equator. Such tesselations would be degenerate in true 3D space as polyhedra. One can show that every regular tessellation of the sphere is characterized by a pair of integers {p, q} with + > . Likewise, a regular tessellation of the plane is characterized by the condition + = . There are three possibilities: In a similar manner, one can consider regular tessellations of the hyperbolic plane. These are characterized by the condition + < . There is an infinite family of such tessellations. Higher dimensions In more than three dimensions, polyhedra generalize to polytopes, with higher-dimensional convex regular polytopes being the equivalents of the three-dimensional Platonic solids. In the mid-19th century the Swiss mathematician Ludwig Schläfli discovered the four-dimensional analogues of the Platonic solids, called convex regular 4-polytopes. There are exactly six of these figures; five are analogous to the Platonic solids : 5-cell as {3,3,3}, 16-cell as {3,3,4}, 600-cell as {3,3,5}, tesseract as {4,3,3}, and 120-cell as {5,3,3}, and a sixth one, the self-dual 24-cell, {3,4,3}. In all dimensions higher than four, there are only three convex regular polytopes: the simplex as {3,3,...,3}, the hypercube as {4,3,...,3}, and the cross-polytope as {3,3,...,4}. In three dimensions, these coincide with the tetrahedron as {3,3}, the cube as {4,3}, and the octahedron as {3,4}. See also Citations General and cited sources Gardner, Martin (1987). The 2nd Scientific American Book of Mathematical Puzzles & Diversions, University of Chicago Press, Chapter 1: The Five Platonic Solids, Haeckel, Ernst, E. (1904). Kunstformen der Natur. Available as Haeckel, E. (1998); Art forms in nature, Prestel USA. . Kepler. Johannes Strena seu de nive sexangula (On the Six-Cornered Snowflake), 1611 paper by Kepler which discussed the reason for the six-angled shape of the snow crystals and the forms and symmetries in nature. Talks about platonic solids. Wildberg, Christian (1988). John Philoponus' Criticism of Aristotle's Theory of Aether. Walter de Gruyter. pp. 11–12. . External links Platonic solids at Encyclopaedia of Mathematics Book XIII of Euclid's Elements. Interactive 3D Polyhedra in Java Platonic Solids in Visual Polyhedra Solid Body Viewer is an interactive 3D polyhedron viewer which allows you to save the model in svg, stl or obj format. Interactive Folding/Unfolding Platonic Solids in Java Paper models of the Platonic solids created using nets generated by Stella software Platonic Solids Free paper models(nets) Teaching Math with Art student-created models Teaching Math with Art teacher instructions for making