Rose-Hulman Talks | Mount Holyoke Talk | ISU talk | Soccer Ball Page

**Abstract:** Two talks on kaleidoscopic tilings, for a general mathematical
audience of students and faculty. The purpose of the talks is to
present an area of intriguing mathematical research, rich with problems
suitable for undergraduate research.

A soccer ball has an attractive pattern of pentagons and hexagons on its surface, with a great deal of symmetry. Baseballs and basketballs also have certain patterns of symmetry which are different from the soccer ball pattern. Though the sportsman might never ask, a mathematician would be intrigued by the possibility of a higher genus soccer ball (a soccer ball with patterned handles). It turns out that they exist in great abundance though we need to give up on having only hexagons and pentagons.

The key to creating and understanding "soccer balls" are kaleidoscopic tilings of the 2-dimensional geometries: the sphere, the euclidean plane and the hyperbolic plane. The sphere tilings, of course, yield the patterns of sports balls. The tilings of the euclidean and hyperbolic planes form beautiful patterns and have their own artistic interest, as in some of the art of Escher. The higher genus soccer balls, though impractical, are a convenient mental hook for generating questions about patterned surfaces, e.g., constructing simple examples.

In the first talk the relation between (higher genus) soccer
balls and tilings will be explored, including an introduction to hyperbolic
geometry. In the second talk I will present some work completed jointly
by undergraduates and myself on divisible kaleidoscopic tilings, i.e.,
simultaneous tilings of the plane by two different kaleidoscopic polygons.
It has a nice interplay between combinatorics (Catalan numbers) and geometry.

- Lecture slides in pdf format - printout of power point file: (18 pages - 93K) soccer.pdf
- Some of the picture referred to in the slides may be found on the soccer ball picture page.

- Lecture Notes in pdf format - printout of power point file: (15 pages - 268K) divisible.pdf

Doorprize/Giveway: One very cool math T-shirt, related to the subject of the talk, will be given away.

**Supporting materials: **The talk outline is similar to the slides
for the Rose-Hulman talk** **soccer.pdf
above. Here is the soccer
ball picture page of the various soccer patterns shown in the talk
and links to Maple scripts used to produce the pictures.

There will be plenty of pictures – see for instance http://www.rose-hulman.edu/~brought/Epubs/soccer/soccerpics.html

**Supporting materials: **

- Slides of the talk: soccerISU.pdf
- Pictures: soccer ball picture page
- Hyperbolic tiling used in the talk t433.pdf

Email: allen.broughton@rose-hulman.edu Webpage: http://www.rose-hulman.edu/~brought/ This page last updated on 15 Nov 06.