## Mathematics and Art

5. Polyhedra, tilings, and dissections

Drawing polyhedra was an early testing ground for ideas related to perspective drawing. Renaissance artists were involved in trying to build on historical references to "Archimedean polyhedra" which were transmitted via the writings of Pappus. What constituted a complete set of convex polyhedra with the property that locally every vertex looked like every other vertex and whose faces were regular polygons, perhaps not all with the same number of sides? Perhaps surprisingly, no complete reconstruction occurred until the work of Kepler (1571-1630), who found 13 such solids, even though one can make a case for there being 14 such solids. (Pappus-Archimedes missed one in ancient times. The modern definition of Archimedean solids defines them as convex polyhedra which have a symmetry group under which all the vertices are alike. Using this definition there are 13 solids, but there is little reason to believe that in ancient Greece geometers were thinking in terms of groups rather than in terms of local vertex equivalence, that is, the pattern of faces around each vertex being identical.)

In more modern times polyhedra have inspired artists and mathematicians with an interest in the arts. Inspired by polyhedra, Stewart Coffin has created a wonderful array of puzzle designs which require putting together pieces he designed made from rare woods to form polyhedra. Coffin's puzzles are remarkable for both their ingenuity as puzzles and their beauty. This beauty is a reflection of the beauty of the polyhedral objects themselves, but also the beauty of the rare woods he used to make his puzzles. Coffin showed creativity in selecting symmetrical variants of well-known polyhedra. Coffin's work, like Escher's, has been an inspiration to others. Good puzzles engender the same sense of wonder that beautiful mathematics inspires. George Hart, whose background is in computer science, is an example of a person who is contributing to the mathematical theory of polyhedra, while at the same time he uses his skills as a sculptor and artist to create original works inspired by polyhedral objects.

(Courtesy of George Hart.)

There is a long tradition of making precise models of polyhedra with regularity properties. It is common at mathematics conferences for geometers to feature a models room where mathematicians who enjoy building models can display the beauties of geometry in a physical form. They complement the beauty of such geometric objects in the mind's eye. The beauty of polyhedral solids in the hands of a skilled model maker results in what are, indeed, works of art. Magnus Wenninger is the author of several books about model making. His models are especially beautiful. Here is a small sample, which only hints at the variety of models that Wenninger has made over many years. His models of "stellated" polyhedra are particularly striking.

(Images courtesy of Magnus Wenninger)

A tiling of the plane is a way of filling up the plane without holes or overlaps with shapes of various kinds. For example, one can tile the plane with congruent copies of any triangle, and, more surprisingly, with congruent copies of any simple quadrilateral, whether convex or not. Tilings are closely related to artistic designs one finds on fabrics, rugs, and wallpaper. Though there were scattered analyses at different ways of tiling the plane that date back to ancient times, there was surprisingly little in the way of a theory for tilings of the plane as compared to what was done to understand polyhedra. Kepler did important work on tilings, but from his time until the late 19th century relatively little work was done. Unfortunately, not only was work on tilings sporadic but often it was incomplete or misleading. The publication of the monumental book by Branko Grünbaum and Geoffrey Shephard, Tilings and Patterns, changed this. Many new tiling problems were addressed and solved and a variety of software tools for creating tilings (and polyhedra and playing games) of different kinds were developed. Daniel Huson and Olaf Fredrichs (RepTiles) and Kevin Lee (Tesselmania) developed very nice tiling programs but some of the locations where this software used to be available are no longer supported.

A more recent source of art inspired by mathematics has been related to dissections. A good starting place for the ideas here is the remarkable theorem known as the Bolyai-Gerwien-Wallace Theorem. It states that two (simple) polygons A and B in the plane have the same area if and only if it is possible to cut one of the polygons up into a finite number of pieces and assemble the pieces to form the other polygon. In one direction this result is straight forward: if one has cut polygon A into pieces which will assemble to form polygon B, then B's area is the same as A's area. The delightful surprise is that if A and B have the same area then one can cut up A into finitely many pieces and reassemble the pieces to get B. Where does the art come in? Given two polygons with the same area, one can ask for two extensions of the Bolyai-Gerwien-Wallace Theorem:

a. Find the smallest number of pieces into which A can be cut and reassembled to form B.

b. Find pieces with appealing properties into which A can be cut and reassembled to from B. These properties might be that all the pieces are congruent, similar, or have edges which are related by some appealing geometric transformation.

Greg Frederickson has collected together a large amount of material about how polygons of one shape can be dissected into other polygons of the same area. These dissections concentrate on dissections of regular polygons (which may be convex or "star-shaped") into other regular polygons. One might expect that the mathematical regularity of the objects leads to aesthetic solutions. This turns out to be the case.

Frederickson also describes how with a suitable mechanism, one can address how to attach the pieces from one polygon and move them so that they create the other polygon. These dissections are known as hinged dissections. The first way that comes to mind to hinge the pieces is to attach the pieces at their vertices. There are lovely examples of hinged dissections of this kind including ones that prove the Pythagorean theorem geometrically by showing how one can cut the squares on the two legs of a right triangle and assemble the pieces to form the square on the hypotenuse. However, there is another ingenious way to do the hinging. This involves hinging the edges so that the polygons that are joined along these two edges can rotate with respect to one another. This type of hinging is known as twist-hinging. Frederickson arranged for several very attractive hinged dissections to be realized physically with polygonal sections of the dissection involved to be made of beautiful woods. For their beauty these physical models draw heavily on the mathematics behind the dissections. For example, in one of the physically realized hinged dissections commissioned by Frederickson, a regular hexagon with a hole is twist-hinge dissected into a hexagram with a hole of the same area.

(Used with the permission of Greg Frederickson)

The mathematics behind this process is a way to dissect a hexagon with a hole into a hexagram with a hole. Based on this dissection Frederickson cleverly produced a hinge-twist dissection. This very attractive object is not very interesting as a puzzle but creates a lovely effect as one watches the unexpected transformation between the two shapes evolve as one manipulates the twist-hinged pieces.