List of mathematical shapes

Following is a list of some mathematically well-defined shapes.

Surfaces in 3-space

Main article: List of surfaces

Algebraic surfaces

See the list of algebraic surfaces.

Regular Polytopes

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.

Polytope elements

The elements of a polytope can be considered according to either their own dimensionality or how many dimensions "down" they are from the body.

• Vertex, a 0-dimensional element
• Edge, a 1-dimensional element
• Face, a 2-dimensional element
• Cell, a 3-dimensional element
• Hypercell or Teron, a 4-dimensional element
• Facet, an (n-1)-dimensional element
• Ridge, an (n-2)-dimensional element
• Peak, an (n-3)-dimensional element

For example, in a polyhedron (3-dimensional polytope), a face is a facet, an edge is a ridge, and a vertex is a peak.

• Vertex figure: not itself an element of a polytope, but a diagram showing how the elements meet.

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.

One-dimensional regular polytope

There is only one polytope in 1 dimension, whose boundaries are the two endpoints of a line segment, represented by the empty Schläfli symbol {}.

2D with 1D surface

Polygons named for their number of sides

Uniform polyhedra

Main article: Uniform polyhedron

• non-convex

Johnson solids

Main article: Johnson solid

Spherical polyhedra

Main article: spherical polyhedron

Honeycombs

Convex uniform honeycomb
Dual uniform honeycomb
Others
Convex uniform honeycombs in hyperbolic space

Regular and uniform compound polyhedra

Polyhedral compound and Uniform polyhedron compound
Convex regular 4-polytope
Abstract regular polytope
Schläfli–Hess 4-polytope (Regular star 4-polytope)
Uniform 4-polytope
Prismatic uniform polychoron

5D with 4D surfaces

Five-dimensional space, 5-polytope and uniform 5-polytope
Prismatic uniform 5-polytope
For each polytope of dimension n, there is a prism of dimension n+1.

Six dimensions

Six-dimensional space, 6-polytope and uniform 6-polytope

Seven dimensions

Seven-dimensional space, uniform 7-polytope

Eight dimension

Eight-dimensional space, uniform 8-polytope

9-polytope

10-polytope

Dimensional families

Regular polytope and List of regular polytopes
Uniform polytope
Honeycombs