# Higher Genus "Soccer Balls"

S. Allen Broughton

### Abstract

This web page contains the notes and supporting scripts for a  talks given in the RoseHulman  Mathematics faculty student seminar and to the Mount Holyoke College Math club.

### Rose-Hulman Mathematics Faculty Student Seminar - Spring 2000

Title: Higher Genus Soccer Balls and Kaleidoscopic Tilings in the Hyperbolic Plane.

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 1: Higher genus soccer balls

• 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.

#### Lecture2: Divisible tilings

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

### Mount Holyoke Math Club - Spring 2001

Abstract:  The soccer ball is a highly symmetric pattern on the sphere, closely related to the icosahedron and the dodecahedron. Using the soccer ball as an easily visualized starting point, we will look at what other "soccer ball patterns" can be put on surfaces such as the torus and so on...  The results of the talk are based on some work by REU students at Rose-Hulman Institute of Technology (tilings.org).

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.

### Indiana State University - Fall 2006

Abstract:  This talk will complement the first talk of the term on Cayley maps given by Robert Jajcay. The underlying link between the two talks will be the beautiful patterns on surfaces and computational power of group theory as a classifying tool. The underlying mathematical question is to classify all finite groups of transformations of closed surfaces. However, the approach to classification presented will be strongly linked to the underlying geometry and maps of the surfaces. The first part of the talk will link the groups of transformations of the sphere and torus to soccer balls, the platonic solids, and tessellations of the plane. The second part of the talk will venture into genus 2 surfaces and beyond.
There will be plenty of pictures – see for instance http://www.rose-hulman.edu/~brought/Epubs/soccer/soccerpics.html

Supporting materials:

Email:  allen.broughton@rose-hulman.edu
Webpage:  http://www.rose-hulman.edu/~brought/