Following is a list of some mathematically well-defined shapes.
Main article: List of curves |
Main article: List of surfaces |
See the list of algebraic surfaces.
Main article: List of fractals by Hausdorff dimension |
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 | 1 | 1 | 0 | 0 | ∞ |
3 | 5 Platonic solids | 4 Kepler–Poinsot solids | 3 tilings | ∞ | ∞ | ∞ | ∞ |
4 | 6 convex polychora | 10 Schläfli–Hess polychora | 1 honeycomb | 4 | 0 | 11 | ∞ |
5 | 3 convex 5-polytopes | 0 | 3 tetracombs | 5 | 4 | 2 | ∞ |
6 | 3 convex 6-polytopes | 0 | 1 pentacombs | 0 | 0 | 5 | ∞ |
7+ | 3 | 0 | 1 | 0 | 0 | 0 | ∞ |
There are no nonconvex Euclidean regular tessellations in any number of dimensions.
The elements of a polytope can be considered according to either their own dimensionality or how many dimensions "down" they are from the body.
For example, in a polyhedron (3-dimensional polytope), a face is a facet, an edge is a ridge, and a vertex is a peak.
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.
There is only one polytope in 1 dimension, whose boundaries are the two endpoints of a line segment, represented by the empty Schläfli symbol {}.
Polygons named for their number of sides
Main article: Uniform polyhedron |
Main article: Johnson solid |
Main article: spherical polyhedron |