Mathematical Tourist Research Articles

The Gallo-Roman Dodecahedron

Submissions should be uploaded to http://tmin.edmgr.com or to be sent directly to Dirk Huylebrouck, huylebrouck@gmail.com A previous contribution to this ‘‘Mathematical Tourist’’ column drew attention to the Gallo-Roman dodecahedron (Greek dxdeja0edqom, from dx0 deja, i.e., ‘‘twelve,’’ and e’dqa, i.e., ‘‘face’’) in Tongeren, Belgium (see [8]). However, the mathematical tourist can visit many more such objects in museums spread all over Europe (see Figs. 1, 2, and 3). Sometimes, the artifacts have even been a source of inspiration for monuments, such as in Homburg a. d. Saar, Germany (see Fig. 4). It all startedon 28 June1739,whenoneMr.North reported to the Society of Antiquaries in London that a remarkable object hadbeendiscovered togetherwith someantique coins in Aston, Hertfordshire, England, namely ‘‘a piece of mixed metal [...] consisting of 12 sides, with an equal number of perforations within them [...]’’ [10]. More than 270 years have passed, and since then 116 similar objects have been unearthed. One of the most recent specimens was found in an archaeological excavation at Clery-sur-Somme, France, in a refuse pit from the third century CE. During thepast fewcenturies some twohundred scientists have come under the spell of these objects, among them Count Leopold Hugo, a nephew of the writer Victor Hugo and a pioneer in dodecahedron research, as well as Julien de Saint-Venant [3], Robert Nouwen [11], and Bernhard A. Greiner [4], who all wrote extensive treatises about GalloRoman dodecahedra. Even such mathematicians as Moritz Cantor (1829–1920), Ferdinandvon Lindemann (1852–1939), and, most recently, Benno Artmann (1933–2010) found them an interesting topic for research (see [1] and [2]).

The Geometry of Christopher Wren and Robert Hooke: A Walking Tour in London

directions so that others may follow in your tracks. L ondon contains a wealth of architectural styles and buildings from many periods; for those interested in architecture it is a wonderful place to explore on foot. After the Great Fire of 1666 destroyed most of the old walled city of London, Sir Christopher Wren and Robert Hooke were appointed to the group charged with rebuilding London. Wren and Hooke, both founding members of the Royal Society, began a lifelong friendship and pattern of collaboration during their student days at Oxford. Until the Great Fire, they were best known for their work in mathematics, physics, and astronomy. Wren was at one time the Savilian Professor of Astronomy at Oxford. As longtime Curator of Experiments for the Royal Society, Hooke had ‘‘to furnish them every day, on which they met, with three or four considerable experiments.’’ However, although the massive effort to rebuild London was led by both men, Wren’s enduring reputation is as an architect and Hooke’s is as a scientist. In the mid-seventeenth century, Inigo Jones brought the notion of ‘‘classical architecture’’ with all of its columns, domes, and Palladian windows to England. Many of these design elements had been rejected in the past as too ‘‘Catholic’’ for Henry VIII’s ‘‘protestant’’ England. But after the Great Fire, Wren and Hooke popularized this very geometric design style, using it in royal, public, and private buildings, as well as in places of worship. Wren has been credited with designing St. Paul’s Cathedral and more than fifty parish churches that were rebuilt in the City of London. Current scholarship indicates that the parish churches were most likely designed by Wren, Hooke, and others who worked under them. It is a difficult matter to determine attributions for most of the churches. Some hints can be found in parish vestry minutes and others in Hooke’s diaries (he was an avid diarist). It is clear from Hooke’s diaries that for decades he and Wren met nearly daily to discuss both architectural and scientific matters, which suggests that the design process was collaborative, with details executed by their apprentices. We know, from diaries, writings, and library lists, that Wren and Hooke had access to the canon of classical architectural texts of their day. These books by Alberti, Vitruvius, and Palladio all advocate geometry, proportion, and symmetry as the basis of architectural design. When seeking connections between Wren’s mathematics and architecture, many authors point to the strong use of geometry in his buildings. Derek Whiteside says of Wren, ‘‘Perhaps his greatest mathematical gift was his visual sensitivity and feeling for form which are obvious in his architectural designs and scientific sketches.’’ Wren himself states in Tract I: Geometrical Figures are naturally more beautiful than other irregular; in this all consent as to a Law of Nature. Of Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mail: huylebrouck@gmail.com

A Mathematical Trip to Princeton

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mail: dirk.huylebrouck@gmail.com T he phone rings and a familiar voice speaks. ‘‘Since you’re going to Princeton this summer, you’ll surely be taking a tour of the Princeton campus, and their math department. When you go there, be sure to take note of a sculpture near the Math Department Commons Room. This piece involves three mathematicians: One was the artist, another commissioned the piece, and the third was the one to whom it was dedicated. When you get back, tell me what you ... what? No time? I’m sorry, you make the time.’’ Orders are orders, so you make the time and tour the beautiful Princeton University Campus. There, you find that sculpture, along with an ivory tower and many other sights worth seeing. The Princeton University mathematics department may or may not be in first place in the mythical USA Mathematics Marathon, but it is certainly up there in the lead pack. Two Fine Halls have housed the Princeton math department, both named for Henry Burchard Fine (1858–1928), the first chair of mathematics at Princeton and President of the American Mathematical Society in 1911–1912. The first building was opened in 1930; it was renamed Jones Hall in 1968 when the second and current Fine Hall was built. In the immediate vicinity of the current Fine Hall are one building and two sculptures worth a second look. The building is the Lewis Science Library [4], which opened in 2008. Designed by architect Frank O. Gehry, it is a mathematical marvel in its own right. (Go inside and look around: It’s time well spent.) Viewed from the Fine Hall tower, it resembles a cell complex (Figure 1). The library is one among many architectural gems at Princeton, as an architect friend from high-school and his historian wife demonstrated during a campus tour. The library stands next to Fine Hall on what was formerly the site of an informal volleyball court used by the math faculty and students. Apparently, when Fields Medalist William Thurston was on the faculty, he supplied the net, which he put up in the morning and took down in the evening. Two steel sculptures of mathematical character stand in the vicinity. The older one is Alexander Calder’s 26-foot sculpture ‘‘Five Disks: One Empty;’’ [3] dedicated in 1971, it stands in the Fine Hall courtyard plaza between the mathematics and physics departments. (Maybe it looks like a horse, and maybe it doesn’t–Modern Art is like that.) The newer one is Richard Serra’s ‘‘The Hedgehog and the Fox,’’ [5] a trio of 90-foot long and 15-foot high nested serpentine steel ribbons standing just east of the library. It’s fun to walk through this huge installation and listen to the echoes. Appropriately, when viewed from the Fine Hall tower, ‘‘The Hedgehog and the Fox’’ looks like a triple integral sign (Figure 2).

A Commemorative Plate for Wilhelm Killing and Karl Weierstraß

Please send all submissions to Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium e-mail: dirk.huylebrouck@gmail.com A search for ‘‘The Greatest Mathematical Paper of All Time’’ on Google, MathSciNet, or ‘‘Zentralblatt fur Mathematik’’ unvaryingly leads to the work of Wilhelm Killing (1847–1923). His classification of simple Lie algebras over the complex numbers is a favorite keyword as well. Killing discovered them while a professor at the Lyceum Hosianum in Braniewo, Poland. In the 19th century, Braunsberg or Braniewo belonged to East Prussia, in Germany, but it became a Polish city after the Second World War, and thus its name changed.

Stefan Banach remembered in Krakow

Does your hometoum have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit an essay to this column. Be sure to include a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.

An excursion around the national mall in Washington DC, USA

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit an essay to this column. Be sure to include a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.

An enigmatic pyramid

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to submit an essay to this column. Be sure to include a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.

Bubble crawling in dublin

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? 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 mathematical significance, and either a map or directions so that others may follow in your tracks.

From fractal geometry to fractured architecture: The federation square of melbourne

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? 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 mathematical significance, and either a map or directions so that others may follow in your tracks.

The home of golden numberism

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? 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 mathematical significance, and either a map or directions so that others may follow in your tracks.

Chronicle of a symmetric tourist in tihany

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? 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 mathematical significance, and either a map or directions so that others may follow in your tracks.

Leray in edelbach

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? 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 mathematical significance, and either a map or directions so that others may follow in your tracks.

Mathematics in the library of el escorial

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? 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 mathematical significance, and either a map or directions so that others may follow in your tracks.

Mathematical tour through the sydney opera house

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the caf e where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? 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 mathematical significance, and either a map or directions so that others may follow in your tracks.

Mathematical tourism in Siberia

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made the desk where the famous initials are scratched, birthplaces, houses, or memorials? 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 mathematical significance, and either a map or directions so that others may follow in your tracks.

A mathematician in lhasa

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? 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 mathematical significance, and either a map or directions so that others may follow in your tracks.

Alphabetic magic square in a medieval Church

Does your home town have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? 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 mathematical significance, and either a map or directions so that others may follow in your tracks.

The Grave of Henry Briggs

Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? 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 mathematical significance, and either a map or directions so that others may follow in your tracks.

The mathematical tourist: new and updated snapshots of modern mathematics, by Ivars Peterson. Pp. 266. £13.95. 1998. ISBN 0 7167 3250 5 (W. H. Freeman).

The mathematical tourist: new and updated snapshots of modern mathematics, by Ivars Peterson. Pp. 266. £13.95. 1998. ISBN 0 7167 3250 5 (W. H. Freeman).

Renaissance area-fillings in the city hall of augsburg

The catapult that Archimedes built, the gambling-houses that Descartes 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 card 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 mathematical significance, and either a map or directions so that others may follow in your tracks. Please send all submissions to the Mathematical Tourist Editor, Ian Stewart.