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 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
- 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
- Polyhedra: Conway notation, symmetric polyhedra,
- Aesthetics of ornament: horror vacui, the sense
of order, perception of symmetry, optimization, randomness
- Applications: Escher tilings,
Islamic star patterns,
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.
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
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!