Editing List of mathematical shapes (section)

==Regular polytopes==
This table shows a summary of regular [[w:polytope|polytope]] counts by dimension.
{| class="wikitable"
|- 
!Dimension
!Convex
!Nonconvex
!Convex<BR>Euclidean<BR>tessellations
!Convex<BR>hyperbolic<BR>tessellations
!Nonconvex<BR>hyperbolic<BR>tessellations
!Hyperbolic Tessellations<BR>with infinite cells<BR>and/or vertex figures
!Abstract<br>Polytopes
|- align=center
|1||1 [[List of regular polytopes#One-dimensional regular polytope|line segment]]||0||1||0||0||0||1
|- align=center
|2||∞ [[#Two-dimensional regular polytopes|polygon]]s||∞ [[#Non-convex|star polygon]]s||1||1||0||0||∞
|- align=center
|3||5 [[#Three-dimensional regular polytopes|Platonic solids]]||4 [[#Non-convex 2|Kepler–Poinsot solid]]s||3 [[#Euclidean tilings|tilings]]||∞|| ∞||∞||∞
|- align=center
|4||6 [[#Four-dimensional regular polytopes|convex polychora]]||10 [[#Non-convex 3|Schläfli–Hess polychora]]||1 [[#Tessellations of Euclidean 3-space|honeycomb]]||4 || 0||11||∞
|- align=center
|5 ||3 [[#Five-dimensional regular polytopes and higher|convex 5-polytopes]]||0|| 3 [[#Tessellations of Euclidean 4-space|tetracombs]]|| 5 || 4||2||∞
|- align=center
|6||3 [[#Five-dimensional regular polytopes and higher|convex 6-polytopes]]||0||1 [[#Tessellations of Euclidean 5-space and higher|pentacombs]]||0||0||5||∞
|- align=center
|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 (geometry)|Vertex]], a 0-dimensional element
*[[Edge (geometry)|Edge]], a 1-dimensional element
*[[Face (geometry)|Face]], a 2-dimensional element
*[[Cell (geometry)|Cell]], a 3-dimensional element
*[[Hypercell]] or Teron, a 4-dimensional element
*[[Facet (geometry)|Facet]], an (''n''-1)-dimensional element
*[[Ridge (geometry)|Ridge]], an (''n''-2)-dimensional element
*[[Peak (geometry)|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 [[tessellation]]s, 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.

===Zero dimension===
*[[Point (geometry)|Point]]

===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]] {}.

=== Two-dimensional regular polytopes ===
*[[Polygon]]
*[[Equilateral]]
*[[Cyclic polygon]]

*[[Convex polygon]]
*[[Star polygon]]
*[[Pentagram]]

====Convex====

*[[Regular polygon]]
*[[Equilateral triangle]]
*[[Simplex]]
*[[Square (geometry)|Square]]
*[[Cross-polytope]]
*[[Hypercube]]
*[[Pentagon]]
*[[Hexagon]]
*[[Heptagon]]
*[[Octagon]]
*[[Enneagon]]
*[[Decagon]]
*[[Hendecagon]]
*[[Dodecagon]]
*[[Tridecagon]]
*[[Tetradecagon]]
*[[Pentadecagon]]
*[[Hexadecagon]]
*[[Heptadecagon]]
*[[Octadecagon]]
*[[Enneadecagon]]
*[[Icosagon]]
*[[Hectogon]]
*[[Chiliagon]]
*[[Regular polygon]]

===== Degenerate (spherical) =====
*[[Monogon]]
*[[Digon]]

====Non-convex====
*[[star polygon]]
*[[Pentagram]]
*[[Heptagram]]
*[[Octagram]]
*[[Enneagram (geometry)|Enneagram]]
*[[Decagram (geometry)|Decagram]]

====Tessellation====
*[[Apeirogon]]

=== Three-dimensional regular polytopes ===
*[[polyhedron]]

====Convex====
*[[Platonic solid]]
**[[Tetrahedron]], the 3-space [[Simplex]]
**[[Cube]], the 3-space [[hypercube]]
**[[Octahedron]], the 3-space [[Cross-polytope]]
**[[Dodecahedron]]
**[[Icosahedron]]

==== Degenerate (spherical) ====
*[[hosohedron]]
*[[dihedron]]
*[[Henagon#In spherical geometry]]

====Non-convex====
*[[Kepler–Poinsot polyhedra]]
**[[Small stellated dodecahedron]]
**[[Great dodecahedron]]
**[[Great stellated dodecahedron]]
**[[Great icosahedron]]

====Tessellations====

===== Euclidean tilings =====

*[[Square tiling]]
*[[Triangular tiling]]
*[[Hexagonal tiling]]
*[[Apeirogon]]
*[[Dihedron]]

===== Hyperbolic tilings =====

*[[Lobachevski plane]]
*[[Hyperbolic tiling]]

===== Hyperbolic star-tilings =====

*[[Order-7 heptagrammic tiling]]
*[[Heptagrammic-order heptagonal tiling]]
*[[Order-9 enneagrammic tiling]]{{Citation needed|reason=See Talk: Placing 'mathematical shapes' that do not have a wikipedia article|date=April 2018}}
*[[Enneagrammic-order enneagonal tiling]]{{Citation needed|reason=See Talk: Placing 'mathematical shapes' that do not have a wikipedia article|date=April 2018}}

=== Four-dimensional regular polytopes ===
*convex [[regular 4-polytope]]
**[[5-cell]], the 4-space [[Simplex]]
**[[8-cell]], the 4-space [[Hypercube]]
**[[16-cell]], the 4-space [[Cross-polytope]]
**[[24-cell]]
**[[120-cell]]
**[[600-cell]]

==== Degenerate (spherical) ====
*[[Ditope]]
*[[Hosotope]]
*[[3-sphere]]

====Non-convex====
*Star or (Schläfli–Hess) [[regular 4-polytope]]
**[[Icosahedral 120-cell]]
**[[Small stellated 120-cell]]
**[[Great 120-cell]]
**[[Grand 120-cell]]
**[[Great stellated 120-cell]]
**[[Grand stellated 120-cell]]
**[[Great grand 120-cell]]
**[[Great icosahedral 120-cell]]
**[[Grand 600-cell]]
**[[Great grand stellated 120-cell]]

====Tessellations of Euclidean 3-space====
*[[Honeycomb (geometry)|Honeycomb]]
*[[Cubic honeycomb]]

====Degenerate tessellations of Euclidean 3-space====
*[[Hosohedron]]
*[[Dihedron]]
*[[Order-2 apeirogonal tiling]]
*[[Apeirogonal hosohedron]]
*[[Order-4 square hosohedral honeycomb]]
*[[Order-6 triangular hosohedral honeycomb]]
*[[Hexagonal hosohedral honeycomb]]{{Citation needed|reason=See Talk: Placing 'mathematical shapes' that do not have a wikipedia article|date=April 2018}}
*[[Order-2 square tiling honeycomb]]
*[[Order-2 triangular tiling honeycomb]]
*[[Order-2 hexagonal tiling honeycomb]]

====Tessellations of hyperbolic 3-space====
*[[Order-4 dodecahedral honeycomb]]
*[[Order-5 dodecahedral honeycomb]]
*[[Order-5 cubic honeycomb]]
*[[Icosahedral honeycomb]]
*[[Order-3 icosahedral honeycomb]]
*[[Order-4 octahedral honeycomb]]
*[[Triangular tiling honeycomb]]
*[[Square tiling honeycomb]]
*[[Order-4 square tiling honeycomb]]
*[[Order-6 tetrahedral honeycomb]]
*[[Order-6 cubic honeycomb]]
*[[Order-6 dodecahedral honeycomb]]
*[[Hexagonal tiling honeycomb]]
*[[Order-4 hexagonal tiling honeycomb]]
*[[Order-5 hexagonal tiling honeycomb]]
*[[Order-6 hexagonal tiling honeycomb]]

=== Five-dimensional regular polytopes and higher ===
*[[5-polytope]]
*[[Honeycomb (geometry)|Honeycomb]]
*[[Tetracomb]]
{|
| [[Simplex]] || [[Hypercube]] || [[Cross-polytope]]
|-
| [[5-simplex]] || [[5-cube]] || [[5-orthoplex]]
|-
| [[6-simplex]] || [[6-cube]] || [[6-orthoplex]]
|-
| [[7-simplex]] || [[7-cube]] || [[7-orthoplex]]
|-
| [[8-simplex]] || [[8-cube]] || [[8-orthoplex]]
|-
| [[9-simplex]] || [[9-cube]] || [[9-orthoplex]]
|-
| [[10-simplex]] || [[10-cube]] || [[10-orthoplex]]
|-
| [[11-simplex]] || [[11-cube]] || [[11-orthoplex]]
|}

====Tessellations of Euclidean 4-space====
*[[Honeycomb (geometry)|honeycombs]]
*[[Tesseractic honeycomb]]
*[[16-cell honeycomb]]
*[[24-cell honeycomb]]

====Tessellations of Euclidean 5-space and higher====
*[[Hypercubic honeycomb]]
*[[Hypercube]]
*[[Square tiling]]
*[[Cubic honeycomb]]
*[[Tesseractic honeycomb]]
*[[5-cube honeycomb]]
*[[6-cube honeycomb]]
*[[7-cube honeycomb]]
*[[8-cube honeycomb]]
*[[Hypercubic honeycomb]]

====Tessellations of hyperbolic 4-space====
*[[Honeycomb (geometry)|honeycombs]]
*[[Order-5 5-cell honeycomb]]
*[[120-cell honeycomb]]
*[[Order-5 tesseractic honeycomb]]
*[[Order-4 120-cell honeycomb]]
*[[Order-5 120-cell honeycomb]]
*[[Order-4 24-cell honeycomb]]
*[[Cubic honeycomb honeycomb]]
*[[Small stellated 120-cell honeycomb]]
*[[Pentagrammic-order 600-cell honeycomb]]
*[[Order-5 icosahedral 120-cell honeycomb]]
*[[Great 120-cell honeycomb]]

====Tessellations of hyperbolic 5-space====

*[[5-orthoplex honeycomb]]
*[[24-cell honeycomb honeycomb]]
*[[16-cell honeycomb honeycomb]]
*[[Order-4 24-cell honeycomb honeycomb]]
*[[Tesseractic honeycomb honeycomb]]

=== Apeirotopes ===

* [[Apeirotope]]
** [[Apeirogon]]
** [[Apeirohedron]]
** [[Regular skew polyhedron]]

=== Abstract polytopes ===

* [[Abstract polytope]]
** [[11-cell]]
** [[57-cell]]