Color the countries of a map on the sphere so that colors of bordering countries are different. What is the maximum number of colors necessary for all possible maps? This famous 4-color problem for maps on the surface of a sphere was solved in 1977.

The analogous problem for surfaces of positive genus g was solved in 1968 by Ringle and Young. (A surface of genus g is the surface of a doughnut with g holes.) In 1890, in a paper pointing out the mistake in an early attempt at solving the 4-color problem, Heawood proved that, for maps on a surface of positive genus g,

Ringle and Young showed that, for every g > 0, there is a map requiring h(g) colors. In 1995, University of Arizona mathematics major Jennifer Widmann took Ringle and Young's analytic description of these maps (for several g) and created models of incredible symmetry and beauty. These models were constructed as part of a final project for Math 430, A Second Course in Geometry, taught by David Gay. Pictures of two of these models are shown below. Jennifer's original models will be on display on the ground floor of the Mathematics Department at The University of Arizona during Math Awareness Week 1997.

	Front of surface of genus 20 with map requiring 19 colors. (58K JPEG)
	Back of surface of genus 20 with map requiring 19 colors. (53K JPEG)
	Front of surface of genus 11 with map requiring 15 colors. (76K JPEG)
	Back of surface of genus 11 with map requiring 15 colors. (68K JPEG)