List of mathematical shapes


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

[Algebraic curve]s

Degree 2

See the list of algebraic surfaces.
This table shows a summary of regular counts by dimension.
DimensionConvexNonconvexConvex
Euclidean
tessellations		Convex
hyperbolic
tessellations		Nonconvex
hyperbolic
tessellations		Hyperbolic Tessellations
with infinite cells
and/or vertex figures		Abstract
Polytopes
11 line segment010001
2∞ polygons∞ star polygons1100∞
35 [|Platonic solids]4 Kepler–Poinsot solids3 [|tilings]∞∞∞∞
46 [|convex polychora]10 [|Schläfli–Hess polychora]1 [|honeycomb]4011∞
53 [|convex 5-polytopes]03 [|tetracombs]542∞
63 [|convex 6-polytopes]01 [|pentacombs]005∞
7+301000∞

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.
For example, in a polyhedron, 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 platonic solids tessellate the 'two'-dimensional 'surface' of the sphere.

Zero dimension

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

Two-dimensional regular polytopes

Euclidean tilings

2D with 1D surface

Polygons named for their number of sides
;Convex uniform honeycomb
;Dual uniform honeycomb
;Others
;Convex uniform honeycombs in hyperbolic space
;Polyhedral compound and Uniform polyhedron compound
;Convex regular 4-polytope
;Abstract regular polytope
;Schläfli–Hess 4-polytope
;Uniform 4-polytope
;Prismatic uniform polychoron
;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.

Honeycombs

;Six-dimensional space, 6-polytope and uniform 6-polytope
;Seven-dimensional space, uniform 7-polytope
;Eight-dimensional space, uniform 8-polytope
;9-polytope
;10-polytope
;Regular polytope and List of regular polytopes
;Uniform polytope
;Honeycombs