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This table shows a summary of regular polytope counts by dimension. Dimension Convex Nonconvex Convex Euclidean tessellations Convex hyperbolic tessellations Nonconvex hyperbolic tessellations Hyperbolic Tessellations with infinite cells and/or vertex figures Abstract Polytopes 1 1 line segment 0 1 0 0 0 1 2 ∞ polygons ∞ star polygons 3 tilings 1 0 0 ∞ 3 5 Platonic solids 4 Kepler–Poinsot solids 1 honeycomb ∞ ∞ 0 ∞ 4 6 convex polychora 10 Schläfli–Hess polychora 3 tessellations 4 0 11 ∞ 5 3 convex 5-polytopes 0 1 tessellation 5 4 2 ∞ 6 3 convex 6-polytopes 0 1 tessellation 0 0 5 ∞ 7+ 3 0 1 0 0 0 ∞ There are no nonconvex Euclidean regular tessellations in any number of dimensions. Tessellations The classical convex polytopes may be considered tessellations, or tilings, of spherical space. Tessellations of euclidean and hyperbolic space may also be considered regular polytopes. Note that an 'n'-dimensional polytope actually tessellates a space of one dimension less. For example, the (three-dimensional) platonic solids tessellate the 'two'-dimensional 'surface' of the sphere. |
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