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Polytopes include polygons (two-dimensional), polyhedra (three-dimensional), polychora (four-dimensional), and their higher dimensional analogs. An n-dimensional polytope is built up from multiple (n-1)-dimensional polytopes. Thus, polyhedra are built up from polygons, and polychora are built up from polyhedra. A regular polytope is composed of regular (n-1)-dimensional polytopes. There are an infinite number of regular convex polygons, five regular convex polyhedra, six regular convex polychora, and three regular convex polytopes for all dimensions five or higher.

Origami_Polyhedra@	Higher Dimensional
Polyhedra	Software
Sculpture and Art

• Guy's Polyhedra Pages Vertex figures, filling, faceting diagrams, stellation, defining polytopes through generators, trimethoric and trisynaptic polyhedra, space-filling polyhedra, lost stellations of the icosahedron, and links.
• Numericana - Polyhedra & Polytopes Enumeration, cartesian coordinates of vertices, naming, and counting the shapes.
• Polygons, Polyhedra, Polytopes Regular, rectified, and truncated polytopes with normal and hidden-detail-removed projections. Images, animations, and links.
• Polyhedra, Platonic Solids, Polytopes Definitions, pictures, templates, and coordinates of the regular 3d and 4d polytopes. Also includes waterman polyhedra, polar and star spheres, and the time star.
• Regular Polytopes Derivation of volume equations for regular polygons, polyhedra, and polytopes, with images.
• The Charged Particles Model Polytopes and optimal packing of p points in n dimensional spheres. Contains a java applet based on a model which allows for generation of multidimensional regular and semi-regular polytopes.
• The Geometry Junkyard Comprehensive list of links to sites on geometric properties of the shapes.
• The Mathematical Atlas - Polytopes and Polyhedra Collection of discussions regarding volume, vertices, dissection, g-holed tori, and other subjects.
• Zome Edu An educational toy and industrial tool to teach geometry and model complex and simple shapes, from cubes to Fullerenes.

 

 

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