Brief Suggestions of Topics

Classification of Archimedean tilings:

This would be a math project explaining how to determine all the different semi-regular tilings, meaning a tiling by regular polygons where every vertex has the same arrangement of regular tiles around it. An alternative would be a programming project which would prepare either PostScript or Maple programs for all the tilings.

Classification of discrete groups of rigid motions of the plane:

This would be a mathematical determination of all these groups: including the rosette groups, strip groups and wallpaper groups. A general reference for all the tiling projects is the big book of tilings [Grünbaum and Shephard, 1987].

Classification of rigid motions in three dimensions:

We have classified the rigid motions in two dimensions as translations, rotations, reflections and glide reflections. This project would be a study of the different types of rigid motions in three dimensions.

Discrete groups of rotations in three dimensions:

This would be a study of rotations of the two-dimensional sphere, leading to a listing of all the possible discrete groups of rotations. It could also involve drawing the tilings of the sphere corresponding to these discrete groups, and the associated regular polyhedra. A short and beautiful paper on the subject is [Senechal, 1990]. You could also look at Klein's Lectures on the Icosahedron [Klein, 1956].

Discrete groups of rigid motions in three dimensions:

This would be a somewhat advanced project, as the variety of discrete groups in three dimensions is much greater than that in two dimensions. This classification is part of the science of crystallography, which is of huge practical importance in chemistry, solid state physics and materials science. One project of the right scale would be just a brief overview of the crystallographic groups.

Aperiodic tilings:

This would be further reading and study of the aperiodic tilings such as the Penrose tiling. there are many possible directions that one can follow here, but for the purpose of this course it would be particularly relevant to study those tilings that come out of an inflation-and-subdivision recipe. The original Penrose article is [Penrose, 1979]. Martin Gardner's columns on it are in the book [Gardner, 1997]. He refers to this theory in his book about the possibilities of artificial intelligence, The Emperor's New Mind [Penrose, 1989]. There is a nice book [Senechal, 1995] on the modern theory of quasicrystals that came out of Penrose's discovery. The big book [Grünbaum and Shephard, 1987] of tilings has an extensive chapter on aperiodic tilings.

$L$-systems:

This would be a study of a general pattern of recursively defined geometric structures first formulated by Aristid Lindenmayer. Our creation of Penrose tilings was an example of such a system. The project would consist of learning what an $L$-system is, and possibly how to create them in PostScript or some other language that allows recursive programming. A good book on this is The Algorithmic Beauty of Plants [Prusinkiewicz and Lindenmayer, 1990]. Another is [Prusinkiewicz and Hanan, 1989]. There is a conference proceedings [Rozenberg and Salomaa, 1992] on lots of different aspects of Lindenmayer systems.

Formal rewriting systems:

A rewriting system consists of some symbols and some rules for rewriting combinations of the symbols. For example the Fibonacci system consists of the symbols $\{a,b\}$ and the rules $a\to b$ and $b\to ba$. If we apply these rules over and over, we get generations: $a$, $b$, $ba$, $bab$, $babba$, $babbabab$, etc. This project would be a description of some of these systems and the properties they possess. See some of the references in the previous project and also [Choffrut, 1992].

Circle Packings:

A packing of circles is an arrangement of disjoint circles in the plane. We can associate a combinatorial graph to such a packing by making each circle a vertex and by connecting two vertices by an edge if the corresponding circles are tangent. This graph is called the nerve of the packing. This project would be a study of basic properties of circle packings including when a circle packing can be found whose nerve is a prescribed graph. A survey is given in the article [Stephenson, 2003]. A method of calculating and drawing circle packings is described in [Collins and Stephenson, 2003].

Conway's Tiling Groups:

This describes a group-theoretic method invented by Conway for deciding if certain regions of the plane may be tiled by a standard shape, such as a domino or a triomino. The main reference is the article [Thurston, 1990]. The goal would be to learn the method and to carry out in some examples.

Automatic groups:

This is described in Chapter 11 of [Mumford et al., 2002]. An automatic group is like a special kind of rewriting system that applies particularly to groups. This project would be to learn the basic definition of a finite state automaton and to calculate some automata for some simple groups.

Schattschneider's problem on classifying patterns:

Doris Schattschneider described a very interesting problem connected with groups of rigid motions in her talk at OSU last fall. It's concerned with some mathematics carried out by the Dutch artist M. C. Escher on the possible patterns he could construct with certain prescribed symmetries. The project would be to learn about the statement of the problem, and to describe some simple results and maybe work on some parts which are still undone. The main reference is Schattschneider's book [Schattschneider, 1990].

Schwarzian differential operator and Möbius transformations:

We will learn about the Möbius transformations $z\mapsto \frac{az+b}{cz+d}$ which are the angle-preserving transformations of the complex plane (more precisely the Riemann sphere). There is an amazing ordinary differential operator $\mathcal{S}$, called the Schwarzian, which has the property that if $\mathcal{S}(f) = 0$, then $f(z)$ is a Möbius transformation. Moreover, if $\mathcal{S}(f)=\mathcal{S}(g)$, then $f(z)=\frac{a \, g(z)+b}{c\,g(z)+d}$ for some $a,b,c,d$. This project would be about learning what the Schwarzian is and why it works in this way. A basic reference for this and the next two projects is the book by Lester Ford [Ford, 1951].

The hyperbolic metric on the upper half-plane:

The Möbius transformations $z\mapsto \frac{az+b}{cz+d}$ where $a,b,c,d$ are all real have the property that they map points in the upper half-plane $\mathbb{H}=\{\Im z>0\}$ back into the upper half-plane. There is a notion of distance between two points in the upper half-plane such that these Möbius transformations are precisely the distance-preserving transformations of the upper half-plane. This project would involve learning the definition of this distance, called the hyperbolic metric, and some of its basic properties. An important point to learn is what are the paths of shortest distance between two points (they are not straight lines in general).

Triangle groups:

A triangle group is defined by beginning with a triangle and considering all possible combinations of reflections in the three sides of the triangle. The condition for this to be a discrete group of rigid motions depends on the angles of the triangle only. The hyperbolic triangles are constructed in the same way except that the sides of the triangle are allowed to be circular arcs, and in place of reflections we consider the inversions in the three circular arcs. If we apply the resulting group of transformations to the original triangle, we get a tiling by triangles. Two examples are Figures 12.3 and 12.4 of [Mumford et al., 2002]. This project would consists of a study of triangle groups possibly culminating in some computer drawings of the associated tilings.

The construction of surfaces: Conway's ZIP proof:

We have discussed how two-dimensional surfaces can be constructed by taking a polygon with an even number of sides and pasting together pairs of the edges. Can all surfaces be constructed in this way? This project would be a study of a proof of that theorem by an ingenious argument of John Conway consisting of dividing a surface first into triangular pieces with zippers along the edges. The main references is the beautifully illustrated article [Francis and Weeks, 1999].