# CS798 Overview

Motivation
Throughout history, geometric patterns have formed an important part of human expression. Ornament can even be found on artifacts dating back into prehistory.

It is only in the last century or two that we have developed the mathematical tools necessary to study the patterns we can created intuitively for millennia. Even more recently, we have computational tools that let us construct these patterns efficiently and painlessly, opening ever wider horizons of art and design.

This course is about the mathematical and computational tools that make it possible to analyze existing patterns and synthesize new ones. The focus is on ornamental design: abstract geometric patterns that adorn human artifacts.

Audience
The course is designed for students who are interested in the relationship between computers and design. It is open to undergraduates and graduates in math, with a particular emphasis on CS graduate students. Undergraduates should talk to an advisor about enrolling.

A computer graphics course (such as CS488/688) is a recommended prerequisite. If you're not sure about taking the course, come talk to me.

Tentative topics
The course will consist of lectures on mathematical and computational tools, mixed in with specific applications of those tools. Here, I have separated the applications into one line, but they will be interleaved with the theoretical topics throughout the term.

This list is far from final. I will add, delete, and reorder topics before and during the term.

• Symmetry theory: group theory, symmetries, frieze and wallpaper groups, extended notions of symmetry
• Tiling theory: tilings, regularity properties, topology of tilings, periodic tilings, monohedral and isohedral tilings
• Aperiodic tilings: rep-tiles, Penrose tilings, Wang tiles
• Geometry: Euclidean and non-Euclidean geometry, models, ruler and protractor postulates, absolute geometry
• Polyhedra: Conway notation, symmetric polyhedra, stellations
• Aesthetics of ornament: horror vacui, the sense of order, perception of symmetry, optimization, randomness
• Applications: Escher tilings, Celtic knotwork, Islamic star patterns, polyhedral sculpture, spherical and hyperbolic designs

Student responsibilities
There will be three assignments and a final project. A small portion of the final mark will also be based on class participation.

Textbook
There is no textbook for this course. If I could choose a book that ought to be the textbook, it would be Tilings and Patterns by Grünbaum and Shephard (W.H. Freeman, 1987). Alas, that book is out of print (in fact, if you see a reasonably-priced copy of the book, buy it immediately).

In the absence of that book, I will shamelessly direct you to my dissertation for the time being. Much of the math to be covered in the course will come from Chapter 2. I will eventually provide a longer list of links and references here.

Another quick reference (more of a personal note). You'll find lots of inspiration in The Grammar of Ornament, by Owen Jones. This is an old book, and good editions can be quite lovely. Even better, you can find electronic copies online in a couple of places!

 Craig S. Kaplan Last updated: