacker. I am also grateful to Dr. G. Betsch and Dr. W. Lagler (both Tfibingen), and last but not least to Professor B. Sz~missy (Debrecen) for several valuable remarks.

References

1. J. W. Dauben, Georg Cantor. His Mathematics and Philoso- phy of the Infinite, Cambridge: Cambridge University Press (1979).

2. A. Fraenkel, Das Leben Georg Cantors, Georg Cantor, Gesammelte Abhandlungen mathematischen und philosophi- schen Inhalts, Berlin: Springer-Verlag (1990) (reprint of the 1932 edition).

3. E. Heine, De aequationibus nonnuUis differentialibus, Berlin, 1842.

4. E. Heine, Handbuch der Kugelfunktionen, Berlin: G. Reimer (1861, 2nd edition, 1881).

5. E. Heine, Die Elemente der Funktionenlehre, J. Reine Angew. Math. 74 (1872), 172-188.

6. F. Hund, Die Geschichte der G6ttinger Physik, G6ttingen: Vandenhoeck & Ruprecht (1987), 22-23.

7. W. Kaiser, Johann Andreas Segner, Leipzig: Teubner Ver- lagsgesellschaft (1977).

8. F. Meyer, Elemente der Arithmetik und Algebra, 2nd ed., Halle: Verlag H. W. Schmidt (1885).

9. G. Riehm, Friedrich Meyer (obituary), Jahresber.DMV (1899), 59--61.

10. G. Sch6nermark, Beschreibende Darstellung der dlteren Bau- und Kunstdenkmdler der Stadt Halle und des Saalkreises, Halle: Verlag O.Hendel (1886).

11. M. Stern, Memorial places of Georg Cantor in Halle, Mathematical Intelligencer 14, No. 4 (1988), 48-49.

12. B. Sz6missy, History of Mathematics in Hungary until the 20th Century, Berlin: Springer-Verlag (1992).

Kiefernweg 8 D-0-4050 Halle Germany

THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2, 1993 49

Kepler, Einstein, and Ulm Dirk J. F. Nonnenmacher,

Theo F. Nonnenmacher, and P. F. Zweifel

The beautiful city of Ulm, Germany lies on the Danube River some 130 km west of MOnchen, just on the bor- der of W(irttemberg and Bavaria. It is worth visiting in its own right, not only for its wonderful 600-year-old

Gothic cathedral (the "MOnster") but also for its beau- tiful old city next to the river, with timbered medieval houses still in use, a spectacular city wall, and spark- ling torrents rushing down to the Danube. The M~in- ster can be seen for miles around, because its 165- meter-high steeple--the tallest church steeple in the world--dominates the countryside. It is possible to climb to the top of the tower, for an unforgettable pan- orama. Be sure also to visit the inside of the cathedral, and especially to study the wonderful carvings which adorn the choir stalls.

Ulm's theatre is the oldest city theatre in Ger- many, dating from 1641. One of the most fa- mous musicians of all t i m e . . . Herbert von Karajan, conducted the Ulm orchestra from 1929 to 1934.

Figure 1. The bust of Einstein on the left was a gift from the people of India; the sculpture to the right marks the location of the house in which Einstein was born.

Figure 2. Kepler's "'Ulmer Kessel" in the Ulm Museum.

It was in this lovely city on March 14, 1879 at 11:30 a.m. that Albert Einstein was born, the first child of Hermann and Pauline (ne~ Koch) Einstein. The family was then living in a house on Bahnhofstrasse, quite near the station, having moved from another house on the M~insterplatz only a few months before. The house on Bahnhofstrasse was destroyed in an air raid in 1944, but today the site of the home is commemorated by a monument; nearby is a sculpture donated by the peo- ple of India (Fig. 1). To arrive at the monuments, leave the train station by the underpass. Shortly after arriv- ing at street level you will see a McDonald's restaurant on your left. This restaurant is just out of view to the left of Einstein's bust in Figure 1.

Einstein was not the only famous scientist to live in Ulm. Johannes Kepler (1571-1630) stayed in Ulm dur- ing 1626/27 to assist in the printing of the tabulae Ru- dolphinae which he had finished in Linz in 1624. Be- gun by Tycho Brahe, they replaced the "Alfonsinean" and "Prutenean" tables, and were to become the most important auxiliary material for astronomical calcula- tions in the forthcoming centuries.

During his stay in Ulm, Kepler had to make money for living expenses, so he accepted a job offered by the city of Ulm to create the famous "Ulmer Kessel" (Fig. 2). In this kettle he unified all the length, weight, and

5 0 THE MATHEMATICAL INTELLIGENCER VOL. 15, NO. 2 �9 1993 Springer-Verlag New York

volume measurements which were in use in Ulm at that time in order to protect them from manipulations by unethical merchants. The kettle can be seen in the Ulm Museum. This museum is not too far from the beautiful Rathaus (worth a visit in its own right), which in turn is a short walk from the M~inster. On the wall of the Rathaus one can see a memorial plaque dedicated to Kepler (Fig. 3).

As is well known, Kepler worked during the last years of his life in the city of Sagan for Albrecht Wal- lenstein, the Duke of Friendland. The famous General of the Empire had also been in Ulm, and a memorial tablet can be found at the Weinhof near the Rathaus (Fig. 4). It is generally believed that Kepler was work- ing for Wallenstein as an astrologer, but this is falla- cious. What Kepler actually did was to make his plan- etary data available to Wallenstein's astrologers---it was they, and not Kepler, who made astrological pre- dictions of the most propitious startling times for some of the battles of the 30-year War (1618-1648).

Finally, we note that Ulm's theater is the oldest city theater in Germany, dating from 1641. One of the most famous musicians of all time (and the most recorded), Herbert von Karajan, conducted the Ulm orchestra from 1929 to 1934, his first professional job. Music was significant both to Einstein, an accomplished violinist, and to Kepler, who while working for Wallenstein de- veloped an astronomical model in which planetary or- bits were related to the Pythagorean ratios of musical tones (hence "Music of the Spheres").

The data on the birth of Albert Einstein are found in the book Subtle is the Lord by Abraham Pais (Oxford University Press, 1982). The data on Kepler and Wal- lenstein are from Johannes Kepler by Walther Gerlach and Martha List (Piper, Mfinchen, 1987).

Figure 3. Memorial tablet on the wall of the Rathaus, com- memorating Kepler's stay in Ulm.

Abteilung Mathematik II Helmholtzstr. 18 University of Ulm Ulm, Germany

Abteilung Mathematische Physik Albert Einstein-Allee 11 University of Ulm Ulm, Germany

Center for Transport Theory and Mathematical Physics Virginia Polytechnic Institute and State University Blacksburg, VA 24061-0435 USA

Figure 4. Memorial tablet dedicated to Wallenstein on the side of the Weinhof.