CS 791 Overview
[Text taken from Winter 2006, subject to some
Throughout history, geometric patterns have formed an important part
of human expression. Ornament can even be found on artifacts dating back
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.
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 CS 488/688) is suggested as a
prerequisite, but not absolutely required.
If you're not sure about taking the course, come talk to me.
The course will consist of lectures on mathematical and computational
tools, mixed in with specific applications of those tools.
This list is far from final. I will add, delete, and reorder topics
before and during the term. I am also open to input from students.
- Halftoning: ordered and randomized dithering,
- NPR Packing: centroidal Voronoi diagrams,
- Geometric texture synthesis: procedural object
- Line art: TSP art, mazes
- Symmetry theory: group theory, symmetries,
frieze and wallpaper groups, extended notions of
- Tiling theory: tilings, regularity properties,
topology of tilings, periodic tilings,
monohedral and isohedral tilings
- Aperiodic tilings: rep-tiles, Penrose tilings,
- Geometry: Euclidean and non-Euclidean geometry,
models, ruler and protractor postulates, absolute geometry
- Aesthetics of ornament: horror vacui, the sense
of order, perception of symmetry, optimization, randomness
- Applications: Escher tilings,
Islamic star patterns
There will be three assignments and a final project. A small portion
of the final mark will also be based on class participation.
Here are some of the best overall references to consult for
In 2009 I put together a short textbook entitled
Introductory Tiling Theory for Computer Graphics. To
the extent that this course has an official textbook, I suppose
this is it. Note that you can find an electronic copy of the
book via the university's library.
The best reference for tiling theory has always been
Tilings and Patterns by Grünbaum and Shephard.
The book went out of print years ago. Last year I was delighted
to discover that the book was scheduled to be re-released by
Dover in December 2009, which would have been perfect for
this course. Alas, I now see that they've moved the release
back to April 2010. Buy a copy when it's out. You'll be glad you
did; it exemplifies everything a math textbook should be.
John Conway, Chaim Goodman-Strauss and Heidi Burgiel.
The Symmetries of Things. A beautiful and idiosyncratic
book that (finally) presents Conway's "topological" foundation
for symmetry theory in terms of orbifolds. Orbifolds offer
a completely re-worked and very appealing description of symmetries
Dorothy Washburn and Don Crowe, Symmetries of Culture.
An excellent discussion on the relationship between symmetry,
patterns, and the decorative traditions of cultures around the
world. Also a good introduction to the history and mathematical
foundations of symmetry theory.
Thomas Strothotte and Stefan Schlechtweg,
Non-photorealistic computer graphics: modeling, rendering, and animation
. A good overall textbook on NPR, at least up to 2002.
Owen Jones. The Grammar of Ornament (various editions). A
classic sourcebook of images of ornament from around the world.
I believe Dover has an edition with just the colour plates from
the original. Happily, the
full version is available online.
There are no doubt many other classic pattern books available
all over the web. It would be great to collect them here.
A good place to watch for pointers is the fun blog
Archibald Christie. Traditional Methods of Pattern
Designing. Oxford University Press, 1929.
A fun look into different classes of patterns and how they
evolved. The 1910 version appears to be
E.H. Gombrich. The Sense of Order: A Study in the
Psychology of Decorative Art. Phaidon Press Limited,
second edition, 1998. A classic study of art and ornamentaion.