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Home > Chaim Goodman-Strauss :: Symmetries
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"Prime colorings of the sphere, Euclidean plane and the hyperbolic plane," by Chaim Goodman-Strauss, University of Arkansas (

Tilings of the sphere, the Euclidean plane and the hyperbolic plane are shown. In each case, we have triangular faces, but on the sphere, the triangles meet in fives; in the Euclidean plane, the triangles meet in sixes, and in the hyperbolic plane, they meet in sevens. To a great degree, this is forced. It is impossible, for example, to have a tiling of the sphere with triangles meeting only in sevens (try it!).

In each case, a prime-fold coloring of the pattern is shown. It is helpful to realize that there are more similarities than differences among the three geometries.

The symmetry of the hyperbolic plane pictured above was known to Felix Klein by 1878, and has a tremendous number of interesting topological, geometric and algebraic properties.

This image is from "The Symmetries of Things"� by John H. Conway, Heidi Burgiel and Chaim Goodman-Strauss (AK Peters, 2008).

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American Mathematical Society