View source for List of regular polytopes and compounds

{{Split|List of regular polytopes|List of regular polytope compounds|date=January 2024}}
{{Short description|None}}
{| class="wikitable floatright"
|+ Example regular polytopes
!colspan=2|Regular (2D) polygons
|-
!Convex
!Star
|- align=center
|[[File:Regular pentagon.svg|150px]]<BR>[[Pentagon|{5}]]
|[[File:Star polygon 5-2.svg|150px]]<BR>[[Pentagram|{5/2}]]
|-
!colspan=2|Regular (3D) polyhedra
|-
!Convex
!Star
|- align=center
|[[File:Dodecahedron.png|150px]]<BR>[[Dodecahedron|{5,3}]]
|[[File:Small stellated dodecahedron.png|150px]]<BR>[[Small stellated dodecahedron|{5/2,5}]]
|-
! colspan="2" |Regular 4D polytopes
|-
!Convex
!Star
|-
|[[File:Schlegel wireframe 120-cell.png|150px]]<BR>[[120-cell|{5,3,3}]]
|[[File:Ortho solid 010-uniform polychoron p53-t0.png|150px]]<BR>[[Small stellated 120-cell|{5/2,5,3}]]
|-
!colspan=2|Regular 2D tessellations
|-
!Euclidean
!Hyperbolic
|- align=center
|[[File:Uniform tiling 44-t0.svg|150px]]<BR>[[Square tiling|{4,4}]]
|[[File:H2-5-4-dual.svg|150px]]<BR>[[Order-4 pentagonal tiling|{5,4}]]
|-
!colspan=2|Regular 3D tessellations
|-
!Euclidean
!Hyperbolic
|- align=center
|[[File:Cubic_honeycomb.png|150px]]<BR>[[Cubic honeycomb|{4,3,4}]]
|[[File:Hyperbolic orthogonal dodecahedral honeycomb.png|150px]]<BR>[[Order-4 dodecahedral honeycomb|{5,3,4}]]
|}
This article lists the [[Regular polytope|regular polytope]]s and regular [[Polytope compound|polytope compound]]s in [[Euclidean geometry|Euclidean]], [[Spherical geometry|spherical]] and [[Hyperbolic geometry|hyperbolic]] spaces.

==Overview==
This table shows a summary of regular polytope counts by rank.

{| class="wikitable"
!rowspan=3|Rank
!rowspan=2 colspan=3|Finite
!rowspan=2 colspan=2|Euclidean
!colspan=3|Hyperbolic
!rowspan=3|Abstract
!rowspan=2 colspan=2|Compounds
|-
!colspan=2|Compact
!Paracompact
|-
!Convex
!Star
!Skew{{efn|name=full rank|Only counting polytopes of full rank. There are more regular polytopes of each rank > 1 in higher dimensions.}}<ref name="RegPolyFull">{{citation|last=McMullen|first=Peter|year=2004|title=Regular polytopes of full rank|journal=Discrete & computational geometry|url=https://link.springer.com/article/10.1007/s00454-004-0848-5}}</ref>
!Convex
!Skew{{efn|name=full rank}}<ref name="RegPolyFull"/>
!Convex
!Star
!Convex
!Convex
!Star
|- align=center
!1
||1||style="color:#999999; font-size:80%"|''none''||style="color:#999999; font-size:80%"|''none''||style="color:#999999; font-size:80%"|''none''||style="color:#999999; font-size:80%"|''none''||style="color:#999999; font-size:80%"|''none''||style="color:#999999; font-size:80%"|''none''||style="color:#999999; font-size:80%"|''none''||1||style="color:#999999; font-size:80%"|''none''||style="color:#999999; font-size:80%"|''none''
|- align=center
!2
||<math>\infty</math>||<math>\infty</math>||style="color:#999999; font-size:80%"|''none''||1||style="color:#999999; font-size:80%"|''none''||1||style="color:#999999; font-size:80%"|''none''||style="color:#999999; font-size:80%"|''none''||<math>\infty</math>||<math>\infty</math>||<math>\infty</math>
|- align=center
!3
||5||4||9||3||3||<math>\infty</math>||<math>\infty</math>||<math>\infty</math>||<math>\infty</math>||5||style="color:#999999; font-size:80%"|''none''
|- align=center
!4
||6||10||18||1||7||4||style="color:#999999; font-size:80%"|''none''||11||<math>\infty</math>||26||20
|- align=center
!5
||3||style="color:#999999; font-size:80%"|''none''||3||3||15||5||4||2||<math>\infty</math>||style="color:#999999; font-size:80%"|''none''||style="color:#999999; font-size:80%"|''none''
|- align=center
!6
||3||style="color:#999999; font-size:80%"|''none''||3||1||7||style="color:#999999; font-size:80%"|''none''||style="color:#999999; font-size:80%"|''none''||5||<math>\infty</math>||style="color:#999999; font-size:80%"|''none''||style="color:#999999; font-size:80%"|''none''
|- align=center
!7
||3||style="color:#999999; font-size:80%"|''none''||3||1||7||style="color:#999999; font-size:80%"|''none''||style="color:#999999; font-size:80%"|''none''||style="color:#999999; font-size:80%"|''none''||<math>\infty</math>||3||style="color:#999999; font-size:80%"|''none''
|- align=center
!8
||3||style="color:#999999; font-size:80%"|''none''||3||1||7||style="color:#999999; font-size:80%"|''none''||style="color:#999999; font-size:80%"|''none''||style="color:#999999; font-size:80%"|''none''||<math>\infty</math>||6||style="color:#999999; font-size:80%"|''none''
|- align=center
!9+
||3||style="color:#999999; font-size:80%"|''none''||3||1||7||style="color:#999999; font-size:80%"|''none''||style="color:#999999; font-size:80%"|''none''||style="color:#999999; font-size:80%"|''none''||<math>\infty</math>
| {{efn|

<math>
\begin{cases}
  2, & \text{if the number of dimensions is of the form } 2^k     \\
  1, & \text{if the number of dimensions is of the form } 2^k-1   \\
  0, & \text{otherwise}                                           \\      
\end{cases}
</math>

}}
|style="color:#999999; font-size:80%"|''none''
|}
{{notelist}}

There are no Euclidean regular star tessellations in any number of dimensions.

==1-polytopes==
{| class=wikitable align=right width=330
|- valign=top
|[[File:Coxeter node markup1.png|frameless|upright]]
|A Coxeter diagram represent mirror "planes" as nodes, and puts a ring around a node if a point is ''not'' on the plane. A ''dion'' {&nbsp;}, {{CDD|node_1}}, is a point ''p'' and its mirror image point ''p''', and the line segment between them.
|}
There is only one polytope of rank 1 (1-polytope), the closed [[Line segment|line segment]] bounded by its two endpoints. Every realization of this 1-polytope is regular. It has the [[Schläfli symbol]] {&nbsp;},{{sfnp|Coxeter|1973|p=129}}{{sfnp|McMullen|Schulte|2002|p=30}} or a Coxeter diagram with a single ringed node, {{CDD|node_1}}. [[Biography:Norman Johnson (mathematician)|Norman Johnson]] calls it a ''dion''<ref>{{cite book |first=N.W. |last=Johnson |title=Geometries and Transformations |year=2018 |isbn=978-1-107-10340-5 |chapter=Chapter 11: Finite symmetry groups |at=11.1 Polytopes and Honeycombs, p. 224}}</ref> and gives it the Schläfli symbol {&nbsp;}.

Although trivial as a polytope, it appears as the [[Edge (geometry)|edges]] of polygons and other higher dimensional polytopes.{{sfnp|Coxeter|1973|p=120}} It is used in the definition of [[Prism (geometry)|uniform prism]]s like Schläfli symbol {&nbsp;}×{p}, or Coxeter diagram {{CDD|node_1|2|node_1|p|node}} as a [[Cartesian product]] of a line segment and a regular polygon.{{sfnp|Coxeter|1973|p=124}}

==2-polytopes (polygons)==

The polytopes of rank 2 (2-polytopes) are called [[Polygon|polygon]]s. Regular polygons are equilateral and cyclic. A {{mvar|p}}-gonal regular polygon is represented by [[Schläfli symbol]] {p}.

Many sources only cosider [[Convex polygon|convex polygon]]s, but [[Star polygon|star polygon]]s, like the [[Pentagram|pentagram]], when considered, can also be regular. They use the same vertices as the convex forms, but connect in an alternate connectivity which passes around the circle more than once to be completed.

===Convex===

The Schläfli symbol {p} represents a [[Regular polygon|regular {{mvar|p}}-gon]].

{| class="wikitable" style="text-align:center; border:0;"
|- bgcolor="#e0e0e0" valign="top"
!Name
![[Equilateral triangle|Triangle]]<br />([[Simplex|2-simplex]])<!--<br/>(trig)-->
![[Square]]<br />([[Cross-polytope|2-orthoplex]])<br />([[Hypercube|2-cube]])
![[Pentagon]]<br />([[Pentagonal polytope|2-pentagonal<br />polytope]])<!--<br/>(peg)-->
![[Hexagon]]<!--<br/>(hig)-->
![[Heptagon]]<!--<br/>(heg)-->
![[Octagon]]<!--<br/>(oc)-->
| style="border:0; background:white;" rowspan="10" | 
|- bgcolor="#ffd0d0"
![[Schläfli symbol|Schläfli]]
|{3}
|{4}
|{5}
|{6}
|{7}
|{8}
|-
!Symmetry
|D<sub>3</sub>, [3]||D<sub>4</sub>, [4]||D<sub>5</sub>, [5]||D<sub>6</sub>, [6]||D<sub>7</sub>, [7]||D<sub>8</sub>, [8]
|-
!Coxeter
|{{CDD|node_1|3|node}}
|{{CDD|node_1|4|node}}
|{{CDD|node_1|5|node}}
|{{CDD|node_1|6|node}}
|{{CDD|node_1|7|node}}
|{{CDD|node_1|8|node}}
|-
!Image
|[[Image:Regular triangle.svg|60px]]
|[[Image:Regular quadrilateral.svg|60px]]
|[[Image:Regular pentagon.svg|60px]]
|[[Image:Regular hexagon.svg|60px]]
|[[Image:Regular heptagon.svg|60px]]
|[[Image:Regular octagon.svg|60px]]
|-
!Name
![[Nonagon]]<br>''(Enneagon)''<!--<br/>(en)-->
![[Decagon]]<!--<br/>(dec)-->
![[Hendecagon]]<!--<br/>(heng)-->
![[Dodecagon]]<!--<br/>(dog)-->
![[Tridecagon]]<!--<br/>(tad)-->
![[Tetradecagon]]<!--<br/>(ted)-->
|- bgcolor="#ffd0d0"
!Schläfli
|{9}
|{10}
|{11}
|{12}
|{13}
|{14}
|-
!Symmetry
|D<sub>9</sub>, [9]||D<sub>10</sub>, [10]||D<sub>11</sub>, [11]||D<sub>12</sub>, [12]||D<sub>13</sub>, [13]||D<sub>14</sub>, [14]
|-
!Dynkin
|{{CDD|node_1|9|node}}
|{{CDD|node_1|10|node}}
|{{CDD|node_1|11|node}}
|{{CDD|node_1|12|node}}
|{{CDD|node_1|13|node}}
|{{CDD|node_1|14|node}}
|-
!Image
|[[Image:Regular nonagon.svg|60px]]
|[[Image:Regular decagon.svg|60px]]
|[[Image:Regular hendecagon.svg|60px]]
|[[Image:Regular dodecagon.svg|60px]]
|[[Image:Regular tridecagon.svg|60px]]
|[[Image:Regular tetradecagon.svg|60px]]
|-
!Name
![[Pentadecagon]]<!--<br/>(ped)-->
![[Hexadecagon]]<!--<br/>(hed)-->
![[Heptadecagon]]
![[Octadecagon]]
![[Enneadecagon]]
![[Icosagon]]
!&emsp;[[Regular polygon|...p-gon]]&emsp;
|- bgcolor="#ffd0d0"
!Schläfli
|{15}
|{16}
|{17}
|{18}
|{19}
|{20}
|{''p''}
|-
!Symmetry
|D<sub>15</sub>, [15]||D<sub>16</sub>, [16]||D<sub>17</sub>, [17]||D<sub>18</sub>, [18]||D<sub>19</sub>, [19]||D<sub>20</sub>, [20]||D<sub>p</sub>, [p]
|-
!Dynkin
|{{CDD|node_1|15|node}}
|{{CDD|node_1|16|node}}
|{{CDD|node_1|17|node}}
|{{CDD|node_1|18|node}}
|{{CDD|node_1|19|node}}
|{{CDD|node_1|20|node}}
|{{CDD|node_1|p|node}}
|-
!Image
|[[Image:Regular pentadecagon.svg|60px]]
|[[Image:Regular hexadecagon.svg|60px]]
|[[Image:Regular heptadecagon.svg|60px]]
|[[Image:Regular octadecagon.svg|60px]]
|[[Image:Regular enneadecagon.svg|60px]]
|[[Image:Regular icosagon.svg|60px]]
|[[Image:Disk 1.svg|60px]]
|}

====Spherical====

The regular [[Digon|digon]] {2} can be considered to be a [[Degeneracy (mathematics)|degenerate]] regular polygon. It can be realized non-degenerately in some non-Euclidean spaces, such as on the surface of a [[Sphere|sphere]] or [[Torus|torus]]. For example,  digon can be realised non-degenerately as a [[Spherical lune|spherical lune]]. A [[Monogon|monogon]] {1} could also be realised on the sphere as a single point with a great circle through it.<ref>Coxeter, ''Regular Complex Polytopes'', p. 9</ref> However, a monogon is not a valid [[Abstract polytope|abstract polytope]] because its single edge is incident to only one vertex rather than two.

{| class="wikitable" style="text-align:center;"
|- bgcolor="#e0e0e0" valign="top"
!Name
|[[Monogon]]
|[[Digon]]
|- bgcolor="#eeeedd"
![[Schläfli symbol]]
|{1}
|{2}
|-
!Symmetry
|D<sub>1</sub>, [ ]
|D<sub>2</sub>, [2]
|-
!Coxeter diagram
|{{CDD|node}} or {{CDD|node_h|2x|node}}
|{{CDD|node_1|2x|node}}
|-
!Image
|[[File:Monogon.svg|60px]]
|[[Image:Digon.svg|60px]]
|}

===Stars===
There exist infinitely many regular star polytopes in two dimensions, whose Schläfli symbols consist of rational numbers {{math|{{{mvar|n}}/{{mvar|m}}{{)}}}}. They are called [[Star polygon|star polygon]]s and share the same [[Vertex arrangement|vertex arrangement]]s of the convex regular polygons.

In general, for any natural number {{mvar|n}}, there are regular {{mvar|n}}-pointed stars with Schläfli symbols {{math|{{{mvar|n}}/{{mvar|m}}{{)}}}} for all {{mvar|m}} such that {{math|{{mvar|m}} < {{mvar|n}}/2}} (strictly speaking {{math|{{{mvar|n}}/{{mvar|m}}}={{{mvar|n}}/({{mvar|n}}−{{mvar|m}}){{)}}}}) and {{mvar|m}} and {{mvar|n}} are coprime (as such, all stellations of a polygon with a prime number of sides will be regular stars). Symbols where {{mvar|m}} and {{mvar|n}} are not coprime may be used to represent compound polygons.

{| class="wikitable" style="text-align:center;"
|- bgcolor="#e0e0e0"
!Name
|[[Pentagram]]
| colspan="2" | [[Heptagram]]s
|[[Octagram]]
| colspan="2" | [[Enneagram (geometry)|Enneagrams]]
|[[Decagram (geometry)|Decagram]]
|...[[Star polygon|n-grams]]
|- bgcolor="#ffd0d0"
![[Schläfli symbol|Schläfli]]
|{5/2}<!--<br/>(star)-->
|{7/2}<!--<br/>(hag)-->
|{7/3}<!--<br/>(gahg)-->
|{8/3}<!--<br/>(og)-->
|{9/2}<!--<br/>(eng)-->
|{9/4}<!--<br/>(geng)-->
|{10/3}<!--<br/>(dag)-->
|{''p/q''}
|-
!Symmetry
|D<sub>5</sub>, [5]||colspan=2|D<sub>7</sub>, [7]||D<sub>8</sub>, [8]||colspan=2|D<sub>9</sub>, [9],||D<sub>10</sub>, [10]||D<sub>p</sub>, [''p'']
|-
!Coxeter
|{{CDD|node_1|5|rat|d2|node}}
|{{CDD|node_1|7|rat|d2|node}}
|{{CDD|node_1|7|rat|d3|node}}
|{{CDD|node_1|8|rat|d3|node}}
|{{CDD|node_1|9|rat|d2|node}}
|{{CDD|node_1|9|rat|d4|node}}
|{{CDD|node_1|10|rat|d3|node}}
|{{CDD|node_1|p|rat|dq|node}}
|-
!Image
|[[File:Star polygon 5-2.svg|75px]]
|[[File:Star polygon 7-2.svg|75px]]
|[[File:Star polygon 7-3.svg|75px]]
|[[File:Star polygon 8-3.svg|75px]]
|[[File:Star polygon 9-2.svg|75px]]
|[[File:Star polygon 9-4.svg|75px]]
|[[File:Star polygon 10-3.svg|75px]]
|&nbsp;
|}

{| class=wikitable
|+ Regular star polygons up to 20 sides
|- align=center
|[[File:Regular_star_polygon_ 11-2.svg|60px]]<BR>'''{11/2}'''
|[[File:Regular_star_polygon_ 11-3.svg|60px]]<BR>{11/3}
|[[File:Regular_star_polygon_ 11-4.svg|60px]]<BR>{11/4}
|[[File:Regular_star_polygon_ 11-5.svg|60px]]<BR>{11/5}
|[[File:Regular_star_polygon_ 12-5.svg|60px]]<BR>'''{12/5}'''
|[[File:Regular_star_polygon_13-2.svg|60px]]<BR>'''{13/2}'''
|[[File:Regular_star_polygon_13-3.svg|60px]]<BR>{13/3}
|[[File:Regular_star_polygon_13-4.svg|60px]]<BR>{13/4}
|[[File:Regular_star_polygon_13-5.svg|60px]]<BR>{13/5}
|[[File:Regular_star_polygon_13-6.svg|60px]]<BR>{13/6}
|- align=center
|[[File:Regular_star_polygon_14-3.svg|60px]]<BR>'''{14/3}'''
|[[File:Regular_star_polygon_14-5.svg|60px]]<BR>{14/5}
|[[File:Regular_star_polygon_15-2.svg|60px]]<BR>'''{15/2}'''
|[[File:Regular_star_polygon_15-4.svg|60px]]<BR>{15/4}
|[[File:Regular_star_polygon_15-7.svg|60px]]<BR>{15/7}
|[[File:Regular_star_polygon_16-3.svg|60px]]<BR>'''{16/3}'''
|[[File:Regular_star_polygon_16-5.svg|60px]]<BR>{16/5}
|[[File:Regular_star_polygon_16-7.svg|60px]]<BR>{16/7}
|- align=center
|[[File:Regular_star_polygon_17-2.svg|60px]]<BR>'''{17/2}'''
|[[File:Regular_star_polygon_17-3.svg|60px]]<BR>{17/3}
|[[File:Regular_star_polygon_17-4.svg|60px]]<BR>{17/4}
|[[File:Regular_star_polygon_17-5.svg|60px]]<BR>{17/5}
|[[File:Regular_star_polygon_17-6.svg|60px]]<BR>{17/6}
|[[File:Regular_star_polygon_17-7.svg|60px]]<BR>{17/7}
|[[File:Regular_star_polygon_17-8.svg|60px]]<BR>{17/8}
|[[File:Regular_star_polygon_18-5.svg|60px]]<BR>'''{18/5}'''
|[[File:Regular_star_polygon_18-7.svg|60px]]<BR>{18/7}
|- align=center
|[[File:Regular_star_polygon_19-2.svg|60px]]<BR>'''{19/2}'''
|[[File:Regular_star_polygon_19-3.svg|60px]]<BR>{19/3}
|[[File:Regular_star_polygon_19-4.svg|60px]]<BR>{19/4}
|[[File:Regular_star_polygon_19-5.svg|60px]]<BR>{19/5}
|[[File:Regular_star_polygon_19-6.svg|60px]]<BR>{19/6}
|[[File:Regular_star_polygon_19-7.svg|60px]]<BR>{19/7}
|[[File:Regular_star_polygon_19-8.svg|60px]]<BR>{19/8}
|[[File:Regular_star_polygon_19-9.svg|60px]]<BR>{19/9}
|[[File:Regular_star_polygon_20-3.svg|60px]]<BR>'''{20/3}'''
|[[File:Regular_star_polygon_20-7.svg|60px]]<BR>{20/7}
|[[File:Regular_star_polygon_20-9.svg|60px]]<BR>{20/9}
|}

Star polygons that can only exist as spherical tilings, similarly to the monogon and digon, may exist (for example: {3/2}, {5/3}, {5/4}, {7/4}, {9/5}), however these do not appear to have been studied in detail.

There also exist failed star polygons, such as the ''piangle'', which do not cover the surface of a circle finitely many times.<ref name=":0">{{Cite web |url=http://chalkdustmagazine.com/blog/fractional-polygons/ |title=Between a square rock and a hard pentagon: Fractional polygons |last=Duncan |first=Hugh |date=28 September 2017 |work=chalkdust}}</ref>

===Skew polygons ===
In addition to the planar regular polygons there are infinitely many regular skew polygons. Skew polygons can be created via the blending operation.

The blend of two polygons {{mvar|P}} and {{mvar|Q}}, written {{math|{{mvar|P}}#{{mvar|Q}}}}, can be constructed as follows:
# take the cartesian product of their vertices {{math|{{mvar|V}}{{Sub|{{mvar|P}}}}&times;{{mvar|V}}{{sub|{{mvar|Q}}}}}}.
# add edges {{math|({{mvar|p}}{{sub|0}}&times;{{mvar|q}}{{sub|0}}, {{mvar|p}}{{sub|1}}&times;{{mvar|q}}{{sub|1}})}} where {{math|({{mvar|p}}{{sub|0}}, {{mvar|p}}{{sub|1}})}} is an edge of {{mvar|P}} and {{math|({{mvar|q}}{{sub|0}}, {{mvar|q}}{{sub|1}})}} is an edge of {{mvar|Q}}.
# select an arbitrary connected component of the result.
Alternatively, the blend is the polygon {{mvar|{{angbr|{{mvar|&rho;}}{{sub|0}}{{mvar|&sigma;}}{{sub|0}}, {{mvar|&rho;}}{{sub|1}}{{mvar|&sigma;}}{{sub|1}}}}}} where {{mvar|&rho;}} and {{mvar|&sigma;}} are the generating mirrors of {{mvar|P}} and {{mvar|Q}} placed in orthogonal subspaces.{{sfn|McMullen|Schulte|2002}}
The blending operation is commutative, associative and idempotent.

Every regular skew polygon can be expressed as the blend of a unique{{efn|up to identity and idempotency}} set of planar polygons.{{sfn|McMullen|Schulte|2002}} If {{mvar|P}} and {{mvar|Q}} share no factors then {{math|Dim({{mvar|P}}#{{mvar|Q}}) {{=}} Dim({{mvar|P}}) + Dim({{mvar|Q}})}}.

====In 3 space====
The regular finite polygons in 3 dimensions are exactly the blends of the planar polygons (dimension 2) with the digon (dimension 1). They have vertices corresponding to a prism ({{math|{{{mvar|n}}/{{mvar|m}}}#{{(}}{{)}}}} where {{mvar|n}} is odd) or an antiprism ({{math|{{{mvar|n}}/{{mvar|m}}}#{{(}}{{)}}}} where {{mvar|n}} is even). All polygons in 3 space have an even number of vertices and edges.

Several of these appear as the Petrie polygons of regular polyhedra.

====In 4 space====
The regular finite polygons in 4 dimensions are exactly the polygons formed as a blend of two distinct planar polygons. They have vertices lying on a [[Clifford torus]] and related by a Clifford displacement. Unlike 3-dimensional polygons, skew polygons on double rotations can include an odd-number of sides.

==3-polytopes (polyhedra)==

Polytopes of rank 3 are called polyhedra:

A regular polyhedron with [[Schläfli symbol]] {{math|{{{mvar|p}},{{mvar|q}}{{)}}}}, Coxeter diagrams {{CDD|node_1|p|node|q|node}}, has a regular face type {{math|{{{mvar|p}}{{)}}}}, and regular [[Vertex figure|vertex figure]] {{math|{{{mvar|1}}{{)}}}}.

A [[Vertex figure|vertex figure]] (of a polyhedron) is a polygon, seen by connecting those vertices which are one edge away from a given vertex. For [[Regular polyhedron|regular polyhedra]], this vertex figure is always a regular (and planar) polygon.

Existence of a regular polyhedron {{math|{{{mvar|p}},{{mvar|q}}{{)}}}} is constrained by an inequality, related to the vertex figure's angle defect:
<math display=block>
\begin{align}
& \frac{1}{p} + \frac{1}{q} > \frac{1}{2} : \text{Polyhedron (existing in Euclidean 3-space)} \\[6pt]
& \frac{1}{p} + \frac{1}{q} = \frac{1}{2} : \text{Euclidean plane tiling} \\[6pt]
& \frac{1}{p} + \frac{1}{q} < \frac{1}{2} : \text{Hyperbolic plane tiling}
\end{align}
</math>

By enumerating the [[Permutation|permutation]]s, we find five convex forms, four star forms and three plane tilings, all with polygons {{math|{{{mvar|p}}{{)}}}} and {{math|{{{mvar|q}}{{)}}}} limited to: {3}, {4}, {5}, {5/2}, and {6}.

Beyond Euclidean space, there is an infinite set of regular hyperbolic tilings.

===Convex===
The five convex regular [[Polyhedron|polyhedra]] are called the [[Platonic solid]]s. The [[Vertex figure|vertex figure]] is given with each vertex count. All these polyhedra have an [[Euler characteristic]] (χ) of 2.

{| class="wikitable" style="text-align:center;"
|-
!Name
![[Schläfli symbol|Schläfli]]<br />{{math|{{{mvar|p}},{{mvar|q}}{{)}}}}
!Coxeter<br />{{CDD|node_1|p|node|q|node}}
!Image<BR>(solid)
!Image<BR>(sphere)
![[Face (geometry)|Faces]]<br />{p}
![[Edge (geometry)|Edges]]
![[Vertex (geometry)|Vertices]]<br />{q}
![[Symmetry group|Symmetry]]
![[Dual polyhedron|Dual]]
|- bgcolor="#e0e0e0"
|[[Tetrahedron]]<br />([[Simplex|3-simplex]])<!--<br/>(tet)-->
|{3,3}
|{{CDD|node_1|3|node|3|node}}
|[[File:Polyhedron 4b.png|76px]]
|[[File:Uniform tiling 332-t2.png|76px]]
|4<br />{3}
|6
|4<br />{3}
|T<sub>d</sub><BR>[3,3]<BR>(*332)
|(self)
|- bgcolor="#ffe0e0"
|Hexahedron<BR>[[Cube]] <br />([[Hypercube|3-cube]])<!--<br/>(cube)-->
|{4,3}
|{{CDD|node_1|4|node|3|node}}
|[[File:Polyhedron 6.png|76px]]
|[[File:Uniform tiling 432-t0.png|76px]]
|6<br />{4}
|12
|8<br />{3}
|O<sub>h</sub><BR>[4,3]<BR>(*432)
|Octahedron
|- bgcolor="#e0e0ff"
|[[Octahedron]]<br />([[Cross-polytope|3-orthoplex]])<!--<br/>(oct)-->
|{3,4}
|{{CDD|node_1|3|node|4|node}}
|[[File:Polyhedron 8.png|76px]]
|[[File:Uniform tiling 432-t2.png|76px]]
|8<br />{3}
|12
|6<br />{4}
|O<sub>h</sub><BR>[4,3]<BR>(*432)
|Cube
|- bgcolor="#ffe0e0"
|[[Dodecahedron]]<!--<br/>(doe)-->
|{5,3}
|{{CDD|node_1|5|node|3|node}}
|[[File:Polyhedron 12.png|76px]]
|[[File:Uniform tiling 532-t0.png|76px]]
|12<br />{5}
|30
|20<br />{3}
|I<sub>h</sub><BR>[5,3]<BR>(*532)
|Icosahedron
|- bgcolor="#e0e0ff"
|[[Icosahedron]]<!--<br/>(ike)-->
|{3,5}
|{{CDD|node_1|3|node|5|node}}
|[[File:Polyhedron 20.png|76px]]
|[[File:Uniform tiling 532-t2.png|76px]]
|20<br />{3}
|30
|12<br />{5}
|I<sub>h</sub><BR>[5,3]<BR>(*532)
|Dodecahedron
|}

====Spherical====

In [[Spherical geometry|spherical geometry]], regular spherical polyhedra ([[Tessellation|tiling]]s of the [[Sphere|sphere]]) exist that would otherwise be degenerate as polytopes. These are the [[Hosohedron|hosohedra]] {2,n} and their dual [[Dihedron|dihedra]] {n,2}. Coxeter calls these cases "improper" tessellations.{{sfnp|Coxeter|1973|pp=66-67}}

The first few cases (n from 2 to 6) are listed below.
{| class="wikitable" style="text-align:center;"
|+ Hosohedra
|- valign="top"
!Name
![[Schläfli symbol|Schläfli]]<br />{2,p}
!Coxeter<BR>diagram
!Image<BR>(sphere)
![[Face (geometry)|Faces]]<br />{2}<sub>&pi;/p</sub>
![[Edge (geometry)|Edges]]
![[Vertex (geometry)|Vertices]]<br />{p}
![[List of spherical symmetry groups|Symmetry]]
![[Dual polyhedron|Dual]]
|- bgcolor="#e0e0e0"
|Digonal hosohedron
|{2,2}
|{{CDD|node_1|2x|node|2x|node}}
|[[File:Spherical digonal hosohedron.svg|75px]]
|2<BR>{2}<sub>&pi;/2</sub>
|2
|2<BR>{2}<sub>&pi;/2</sub>
|D<sub>2h</sub><BR>[2,2]<BR>(*222)
|Self
|- bgcolor="#ffe0e0"
|Trigonal hosohedron
|{2,3}
|{{CDD|node_1|2x|node|3|node}}
|[[File:Spherical trigonal hosohedron.svg|75px]]
|3<BR>{2}<sub>&pi;/3</sub>
|3
|2<BR>{3}
|D<sub>3h</sub><BR>[2,3]<BR>(*322)
|Trigonal dihedron
|- bgcolor="#ffe0e0"
|Square hosohedron
|{2,4}
|{{CDD|node_1|2x|node|4|node}}
|[[File:Spherical_square_hosohedron.svg|75px]]
|4<BR>{2}<sub>&pi;/4</sub>
|4
|2<BR>{4}
|D<sub>4h</sub><BR>[2,4]<BR>(*422)
|Square dihedron
|- bgcolor="#ffe0e0"
|Pentagonal hosohedron
|{2,5}
|{{CDD|node_1|2x|node|5|node}}
|[[File:Spherical pentagonal hosohedron.svg|75px]]
|5<BR>{2}<sub>&pi;/5</sub>
|5
|2<BR>{5}
|D<sub>5h</sub><BR>[2,5]<BR>(*522)
|Pentagonal dihedron
|- bgcolor="#ffe0e0"
|Hexagonal hosohedron
|{2,6}
|{{CDD|node_1|2x|node|6|node}}
|[[File:Spherical hexagonal hosohedron.svg|75px]]
|6<BR>{2}<sub>&pi;/6</sub>
|6
|2<BR>{6}
|D<sub>6h</sub><BR>[2,6]<BR>(*622)
|Hexagonal dihedron
|}

{| class="wikitable" style="text-align:center;"
|+ Dihedra
|- valign="top"
!Name
![[Schläfli symbol|Schläfli]]<br />{p,2}
!Coxeter<BR>diagram
!Image<BR>(sphere)
![[Face (geometry)|Faces]]<br />{p}
![[Edge (geometry)|Edges]]
![[Vertex (geometry)|Vertices]]<br />{2}
![[List of spherical symmetry groups|Symmetry]]
![[Dual polyhedron|Dual]]
|- bgcolor="#e0e0e0"
|Digonal dihedron
|{2,2}
|{{CDD|node_1|2x|node|2x|node}}
|[[File:Digonal dihedron.png|75px]]
|2<BR>{2}<sub>&pi;/2</sub>
|2
|2<BR>{2}<sub>&pi;/2</sub>
|D<sub>2h</sub><BR>[2,2]<BR>(*222)
|Self
|- bgcolor="#e0e0ff"
|Trigonal dihedron
|{3,2}
|{{CDD|node_1|3|node|2x|node}}
|[[File:Trigonal dihedron.png|75px]]
|2<BR>{3}
|3
|3<BR>{2}<sub>&pi;/3</sub>
|D<sub>3h</sub><BR>[3,2]<BR>(*322)
|Trigonal hosohedron
|- bgcolor="#e0e0ff"
|Square dihedron
|{4,2}
|{{CDD|node_1|4|node|2x|node}}
|[[File:Tetragonal_dihedron.png|75px]]
|2<BR>{4}
|4
|4<BR>{2}<sub>&pi;/4</sub>
|D<sub>4h</sub><BR>[4,2]<BR>(*422)
|Square hosohedron
|- bgcolor="#e0e0ff"
|Pentagonal dihedron
|{5,2}
|{{CDD|node_1|5|node|2x|node}}
|[[File:Pentagonal dihedron.png|75px]]
|2<BR>{5}
|5
|5<BR>{2}<sub>&pi;/5</sub>
|D<sub>5h</sub><BR>[5,2]<BR>(*522)
|Pentagonal hosohedron
|- bgcolor="#e0e0ff"
|Hexagonal dihedron
|{6,2}
|{{CDD|node_1|6|node|2x|node}}
|[[File:Hexagonal dihedron.png|75px]]
|2<BR>{6}
|6
|6<BR>{2}<sub>&pi;/6</sub>
|D<sub>6h</sub><BR>[6,2]<BR>(*622)
|Hexagonal hosohedron
|}
Star-dihedra and hosohedra {''p''/''q'',2} and {2,''p''/''q''} also exist for any star polygon {''p''/''q''}.

===Stars===
The regular [[Star polyhedron|star polyhedra]] are called the Kepler–Poinsot polyhedra and there are four of them, based on the [[Vertex arrangement|vertex arrangement]]s of the [[Dodecahedron|dodecahedron]] {5,3} and [[Icosahedron|icosahedron]] {3,5}:

As spherical tilings, these star forms overlap the sphere multiple times, called its ''density'', being 3 or 7 for these forms. The tiling images show a single spherical polygon face in yellow.
{| class="wikitable"
|-
!Name
!Image<BR>(skeletonic)
!Image<BR>(solid)
!Image<BR>(sphere)
![[Stellation]]<BR>diagram
![[Schläfli symbol|Schläfli]]<br />{p,q} and<br />Coxeter
!Faces<br />{p}
!Edges
!Vertices<br />{q}<br />[[Vertex figure|verf.]]
![[Euler characteristic|χ]]
![[Density (polytope)|Density]]
![[Symmetry group|Symmetry]]
![[Dual polyhedron|Dual]]
|- BGCOLOR="#ffe0e0" align=center
|[[Small stellated dodecahedron]]<!--<br/>(sissid)-->
|[[Image:Skeleton St12, size m.png|80px]]
|[[Image:Small stellated dodecahedron (gray with yellow face).svg|80px]]
|[[Image:Small stellated dodecahedron tiling.png|80px]]
|[[File:First stellation of dodecahedron facets.svg|80px]]
|{5/2,5}<br />{{CDD|node|5|node|5|rat|d2|node_1}}
|12<br />{5/2}<br />[[Image:Star polygon 5-2.svg|30px]]
|30||12<br />{5}<br />[[Image:Regular pentagon.svg|30px]]
| −6||3||I<sub>h</sub><BR>[5,3]<BR>(*532)||Great dodecahedron
|- BGCOLOR="#e0e0ff" align=center
|[[Great dodecahedron]]<!--<br/>(gad)-->
|[[Image:Skeleton Gr12, size m.png|80px]]
|[[Image:Great dodecahedron (gray with yellow face).svg|80px]]
|[[Image:Great dodecahedron tiling.svg|80px]]
|[[File:Second stellation of dodecahedron facets.svg|80px]]
|{5,5/2}<br />{{CDD|node_1|5|node|5|rat|d2|node}}
|12<br />{5}<br />[[Image:Regular pentagon.svg|30px]]
|30||12<br />{5/2}<br />[[Image:Star polygon 5-2.svg|30px]]
| −6||3||I<sub>h</sub><BR>[5,3]<BR>(*532)||Small stellated dodecahedron
|- BGCOLOR="#ffe0e0" align=center
|[[Great stellated dodecahedron]]<!--<br/>(gissid)-->
|[[Image:Skeleton GrSt12, size s.png|80px]]
|[[Image:Great stellated dodecahedron (gray with yellow face).svg|80px]]
|[[Image:Great stellated dodecahedron tiling.svg|80px]]
|[[File:Third stellation of dodecahedron facets.svg|80px]]
|{5/2,3}<br />{{CDD|node|3|node|5|rat|d2|node_1}}
|12<br />{5/2}<br />[[Image:Star polygon 5-2.svg|30px]]
|30||20<br />{3}<br />[[Image:Regular triangle.svg|30px]]
|2||7||I<sub>h</sub><BR>[5,3]<BR>(*532)||Great icosahedron
|- BGCOLOR="#e0e0ff" align=center
|[[Great icosahedron]]<!--<br/>(gike)-->
|[[Image:Skeleton Gr20, size m.png|80px]]
|[[Image:Great icosahedron (gray with yellow face).svg|80px]]
|[[Image:Great icosahedron tiling.svg|80px]]
|[[File:Great icosahedron stellation facets.svg|80px]]
|{3,5/2}<br />{{CDD|node_1|3|node|5|rat|d2|node}}
|20<br />{3}<br />[[Image:Regular triangle.svg|30px]]
|30||12<br />{5/2}<br />[[Image:Star polygon 5-2.svg|30px]]
|2||7||I<sub>h</sub><BR>[5,3]<BR>(*532)||Great stellated dodecahedron
|}

There are infinitely many failed star polyhedra. These are also spherical tilings with star polygons in their Schläfli symbols, but they do not cover a sphere finitely many times. Some examples are {5/2,4}, {5/2,9}, {7/2,3}, {5/2,5/2}, {7/2,7/3}, {4,5/2}, and {3,7/3}.

=== Skew polyhedra ===
[[Regular skew polyhedron|Regular skew polyhedra]] are generalizations to the set of [[Regular polyhedron|regular polyhedron]] which include the possibility of nonplanar [[Vertex figure|vertex figure]]s.

For 4-dimensional skew polyhedra, Coxeter offered a modified [[Schläfli symbol]] {l,m|n} for these figures, with {l,m} implying the [[Vertex figure|vertex figure]], ''m'' l-gons around a vertex, and {{mvar|n}}-gonal holes. Their vertex figures are [[Skew polygon|skew polygon]]s, zig-zagging between two planes.

The regular skew polyhedra, represented by {l,m|n}, follow this equation:
: 2 sin(π/l) sin(π/m) = cos(π/n)

Four of them can be seen in 4-dimensions as a subset of faces of four [[Regular 4-polytope|regular 4-polytope]]s, sharing the same [[Vertex arrangement|vertex arrangement]] and edge arrangement:

{| class=wikitable 
|[[File:4-simplex t03.svg|100px]]
|[[File:4-simplex t12.svg|100px]]
|[[File:24-cell t03 F4.svg|100px]]
|[[File:24-cell t12 F4.svg|100px]]
|-
!{4, 6 &#124; 3}
!{6, 4 &#124; 3}
!{4, 8 &#124; 3}
!{8, 4 &#124; 3}
|}

==4-polytopes==

Regular 4-polytopes with [[Schläfli symbol]] <math>\{p,q,r\}</math> have cells of type <math>\{p,q\}</math>, faces of type <math>\{p\}</math>, edge figures
<math>\{r\}</math>, and vertex figures <math>\{q,r\}</math>.
* A [[Vertex figure|vertex figure]] (of a 4-polytope) is a polyhedron, seen by the arrangement of neighboring vertices around a given vertex. For regular 4-polytopes, this vertex figure is a regular polyhedron.
* An [[Vertex figure#Edge figure|edge figure]] is a polygon, seen by the arrangement of faces around an edge. For regular 4-polytopes, this edge figure will always be a regular polygon.

The existence of a regular 4-polytope <math>\{p,q,r\}</math> is constrained by the existence of the regular polyhedra <math>\{p,q\}, \{q,r\}</math>. A suggested name for 4-polytopes is "polychoron".<ref>{{cite conference |conference=Convex and Abstract Polytopes (May 19–21, 2005) and Polytopes Day in Calgary (May 22, 2005) |title=Abstracts |url=http://www.mit.edu/~hlb/Associahedron/program.pdf}}</ref>

Each will exist in a space dependent upon this expression:
: <math>\sin \left ( \frac{\pi}{p} \right ) \sin \left(\frac{\pi}{r}\right) - \cos\left(\frac{\pi}{q}\right)</math>
:: <math>> 0</math> : Hyperspherical 3-space honeycomb or 4-polytope
:: <math>= 0</math> : Euclidean 3-space honeycomb
:: <math>< 0</math> : Hyperbolic 3-space honeycomb

These constraints allow for 21 forms: 6 are convex, 10 are nonconvex, ''one'' is a Euclidean 3-space honeycomb, and 4 are hyperbolic honeycombs.

The [[Euler characteristic]] <math>\chi</math> for convex 4-polytopes is zero:
<math>\chi = V+F-E-C = 0</math>

===Convex===

The 6 convex [[Regular 4-polytope|regular 4-polytope]]s are shown in the table below. All these 4-polytopes have an [[Euler characteristic]] (χ) of 0.

{| class="wikitable"
! Name<br />
! [[Schläfli symbol|Schläfli]]<br />{p,q,r}
! Coxeter<br />{{CDD|node|p|node|q|node|r|node}}
! Cells<br /> {p,q}
! [[Face (geometry)|Faces]]<br /> {p}
! [[Edge (geometry)|Edges]]<br /> {r}
! [[Vertex (geometry)|Vertices]]<br /> {q,r}
! Dual<br /> {r,q,p}
|- BGCOLOR="#e0e0e0" align=center
| [[5-cell|5-cell]]<br />([[Simplex|4-simplex]])
| {3,3,3}
|{{CDD|node_1|3|node|3|node|3|node}}
| 5<br />{3,3}
| 10<br />{3}
| 10<br /> {3}
| 5<br /> {3,3}
| (self)
|- BGCOLOR="#ffe0e0" align=center
| 8-cell<br />([[Hypercube|4-cube]])<br />(Tesseract)
| {4,3,3}
|{{CDD|node_1|4|node|3|node|3|node}}
| 8<br /> {4,3}
| 24<br /> {4}
| 32<br /> {3}
| 16<br /> {3,3}
| 16-cell
|- BGCOLOR="#e0e0ff" align=center
| [[16-cell|16-cell]]<br />([[Cross-polytope|4-orthoplex]])
| {3,3,4}
|{{CDD|node_1|3|node|3|node|4|node}}
| 16<br /> {3,3}
| 32<br /> {3}
| 24<br /> {4}
| 8<br /> {3,4}
| Tesseract
|- BGCOLOR="#e0e0e0" align=center
| [[24-cell|24-cell]]
| {3,4,3}
|{{CDD|node_1|3|node|4|node|3|node}}
| 24<br /> {3,4}
| 96<br /> {3}
| 96<br /> {3}
| 24<br /> {4,3}
| (self)
|- BGCOLOR="#ffe0e0" align=center
| [[120-cell|120-cell]]
| {5,3,3}
|{{CDD|node_1|5|node|3|node|3|node}}
| 120<br /> {5,3}
| 720<br /> {5}
| 1200<br /> {3}
| 600<br /> {3,3}
| 600-cell
|- BGCOLOR="#e0e0ff" align=center
| [[600-cell|600-cell]]
| {3,3,5}
|{{CDD|node_1|3|node|3|node|5|node}}
| 600<br /> {3,3}
| 1200<br /> {3}
| 720<br /> {5}
| 120<br /> {3,5}
| 120-cell
|}

{| class="wikitable"
|-
! [[5-cell|5-cell]] || 8-cell || [[16-cell|16-cell]] || [[24-cell|24-cell]] || [[120-cell|120-cell]] || [[600-cell|600-cell]]
|-
! {3,3,3} || {4,3,3} || {3,3,4} || {3,4,3} || {5,3,3} || {3,3,5}
|-
!colspan=6|Wireframe ([[Petrie polygon]]) skew [[Orthographic projection|orthographic projection]]s
|-
| [[File:Complete graph K5.svg|105px]]
| [[File:4-cube graph.svg|105px]]
| [[File:4-orthoplex.svg|105px]]
| [[File:24-cell graph F4.svg|105px]]
| [[File:Cell120Petrie.svg|105px]]
| [[File:Cell600Petrie.svg|105px]]
|-
!colspan=6|Solid [[Orthographic projection|orthographic projection]]s
|-
| [[Image:Tetrahedron.png|105px]]<BR>[[Tetrahedron|tetrahedral<BR>envelope]] (cell/<BR>vertex-centered)
| [[Image:Hexahedron.png|105px]]<BR>[[Cube|cubic envelope]]<BR>(cell-centered)
| [[File:16-cell ortho cell-centered.png|105px]]<BR>[[Cube|cubic envelope]]<BR>(cell-centered)
| [[Image:Ortho solid 24-cell.png|105px]]<BR>[[Cuboctahedron|cuboctahedral<BR>envelope]]<BR>(cell-centered)
| [[Image:Ortho solid 120-cell.png|105px]]<BR>truncated rhombic<BR>triacontahedron<BR>envelope<BR>(cell-centered)
| [[Image:Ortho solid 600-cell.png|105px]]<BR>Pentakis<BR>icosidodecahedral<BR>envelope<BR>(vertex-centered)
|-
!colspan=6|Wireframe [[Schlegel diagram]]s (Perspective projection)
|-
| [[Image:Schlegel wireframe 5-cell.png|105px]]<BR>(cell-centered)
| [[Image:Schlegel wireframe 8-cell.png|105px]]<BR>(cell-centered)
| [[Image:Schlegel wireframe 16-cell.png|105px]]<BR>(cell-centered)
| [[Image:Schlegel wireframe 24-cell.png|105px]]<BR>(cell-centered)
| [[Image:Schlegel wireframe 120-cell.png|105px]]<BR>(cell-centered)
| [[Image:Schlegel wireframe 600-cell vertex-centered.png|105px]]<BR>(vertex-centered)
|-
!colspan=6|Wireframe [[Stereographic projection|stereographic projection]]s ([[3-sphere|Hyperspherical]])
|-
| [[Image:Stereographic polytope 5cell.png|105px]]
| [[Image:Stereographic polytope 8cell.png|105px]]
| [[Image:Stereographic polytope 16cell.png|105px]]
| [[Image:Stereographic polytope 24cell.png|105px]]
| [[Image:Stereographic polytope 120cell.png|105px]]
| [[Image:Stereographic polytope 600cell.png|105px]]
|}

====Spherical====

Di-4-topes and hoso-4-topes exist as regular tessellations of the [[3-sphere|3-sphere]].

Regular '''di-4-topes''' (2 facets) include: {3,3,2}, {3,4,2}, {4,3,2}, {5,3,2}, {3,5,2}, {p,2,2}, and their '''hoso-4-tope''' duals (2 vertices): {2,3,3}, {2,4,3}, {2,3,4}, {2,3,5}, {2,5,3}, {2,2,''p''}. 4-polytopes of the form {2,''p'',2} are the same as {2,2,''p''}. There are also the cases {''p'',2,''q''} which have dihedral cells and hosohedral vertex figures.

{| class="wikitable" style="text-align:center;" width=640
|+ Regular hoso-4-topes as [[3-sphere|3-sphere]] honeycombs
|-
![[Schläfli symbol|Schläfli]]<br />{2,''p'',''q''}
!Coxeter<br />{{CDD|node_1|2x|node|p|node|q|node}}
!Cells<br />{2,''p''}<sub>&pi;/''q''</sub>
![[Face (geometry)|Faces]]<br />{2}<sub>&pi;/''p'',&pi;/''q''</sub>
![[Edge (geometry)|Edges]]
![[Vertex (geometry)|Vertices]]
![[Vertex figure]]<BR>{''p'',''q''}
![[Symmetry group|Symmetry]]
![[Dual polyhedron|Dual]]
|- bgcolor="#e0e0e0"
|{2,3,3}
|{{CDD|node_1|2x|node|3|node|3|node}}
|4<BR>{2,3}<sub>&pi;/3</sub><BR>[[File:Spherical trigonal hosohedron.svg|50px]]
|6<br />{2}<sub>&pi;/3,&pi;/3</sub>
|4
|2
|{3,3}<BR>[[File:Uniform tiling 332-t0-1-.png|75px]]
|[2,3,3]
|{3,3,2}
|- bgcolor="#ffe0e0"
|{2,4,3}
|{{CDD|node_1|2x|node|4|node|3|node}}
|6<BR>{2,4}<sub>&pi;/3</sub><BR>[[File:Spherical square hosohedron.svg|50px]]
|12<br />{2}<sub>&pi;/4,&pi;/3</sub>
|8
|2
|{4,3}<BR>[[File:Uniform tiling 432-t0.png|75px]]
|[2,4,3]
|{3,4,2}
|- bgcolor="#e0e0ff"
|{2,3,4}
|{{CDD|node_1|2x|node|3|node|4|node}}
|8<BR>{2,3}<sub>&pi;/4</sub><BR>[[File:Spherical trigonal hosohedron.svg|50px]]
|12<br />{2}<sub>&pi;/3,&pi;/4</sub>
|6
|2
|{3,4}<BR>[[File:Uniform tiling 432-t2.png|75px]]
|[2,4,3]
|{4,3,2}
|- bgcolor="#ffe0e0"
|{2,5,3}
|{{CDD|node_1|2x|node|5|node|3|node}}
|12<BR>{2,5}<sub>&pi;/3</sub><BR>[[File:Spherical trigonal hosohedron.svg|50px]]
|30<br />{2}<sub>&pi;/5,&pi;/3</sub>
|20
|2
|{5,3}<BR>[[File:Uniform tiling 532-t0.png|75px]]
|[2,5,3]
|{3,5,2}
|- bgcolor="#e0e0ff"
|{2,3,5}
|{{CDD|node_1|2x|node|3|node|5|node}}
|20<BR>{2,3}<sub>&pi;/5</sub><BR>[[File:Spherical pentagonal hosohedron.svg|50px]]
|30<br />{2}<sub>&pi;/3,&pi;/5</sub>
|12
|2
|{3,5}<BR>[[File:Uniform tiling 532-t2.png|75px]]
|[2,5,3]
|{5,3,2}
|}

===Stars===
There are ten regular star 4-polytopes, which are called the Schläfli–Hess 4-polytopes.  Their vertices are based on the convex [[120-cell|120-cell]] ''{5,3,3}'' and [[600-cell|600-cell]] ''{3,3,5}''.

[[Biography:Ludwig Schläfli|Ludwig Schläfli]] found four of them and skipped the last six because he would not allow forms that failed the [[Euler characteristic]] on cells or vertex figures (for zero-hole tori: F+V−E=2). Edmund Hess (1843–1903) completed the full list of ten in his German book ''Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder'' (1883)[http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=ABN8623.0001.001].

There are 4 unique edge arrangements and 7 unique face arrangements from these 10 regular star 4-polytopes, shown as orthogonal projections:

{| class="wikitable"
! Name<BR>
! Wireframe
! Solid
! [[Schläfli symbol|Schläfli]]<BR>{p, q, r}<BR>Coxeter
! Cells<BR>{p, q}
! Faces<BR>{p}
! Edges<BR>{r}
! Vertices<BR>{q, r}
![[Density (polytope)|Density]]
! [[Euler characteristic|χ]]
![[Coxeter group|Symmetry group]]
! Dual<BR>{r, q,p}
|- align=center BGCOLOR="#e0e0ff"
| [[Icosahedral 120-cell]]<br>(faceted 600-cell)
| [[Image:Schläfli-Hess polychoron-wireframe-3.png|75px]]
| [[Image:ortho solid 007-uniform polychoron 35p-t0.png|75px]]
| {3,5,5/2}<BR>{{CDD|node_1|3|node|5|node|5|rat|d2|node}}
| 120<BR>[[Icosahedron|{3,5}]]<BR>[[Image:Icosahedron.png|25px]]
| 1200<BR>[[Triangle|{3}]]<BR>[[Image:Regular triangle.svg|25px]]
| 720<BR>[[Pentagram|{5/2}]]<BR>[[Image:Star polygon 5-2.svg|25px]]
| 120<BR>[[Great dodecahedron|{5,5/2}]]<BR>[[Image:Great dodecahedron.png|25px]]
| 4
| 480
| ''H''<sub>4</sub><BR>[5,3,3]
| Small stellated 120-cell
|- align=center BGCOLOR="#ffe0e0"
| [[Small stellated 120-cell]]
| [[Image:Schläfli-Hess polychoron-wireframe-2.png|75px]]
| [[Image:ortho solid 010-uniform polychoron p53-t0.png|75px]]
| {5/2,5,3}<BR>{{CDD|node|3|node|5|node|5|rat|d2|node_1}}
| 120<BR>[[Small stellated dodecahedron|{5/2,5}]]<BR>[[Image:Small stellated dodecahedron.png|25px]]
| 720<BR>[[Pentagram|{5/2}]]<BR>[[Image:Star polygon 5-2.svg|25px]]
| 1200<BR>[[Triangle|{3}]]<BR>[[Image:Regular triangle.svg|25px]]
| 120<BR>[[Dodecahedron|{5,3}]]<BR>[[Image:Dodecahedron.png|25px]]
| 4
| −480
| ''H''<sub>4</sub><BR>[5,3,3]
| Icosahedral 120-cell
|- align=center BGCOLOR="#e0ffe0"
| [[Great 120-cell]]
| [[Image:Schläfli-Hess polychoron-wireframe-3.png|75px]]
| [[Image:ortho solid 008-uniform polychoron 5p5-t0.png|75px]]
| {5,5/2,5}<BR>{{CDD|node_1|5|node|5|rat|d2|node|5|node}}
| 120<BR>[[Great dodecahedron|{5,5/2}]]<BR>[[Image:Great dodecahedron.png|25px]]
| 720<BR>[[Pentagon|{5}]]<BR>[[Image:Regular pentagon.svg|25px]]
| 720<BR>[[Pentagon|{5}]]<BR>[[Image:Regular pentagon.svg|25px]]
| 120<BR>[[Small stellated dodecahedron|{5/2,5}]]<BR>[[Image:Small stellated dodecahedron.png|25px]]
| 6
| 0
| ''H''<sub>4</sub><BR>[5,3,3]
| Self-dual
|- align=center BGCOLOR="#e0e0ff"
| [[Grand 120-cell]]
| [[Image:Schläfli-Hess polychoron-wireframe-3.png|75px]]
| [[Image:ortho solid 009-uniform polychoron 53p-t0.png|75px]]
| {5,3,5/2}<BR>{{CDD|node_1|5|node|3|node|5|rat|d2|node}}
| 120<BR>[[Dodecahedron|{5,3}]]<BR>[[Image:Dodecahedron.png|25px]]
| 720<BR>[[Pentagon|{5}]]<BR>[[Image:Regular pentagon.svg|25px]]
| 720<BR>[[Pentagram|{5/2}]]<BR>[[Image:Star polygon 5-2.svg|25px]]
| 120<BR>[[Great icosahedron|{3,5/2}]]<BR>[[Image:Great icosahedron.png|25px]]
| 20
| 0
| ''H''<sub>4</sub><BR>[5,3,3]
| Great stellated 120-cell
|- align=center BGCOLOR="#ffe0e0"
| [[Great stellated 120-cell]]
| [[Image:Schläfli-Hess polychoron-wireframe-4.png|75px]]
| [[Image:ortho solid 012-uniform polychoron p35-t0.png|75px]]
| {5/2,3,5}<BR>{{CDD|node|5|node|3|node|5|rat|d2|node_1}}
| 120<BR>[[Great stellated dodecahedron|{5/2,3}]]<BR>[[Image:Great stellated dodecahedron.png|25px]]
| 720<BR>[[Pentagram|{5/2}]]<BR>[[Image:Star polygon 5-2.svg|25px]]
| 720<BR>[[Pentagon|{5}]]<BR>[[Image:Regular pentagon.svg|25px]]
| 120<BR>[[Icosahedron|{3,5}]]<BR>[[Image:Icosahedron.png|25px]]
| 20
| 0
| ''H''<sub>4</sub><BR>[5,3,3]
| Grand 120-cell
|- align=center BGCOLOR="#e0ffe0"
| [[Grand stellated 120-cell]]
| [[Image:Schläfli-Hess polychoron-wireframe-4.png|75px]]
| [[Image:ortho solid 013-uniform polychoron p5p-t0.png|75px]]
| {5/2,5,5/2}<BR>{{CDD|node_1|5|rat|d2|node|5|node|5|rat|d2|node}}
| 120<BR>[[Small stellated dodecahedron|{5/2,5}]]<BR>[[Image:Small stellated dodecahedron.png|25px]]
| 720<BR>[[Pentagram|{5/2}]]<BR>[[Image:Star polygon 5-2.svg|25px]]
| 720<BR>[[Pentagram|{5/2}]]<BR>[[Image:Star polygon 5-2.svg|25px]]
| 120<BR>[[Great dodecahedron|{5,5/2}]]<BR>[[Image:Great dodecahedron.png|25px]]
| 66
| 0
| ''H''<sub>4</sub><BR>[5,3,3]
| Self-dual
|- align=center BGCOLOR="#e0e0ff"
| [[Great grand 120-cell]]
| [[Image:Schläfli-Hess polychoron-wireframe-2.png|75px]]
| [[Image:ortho solid 011-uniform polychoron 53p-t0.png|75px]]
| {5,5/2,3}<BR>{{CDD|node_1|5|node|5|rat|d2|node|3|node}}
| 120<BR>[[Great dodecahedron|{5,5/2}]]<BR>[[Image:Great dodecahedron.png|25px]]
| 720<BR>[[Pentagon|{5}]]<BR>[[Image:Regular pentagon.svg|25px]]
| 1200<BR>[[Triangle|{3}]]<BR>[[Image:Regular triangle.svg|25px]]
| 120<BR>[[Great stellated dodecahedron|{5/2,3}]]<BR>[[Image:Great stellated dodecahedron.png|25px]]
| 76
| −480
| ''H''<sub>4</sub><BR>[5,3,3]
| Great icosahedral 120-cell
|- align=center BGCOLOR="#ffe0e0"
| [[Great icosahedral 120-cell]]<br>(great faceted 600-cell)
| [[Image:Schläfli-Hess polychoron-wireframe-4.png|75px]]
| [[Image:ortho solid 014-uniform polychoron 3p5-t0.png|75px]]
| {3,5/2,5}<BR>{{CDD|node|5|node|5|rat|d2|node|3|node_1}}
| 120<BR>[[Great icosahedron|{3,5/2}]]<BR>[[Image:Great icosahedron.png|25px]]
| 1200<BR>[[Triangle|{3}]]<BR>[[Image:Regular triangle.svg|25px]]
| 720<BR>[[Pentagon|{5}]]<BR>[[Image:Regular pentagon.svg|25px]]
| 120<BR>[[Small stellated dodecahedron|{5/2,5}]]<BR>[[Image:Small stellated dodecahedron.png|25px]]
| 76
| 480
| ''H''<sub>4</sub><BR>[5,3,3]
| Great grand 120-cell
|- align=center BGCOLOR="#e0e0ff"
| [[Grand 600-cell]]
| [[Image:Schläfli-Hess polychoron-wireframe-4.png|75px]]
| [[Image:ortho solid 015-uniform polychoron 33p-t0.png|75px]]
| {3,3,5/2}<BR>{{CDD|node_1|3|node|3|node|5|rat|d2|node}}
| 600<BR>[[Tetrahedron|{3,3}]]<BR>[[Image:Tetrahedron.png|25px]]
| 1200<BR>[[Triangle|{3}]]<BR>[[Image:Regular triangle.svg|25px]]
| 720<BR>[[Pentagram|{5/2}]]<BR>[[Image:Star polygon 5-2.svg|25px]]
| 120<BR>[[Great icosahedron|{3,5/2}]]<BR>[[Image:Great icosahedron.png|25px]]
| 191
| 0
| ''H''<sub>4</sub><BR>[5,3,3]
| Great grand stellated 120-cell
|- align=center BGCOLOR="#ffe0e0"
| [[Great grand stellated 120-cell]]
| [[Image:Schläfli-Hess polychoron-wireframe-1.png|75px]]
| [[Image:ortho solid 016-uniform polychoron p33-t0.png|75px]]
| {5/2,3,3}<BR>{{CDD|node|3|node|3|node|5|rat|d2|node_1}}
| 120<BR>[[Great stellated dodecahedron|{5/2,3}]]<BR>[[Image:Great stellated dodecahedron.png|25px]]
| 720<BR>[[Pentagram|{5/2}]]<BR>[[Image:Star polygon 5-2.svg|25px]]
| 1200<BR>[[Triangle|{3}]]<BR>[[Image:Regular triangle.svg|25px]]
| 600<BR>[[Tetrahedron|{3,3}]]<BR>[[Image:Tetrahedron.png|25px]]
| 191
| 0
| ''H''<sub>4</sub><BR>[5,3,3]
| Grand 600-cell
|}
There are 4 ''failed'' potential regular star 4-polytopes permutations: {3,5/2,3}, {4,3,5/2}, {5/2,3,4}, {5/2,3,5/2}. Their cells and vertex figures exist, but they do not cover a hypersphere with a finite number of repetitions.

===Skew 4-polytopes===
In addition to the 16 planar 4-polytopes above there are 18 finite skew polytopes.{{sfnp|McMullen|2004}} One of these is obtained as the Petrial of the tesseract, and the other 17 can be formed by applying the kappa operation to the planar polytopes and the Petrial of the tesseract.

==Ranks 5 and higher==
5-polytopes can be given the symbol <math>\{p,q,r,s\}</math> where <math>\{p,q,r\}</math> is the 4-face type, <math>\{p,q\}</math> is the cell type, <math>\{p\}</math> is the face type, and <math>\{s\}</math> is the face figure, <math>\{r,s\}</math> is the edge figure, and <math>\{q,r,s\}</math> is the vertex figure.

: A [[Vertex figure|vertex figure]] (of a 5-polytope) is a 4-polytope, seen by the arrangement of neighboring vertices to each vertex.
: An edge figure (of a 5-polytope) is a polyhedron, seen by the arrangement of faces around each edge.
: A face figure (of a 5-polytope) is a polygon, seen by the arrangement of cells around each face.

A regular 5-polytope <math>\{p,q,r,s\}</math> exists only if <math>\{p,q,r\}</math> and <math>\{q,r,s\}</math> are regular 4-polytopes.

The space it fits in is based on the expression:
: <math>\frac{\cos^2\left(\frac{\pi}{q}\right)}{\sin^2\left(\frac{\pi}{p}\right)} + \frac{\cos^2\left(\frac{\pi}{r}\right)}{\sin^2\left(\frac{\pi}{s}\right)}</math>
:: <math>< 1</math> : Spherical 4-space tessellation or 5-space polytope
:: <math>= 1</math> : Euclidean 4-space tessellation
:: <math>> 1</math> : hyperbolic 4-space tessellation

Enumeration of these constraints produce 3 convex polytopes, no star polytopes, 3 tessellations of Euclidean 4-space, and ''5'' tessellations of paracompact hyperbolic 4-space. The only no non-convex regular polytopes for ranks 5 and higher are skews.

===Convex===
In dimensions 5 and higher, there are only three kinds of convex regular polytopes.{{sfnp|Coxeter|1973|loc=Table I: Regular polytopes, (iii) The three regular polytopes in {{mvar|n}} dimensions (n>=5), pp. 294–295}}

{| class="wikitable"
|-
!Name
![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p<sub>1</sub>,...,p<sub><var>n</var>−1</sub>}
!Coxeter
!<var>k</var>-faces
!Facet<BR>type
![[Vertex figure|Vertex<BR>figure]]
!Dual
|- BGCOLOR="#e0e0e0" align=center
|[[Simplex|''n''-simplex]]||{3<sup><var>n</var>−1</sup>}||{{CDD|node_1|3|node|3}}...{{CDD|3|node|3|node}}||[[Binomial coefficient|<math>{{n+1} \choose {k+1}}</math>]]||{3<sup><var>n</var>−2</sup>}||{3<sup><var>n</var>−2</sup>}||Self-dual
|- BGCOLOR="#ffe0e0" align=center
|[[Hypercube|<var>n</var>-cube]]||{4,3<sup><var>n</var>−2</sup>}||{{CDD|node_1|4|node|3}}...{{CDD|3|node|3|node}}||<math>2^{n-k}{n \choose k}</math>||{4,3<sup><var>n</var>−3</sup>}||{3<sup><var>n</var>−2</sup>}||<var>n</var>-orthoplex
|- BGCOLOR="#e0e0ff" align=center
|[[Cross-polytope|<var>n</var>-orthoplex]]||{3<sup><var>n</var>−2</sup>,4}||{{CDD|node_1|3|node|3}}...{{CDD|3|node|4|node}}||<math>2^{k+1}{n \choose {k+1}}</math>||{3<sup><var>n</var>−2</sup>}||{3<sup><var>n</var>−3</sup>,4}||<var>n</var>-cube
|}

There are also improper cases where some numbers in the Schläfli symbol are 2. For example, {p,q,r,...2} is an improper regular spherical polytope whenever {p,q,r...} is a regular spherical polytope, and {2,...p,q,r} is an improper regular spherical polytope whenever {...p,q,r} is a regular spherical polytope. Such polytopes may also be used as facets, yielding forms such as {p,q,...2...y,z}.

====5 dimensions====
{| class="wikitable"
|-
!Name
![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p,q,r,s}<BR>Coxeter
!Facets<BR>{p,q,r}
!Cells<BR>{p,q}
!Faces<BR>{p}
!Edges
!Vertices
!Face<BR>figure<BR>{s}
!Edge<BR>figure<BR>{r,s}
![[Vertex figure|Vertex<BR>figure]]<BR>{q,r,s}
|- BGCOLOR="#e0e0e0" align=center
|[[5-simplex|5-simplex]]
|{3,3,3,3}<BR>{{CDD|node_1|3|node|3|node|3|node|3|node}}||6<BR>{3,3,3}||15<BR>{3,3}||20<BR>{3}||15||6||{3}||{3,3}||{3,3,3}
|- BGCOLOR="#ffe0e0" align=center
|[[5-cube|5-cube]]
|{4,3,3,3}<BR>{{CDD|node_1|4|node|3|node|3|node|3|node}}||10<BR>{4,3,3}||40<BR>{4,3}||80<BR>{4}||80||32||{3}||{3,3}||{3,3,3}
|- BGCOLOR="#e0e0ff" align=center
|[[5-orthoplex|5-orthoplex]]
|{3,3,3,4}<BR>{{CDD|node_1|3|node|3|node|3|node|4|node}}||32<BR>{3,3,3}||80<BR>{3,3}||80<BR>{3}||40||10||{4}||{3,4}||{3,3,4}
|}

{| class=wikitable
|- align=center valign=top
|[[Image:5-simplex t0.svg|150px]]<BR>[[5-simplex|5-simplex]]
|[[Image:5-cube graph.svg|150px]]<BR>[[5-cube|5-cube]]
|[[File:5-orthoplex.svg|150px]]<BR>[[5-orthoplex|5-orthoplex]]
|}

====6 dimensions====
{| class=wikitable
!Name!![[Schläfli symbol|Schläfli]]!!Vertices!!Edges!!Faces!!Cells!!4-faces||5-faces!!χ
|- BGCOLOR="#e0e0e0" align=center
|[[6-simplex|6-simplex]]||{3,3,3,3,3}||7||21||35||35||21||7||0
|- BGCOLOR="#ffe0e0" align=center
|[[6-cube|6-cube]]||{4,3,3,3,3}||64||192||240||160||60||12||0
|- BGCOLOR="#e0e0ff" align=center
|[[6-orthoplex|6-orthoplex]]||{3,3,3,3,4}||12||60||160||240||192||64||0
|}

{| class=wikitable
|- align=center valign=top
|[[Image:6-simplex t0.svg|150px]]<BR>[[6-simplex|6-simplex]]
|[[Image:6-cube graph.svg|150px]]<BR>[[6-cube|6-cube]]
|[[File:6-orthoplex.svg|150px]]<BR>[[6-orthoplex|6-orthoplex]]
|}

====7 dimensions====
{| class=wikitable
!Name!![[Schläfli symbol|Schläfli]]!!Vertices!!Edges!!Faces!!Cells!!4-faces||5-faces!!6-faces!!χ
|- BGCOLOR="#e0e0e0" align=center
|[[7-simplex|7-simplex]]||{3,3,3,3,3,3}||8||28||56||70||56||28||8||2
|- BGCOLOR="#ffe0e0" align=center
|[[7-cube|7-cube]]||{4,3,3,3,3,3}||128||448||672||560||280||84||14||2
|- BGCOLOR="#e0e0ff" align=center
|[[7-orthoplex|7-orthoplex]]||{3,3,3,3,3,4}||14||84||280||560||672||448||128||2
|}

{| class=wikitable
|- align=center valign=top
|[[Image:7-simplex t0.svg|150px]]<BR>[[7-simplex|7-simplex]]
|[[Image:7-cube graph.svg|150px]]<BR>[[7-cube|7-cube]]
|[[File:7-orthoplex.svg|150px]]<BR>[[7-orthoplex|7-orthoplex]]
|}

====8 dimensions====
{| class=wikitable
!Name!![[Schläfli symbol|Schläfli]]!!Vertices!!Edges!!Faces!!Cells!!4-faces||5-faces!!6-faces!!7-faces!!χ
|- BGCOLOR="#e0e0e0" align=center
|8-simplex||{3,3,3,3,3,3,3}||9||36||84||126||126||84||36||9||0
|- BGCOLOR="#ffe0e0" align=center
|8-cube||{4,3,3,3,3,3,3}||256||1024||1792||1792||1120||448||112||16||0
|- BGCOLOR="#e0e0ff" align=center
|8-orthoplex||{3,3,3,3,3,3,4}||16||112||448||1120||1792||1792||1024||256||0
|}

{| class=wikitable
|- align=center valign=top
|[[Image:8-simplex t0.svg|150px]]<BR>8-simplex
|[[Image:8-cube.svg|150px]]<BR>8-cube
|[[File:8-orthoplex.svg|150px]]<BR>8-orthoplex
|}

====9 dimensions====
{| class=wikitable
!Name!![[Schläfli symbol|Schläfli]]!!Vertices!!Edges!!Faces!!Cells!!4-faces||5-faces!!6-faces!!7-faces!!8-faces!!χ
|- BGCOLOR="#e0e0e0" align=center
|[[9-simplex|9-simplex]]||{3<sup>8</sup>}||10||45||120||210||252||210||120||45||10||2
|- BGCOLOR="#ffe0e0" align=center
|[[9-cube|9-cube]]||{4,3<sup>7</sup>}||512||2304||4608||5376||4032||2016||672||144||18||2
|- BGCOLOR="#e0e0ff" align=center
|[[9-orthoplex|9-orthoplex]]||{3<sup>7</sup>,4}||18||144||672||2016||4032||5376||4608||2304||512||2
|}

{| class=wikitable
|- align=center valign=top
|[[Image:9-simplex t0.svg|150px]]<BR>[[9-simplex|9-simplex]]
|[[Image:9-cube.svg|150px]]<BR>[[9-cube|9-cube]]
|[[File:9-orthoplex.svg|150px]]<BR>[[9-orthoplex|9-orthoplex]]
|}

====10 dimensions====
{| class=wikitable
!Name!![[Schläfli symbol|Schläfli]]!!Vertices!!Edges!!Faces!!Cells!!4-faces||5-faces!!6-faces!!7-faces!!8-faces!!9-faces!!χ
|- BGCOLOR="#e0e0e0" align=center
|[[10-simplex|10-simplex]]||{3<sup>9</sup>}||11||55||165||330||462||462||330||165||55||11||0
|- BGCOLOR="#ffe0e0" align=center
|[[10-cube|10-cube]]||{4,3<sup>8</sup>}||1024||5120||11520||15360||13440||8064||3360||960||180||20||0
|- BGCOLOR="#e0e0ff" align=center
|[[10-orthoplex|10-orthoplex]]||{3<sup>8</sup>,4}||20||180||960||3360||8064||13440||15360||11520||5120||1024||0
|}

{| class=wikitable
|- align=center valign=top
|[[Image:10-simplex t0.svg|150px]]<BR>[[10-simplex|10-simplex]]
|[[File:10-cube.svg|150px]]<BR>[[10-cube|10-cube]]
|[[File:10-orthoplex.svg|150px]]<BR>[[10-orthoplex|10-orthoplex]]
|}

===Star polytopes===

There are no regular star polytopes of rank 5 or higher, with the exception of degenerate polytopes created by the star product of lower rank star polytopes. {{abbr|e.g.}} hosotopes and ditopes.

== Regular projective polytopes ==
A projective regular {{math|({{mvar|n}}+1)}}-polytope exists when an original regular {{mvar|n}}-spherical tessellation, {p,q,...}, is centrally symmetric. Such a polytope is named hemi-<nowiki/>{p,q,...}, and contain half as many elements. Coxeter gives a symbol {p,q,...}/2, while McMullen writes {p,q,...}<sub>h/2</sub> with ''h'' as the coxeter number.{{sfnp|McMullen|Schulte|2002|loc=[https://books.google.com/books?id=JfmlMYe6MJgC&pg=PA162 "6C Projective Regular Polytopes" pp. 162-165]}}

Even-sided [[Regular polygon|regular polygon]]s have hemi-''2n''-gon projective polygons, {2p}/2.

There are 4 regular projective polyhedra related to 4 of 5 [[Platonic solid]]s.

The hemi-cube and hemi-octahedron generalize as hemi-[[Hypercube|{{mvar|n}}-cubes]] and hemi-{{mvar|n}}-orthoplexes to any rank.

=== Regular projective polyhedra===
{| class=wikitable
|+ rank 3 regular hemi-polytopes
!Name||Coxeter<BR>McMullen||Image||Faces||Edges||Vertices||[[Euler characteristic|&chi;]]
|- align=center
!Hemi-cube||{4,3}/2<BR>{4,3}<sub>3</sub>
|[[File:Hemicube.svg|60px]]||3||6||4||1
|- align=center
!Hemi-octahedron||{3,4}/2<BR>{3,4}<sub>3</sub>
|[[File:Hemi-octahedron2.png|60px]]||4||6||3||1
|- align=center
!Hemi-dodecahedron||{5,3}/2<BR>{5,3}<sub>5</sub>
|[[File:Hemi-dodecahedron.png|60px]]||6||15||10||1
|- align=center
!Hemi-icosahedron||{3,5}/2<BR>{3,5}<sub>5</sub>
|[[File:Hemi-icosahedron2.png|60px]]||10||15||6||1
|}

=== Regular projective 4-polytopes===
5 of 6 convex regular 4-polytopes are centrally symmetric generating projective 4-polytopes. The 3 special cases are hemi-24-cell, hemi-600-cell, and hemi-120-cell.

{| class=wikitable
|+ Rank 4 regular hemi-polytopes
!Name||Coxeter<BR>symbol||McMullen<BR>Symbol||Cells||Faces||Edges||Vertices||[[Euler characteristic|&chi;]]
|- align=center
!Hemi-[[Tesseract|tesseract]]
||{4,3,3}/2||{4,3,3}<sub>4</sub>
|4 ||12 ||16 || 8 || 0
|- align=center
!Hemi-[[16-cell|16-cell]]
||{3,3,4}/2||{3,3,4}<sub>4</sub>
|8 ||16 ||12 || 4 || 0
|- align=center
!Hemi-[[24-cell|24-cell]]
||{3,4,3}/2||{3,4,3}<sub>6</sub>
|12 ||48 ||48 ||12 || 0
|- align=center
!Hemi-[[120-cell|120-cell]]
||{5,3,3}/2||{5,3,3}<sub>15</sub>
|60 ||360 ||600 || 300 || 0
|- align=center
!Hemi-[[600-cell|600-cell]]
||{3,3,5}/2||{3,3,5}<sub>15</sub>
|300 ||600 ||360 || 60 || 0
|}

===Regular projective 5-polytopes===
Only 2 of 3 regular spereical polytopes are centrally symmetric for ranks 5 or higher: they are the hemi versions of the regular hypercube and orthoplex. They are tabulated below for rank 5, for example:

{| class="wikitable"
|-
!Name
||[[Schläfli symbol|Schläfli]]||4-faces||Cells||Faces||Edges||Vertices||[[Euler characteristic|&chi;]]
|- align=center
!hemi-penteract
|{4,3,3,3}/2||5||20||40||40||16||1
|- align=center
!hemi-pentacross
|{3,3,3,4}/2||16||40||40||20||5||1
|}

==Apeirotopes==
An '''[[Apeirotope|apeirotope]]''' or '''infinite polytope''' is a [[Polytope|polytope]] which has infinitely many [[Facet (geometry)|facets]]. An {{mvar|n}}-apeirotope is an infinite {{mvar|n}}-polytope: a 2-apeirotope or apeirogon is an infinite polygon, a 3-apeirotope or apeirohedron is an infinite polyhedron, etc.

There are two main geometric classes of apeirotope:<ref>{{cite journal |last=Grünbaum |first=B. |title=Regular Polyhedra—Old and New |journal=Aequationes Mathematicae |volume=16 |date=1977 |issue=1–2 |pages=1–20 |doi=10.1007/BF01836414|s2cid=125049930 }}</ref>
*Regular [[Honeycomb (geometry)|honeycombs]] in {{mvar|n}} dimensions, which completely fill an {{mvar|n}}-dimensional space.
*Regular skew apeirotopes, comprising an {{mvar|n}}-dimensional manifold in a higher space.

===2-apeirotopes (apeirogons)===
The straight [[Apeirogon|apeirogon]] is a regular tessellation of the line, subdividing it into infinitely many equal segments. It has infinitely many vertices and edges. Its [[Schläfli symbol]] is {∞}, and Coxeter diagram {{CDD|node_1|infin|node}}.

...[[File:Regular apeirogon.svg|320px]]...

It exists as the limit of the {{mvar|p}}-gon as {{mvar|p}} tends to infinity, as follows:

{| class="wikitable" style="text-align:center;"
|- bgcolor="#e0e0e0" valign="top"
!Name
|''[[Monogon]]''
|''[[Digon]]''
|''[[Equilateral triangle|Triangle]]''
|''[[Square]]''
|''[[Pentagon]]''
|''[[Hexagon]]''
|''[[Heptagon]]''
|''[[Regular polygon|p-gon]]''
|''[[Apeirogon]]''
|- bgcolor="#ffd0d0"
![[Schläfli symbol|Schläfli]]
| bgcolor="#eeeedd" |{1}
| bgcolor="#eeeedd" |{2}
|{3}
|{4}
|{5}
|{6}
|{7}
|{''p''}
| bgcolor="#c0ffc0" |{∞}
|-
!Symmetry
|D<sub>1</sub>, [ ]
|D<sub>2</sub>, [2]
|D<sub>3</sub>, [3]||D<sub>4</sub>, [4]||D<sub>5</sub>, [5]||D<sub>6</sub>, [6]||D<sub>7</sub>, [7]||[p]
|-
!Coxeter
|{{CDD|node}} or {{CDD|node_h|2x|node}}
|{{CDD|node_1|2x|node}}
|{{CDD|node_1|3|node}}
|{{CDD|node_1|4|node}}
|{{CDD|node_1|5|node}}
|{{CDD|node_1|6|node}}
|{{CDD|node_1|7|node}}
|{{CDD|node_1|p|node}}
|{{CDD|node_1|infin|node}}
|-
!Image
|[[File:Monogon.svg|60px]]
|[[File:Digon.svg|60px]]
|[[File:Regular triangle.svg|60px]]
|[[File:Regular quadrilateral.svg|60px]]
|[[File:Regular pentagon.svg|60px]]
|[[File:Regular hexagon.svg|60px]]
|[[File:Regular heptagon.svg|60px]]
|
|[[File:Regular apeirogon.svg|60px]]
|}

Apeirogons in the hyperbolic plane, most notably the ''regular apeirogon'', {∞}, can have a curvature just like finite polygons of the Euclidean plane, with the vertices circumscribed by [[Horocycle|horocycle]]s or [[Hypercycle (geometry)|hypercycles]] rather than [[Circle|circle]]s.

Regular apeirogons that are scaled to converge at infinity have the symbol {∞} and exist on horocycles, while more generally they can exist on hypercycles.

{| class=wikitable
!{∞}
!{πi/λ}
|-
|[[File:Hyperbolic apeirogon example.png|160px]]<BR>Apeirogon on [[Horocycle|horocycle]]
|[[File:Pseudogon example.png|160px]]<BR>Apeirogon on hypercycle
|}

Above are two regular hyperbolic apeirogons in the [[Poincaré disk model]], the right one shows perpendicular reflection lines of divergent [[Fundamental domain|fundamental domain]]s, separated by length λ.

==== Skew apeirogons ====
A skew apeirogon in two dimensions forms a zig-zag line in the plane. If the zig-zag is even and symmetrical, then the apeirogon is regular.

Skew apeirogons can be constructed in any number of dimensions. In three dimensions, a regular [[Skew apeirogon#Helical apeirogons in 3-dimensions|skew apeirogon]] traces out a helical spiral and may be either left- or right-handed.

{| class=wikitable
!2-dimensions
!3-dimensions
|- align=center
|[[File:Regular_zig-zag.svg|400px]]<BR>Zig-zag apeirogon
|[[File:Triangular helix.png|160px]]<BR>Helix apeirogon
|}

===2-apeirotopes (apeirohedra)===

====Euclidean tilings====

There are three regular tessellations of the plane. All three have an [[Euler characteristic]] (χ) of 0.
{| class=wikitable
!Name
![[Square tiling]]<BR>(quadrille)<!--<br/>(squat)-->
![[Triangular tiling]]<BR>(deltille)<!--<br/>(trat)-->
![[Hexagonal tiling]]<BR>(hextille)<!--<br/>(hexat)-->
|- align=center
![[List of planar symmetry groups#Wallpaper groups|Symmetry]]
|p4m, [4,4], (*442)
|colspan=2|p6m, [6,3], (*632)
|- align=center
![[Schläfli symbol|Schläfli]] {p,q}
|{4,4}
|{3,6}
|{6,3}
|- align=center
!Coxeter diagram
|{{CDD|node_1|4|node|4|node}}
|{{CDD|node|6|node|3|node_1}}
|{{CDD|node_1|6|node|3|node}}
|- align=center
!Image
|[[File:Uniform tiling 44-t0.png|100px]]
|[[File:Uniform tiling 63-t2.png|100px]]
|[[File:Uniform tiling 63-t0.png|100px]]
|}

There are two improper regular tilings: {∞,2}, an apeirogonal [[Dihedron|dihedron]], made from two [[Apeirogon|apeirogon]]s, each filling half the plane; and secondly, its dual, {2,∞}, an apeirogonal [[Hosohedron|hosohedron]], seen as an infinite set of parallel lines.

{| class=wikitable
|- align=center
|[[File:Apeirogonal tiling.png|210px]]<BR>[[Order-2 apeirogonal tiling|{∞,2}]], {{CDD|node_1|infin|node|2|node}}
|[[File:Apeirogonal hosohedron.png|240px]]<BR>[[Apeirogonal hosohedron|{2,∞}]], {{CDD|node_1|2|node|infin|node}}
|}

====Euclidean star-tilings====

There are no regular plane tilings of [[Star polygon|star polygon]]s. There are many enumerations that fit in the plane (1/''p'' + 1/''q'' = 1/2), like {8/3,8}, {10/3,5}, {5/2,10}, {12/5,12}, etc., but none repeat periodically.

====Hyperbolic tilings====

Tessellations of hyperbolic 2-space are ''hyperbolic tilings''. There are infinitely many regular tilings in H<sup>2</sup>. As stated above, every positive integer pair {''p'',''q''} such that 1/''p''&nbsp;+&nbsp;1/''q'' < 1/2 gives a hyperbolic tiling. In fact, for the general [[Schwarz triangle]] (''p'',&nbsp;''q'',&nbsp;''r'') the same holds true for 1/''p''&nbsp;+&nbsp;1/''q''&nbsp;+&nbsp;1/''r'' < 1.

There are a number of different ways to display the hyperbolic plane, including the Poincaré disc model which maps the plane into a circle, as shown below. It should be recognized that all of the polygon faces in the tilings below are equal-sized and only appear to get smaller near the edges due to the projection applied, very similar to the effect of a camera [[Engineering:Fisheye lens|fisheye lens]].

There are infinitely many flat regular 3-apeirotopes (apeirohedra) as regular tilings of the hyperbolic plane, of the form {p,q}, with p+q<pq/2. (previously listed above as tessellations)
* {3,7}, {3,8}, {3,9} ... {3,∞}
* {4,5}, {4,6}, {4,7} ... {4,∞}
* {5,4}, {5,5}, {5,6} ... {5,∞}
* {6,4}, {6,5}, {6,6} ... {6,∞}
* {7,3}, {7,4}, {7,5} ... {7,∞}
* {8,3}, {8,4}, {8,5} ... {8,∞}
* {9,3}, {9,4}, {9,5} ... {9,∞}
* ...
* {∞,3}, {∞,4}, {∞,5} ... {∞,∞}

A sampling:

The tilings {p, ∞} have [[Ideal point|ideal]] vertices, on the ''edge'' of the Poincaré disc model. Their duals {∞, p} have ideal [[Apeirogon|apeirogon]]al faces, meaning that they are inscribed in [[Horocycle|horocycle]]s. One could go further (as is done in the table above) and find tilings with ultra-ideal vertices, outside the Poincaré disc, which are dual to tiles inscribed in [[Hypercycle (geometry)|hypercycles]]; in what is symbolised {p, iπ/λ} above, infinitely many tiles still fit around each ultra-ideal vertex.<ref name=hyphon>Roice Nelson and Henry Segerman, [https://arxiv.org/abs/1511.02851 Visualizing Hyperbolic Honeycombs]</ref> (Parallel lines in extended hyperbolic space meet at an ideal point; ultraparallel lines meet at an ultra-ideal point.)<ref>Irving Adler, ''A New Look at Geometry'' (2012 Dover edition), [https://books.google.com/books?id=EbIAAQAAQBAJ&dq=ultra-ideal+hyperbolic+geometry&pg=PA233 p.233]</ref>

====Hyperbolic star-tilings====
There are 2 infinite forms of hyperbolic tilings whose [[Face (geometry)|faces]] or [[Vertex figure|vertex figure]]s are star polygons: {''m''/2, ''m''} and their duals {''m'', ''m''/2} with ''m'' = 7, 9, 11, .... The {''m''/2, ''m''} tilings are [[Stellation|stellation]]s of the {''m'', 3} tilings while the {''m'', ''m''/2} dual tilings are [[Faceting|faceting]]s of the {3, ''m''} tilings and greatenings of the {''m'', 3} tilings.

The patterns {''m''/2, ''m''} and {''m'', ''m''/2} continue for odd ''m'' &lt; 7 as [[Polyhedron|polyhedra]]: when ''m'' = 5, we obtain the [[Small stellated dodecahedron|small stellated dodecahedron]] and [[Great dodecahedron|great dodecahedron]], and when ''m'' = 3, the case degenerates to a [[Tetrahedron|tetrahedron]]. The other two Kepler–Poinsot polyhedra (the [[Great stellated dodecahedron|great stellated dodecahedron]] and [[Great icosahedron|great icosahedron]]) do not have regular hyperbolic tiling analogues. If ''m'' is even, depending on how we choose to define {''m''/2}, we can either obtain degenerate double covers of other tilings or compound tilings.

{| class="wikitable" style="text-align:center;"
|-
! Name
! [[Schläfli symbol|Schläfli]]
! Coxeter diagram
! Image
! Face type<br />{p}
! [[Vertex figure]]<br />{q}
! [[Density (polytope)|Density]]
! [[Symmetry group|Symmetry]]
! Dual
|- BGCOLOR="#ffe0e0" align=center
|[[Order-7 heptagrammic tiling]]
|{7/2,7}
|{{CDD|node_1|7|rat|d2|node|7|node}}
|[[Image:Hyperbolic tiling 7-2 7.png|60px]]
|{7/2}<br />[[Image:Star polygon 7-2.svg|30px]]
|{7}<br />[[Image:Regular heptagon.svg|30px]]
| 3
| *732<BR>[7,3]
| Heptagrammic-order heptagonal tiling
|- BGCOLOR="#e0e0ff" align=center
|[[Heptagrammic-order heptagonal tiling]]
|{7,7/2}
|{{CDD|node_1|7|node|7|rat|d2|node}}
|[[Image:Hyperbolic tiling 7 7-2.png|60px]]
|{7}<br />[[Image:Regular heptagon.svg|30px]]
|{7/2}<br />[[Image:Star polygon 7-2.svg|30px]]
| 3
| *732<BR>[7,3]
| Order-7 heptagrammic tiling
|- BGCOLOR="#ffe0e0" align=center
| Order-9 enneagrammic tiling
|{9/2,9}
|{{CDD|node_1|9|rat|d2|node|9|node}}
|[[Image:Hyperbolic tiling 9-2 9.png|60px]]
|{9/2}<br />[[Image:Star polygon 9-2.svg|30px]]
|{9}<br />[[Image:Regular nonagon.svg|30px]]
| 3
| *932<BR>[9,3]
| Enneagrammic-order enneagonal tiling
|- BGCOLOR="#e0e0ff" align=center
| Enneagrammic-order enneagonal tiling
|{9,9/2}
|{{CDD|node_1|9|node|9|rat|d2|node}}
|[[Image:Hyperbolic tiling 9 9-2.png|60px]]
|{9}<br />[[Image:Regular nonagon.svg|30px]]
|{9/2}<br />[[Image:Star polygon 9-2.svg|30px]]
| 3
| *932<BR>[9,3]
| Order-9 enneagrammic tiling
|- BGCOLOR="#ffe0e0" align=center
| Order-11 hendecagrammic tiling
|{11/2,11}
|{{CDD|node_1|11|rat|d2|node|11|node}}
|[[Image:Order-11 hendecagrammic tiling.png|60px]]
|{11/2}<br />[[Image:Star polygon 11-2.svg|30px]]
|{11}<br />[[Image:Regular hendecagon.svg|30px]]
| 3
| *11.3.2<BR>[11,3]
| Hendecagrammic-order hendecagonal tiling
|- BGCOLOR="#e0e0ff" align=center
| Hendecagrammic-order hendecagonal tiling
|{11,11/2}
|{{CDD|node_1|11|node|11|rat|d2|node}}
|[[Image:Hendecagrammic-order hendecagonal tiling.png|60px]]
|{11}<br />[[Image:Regular hendecagon.svg|30px]]
|{11/2}<br />[[Image:Star polygon 11-2.svg|30px]]
| 3
| *11.3.2<BR>[11,3]
| Order-11 hendecagrammic tiling

|- BGCOLOR="#ffe0e0" align=center
| Order-''p'' ''p''-grammic tiling
|{''p''/2,''p''}
|{{CDD|node_1|p|rat|d2|node|p|node}}
|&nbsp;
|{''p''/2}
|{''p''}
| 3
| *''p''32<BR>[p,3]
| ''p''-grammic-order ''p''-gonal tiling
|- BGCOLOR="#e0e0ff" align=center
| ''p''-grammic-order ''p''-gonal tiling
|{''p'',''p''/2}
|{{CDD|node_1|p|node|p|rat|d2|node}}
|&nbsp;
|{''p''}
|{''p''/2}
| 3
| *''p''32<BR>[p,3]
| Order-''p'' ''p''-grammic tiling
|}

====Skew apeirohedra in Euclidean 3-space====
There are three [[Regular skew apeirohedron|regular skew apeirohedra]] in Euclidean 3-space, with planar faces.<ref>{{cite journal |last=Coxeter |first=H.S.M. |title=Regular Skew Polyhedra in Three and Four Dimensions |journal=Proc. London Math. Soc. |series=2 |volume=43 |pages=33–62 |date=1938 |doi=10.1112/plms/s2-43.1.33}}</ref><ref>{{cite journal |last=Coxeter |first=H.S.M. |date=1985 |title=Regular and semi-regular polytopes II |journal=Mathematische Zeitschrift |volume=188 |issue=4 |pages=559–591 |doi=10.1007/BF01161657|s2cid=120429557 }}</ref><ref>{{cite book |last1=Conway |first1=John H. |last2=Burgiel |first2=Heidi |last3=Goodman-Strauss |first3=Chaim |title=The Symmetries of Things |year=2008 |chapter=Chapter 23: Objects with Primary Symmetry, Infinite Platonic Polyhedra |pages=333–335 |publisher=Taylor & Francis |isbn=978-1-568-81220-5}}</ref> They share the same [[Vertex arrangement|vertex arrangement]] and edge arrangement of 3 [[Convex uniform honeycomb|convex uniform honeycomb]]s.
* 6 squares around each vertex: {4,6|4}
* 4 hexagons around each vertex: {6,4|4}
* 6 hexagons around each vertex: {6,6|3}
[[File:Pure 3-dimensional apeirohedra chart.png|upright=1.5|thumb|12 "pure" apeirohedra in Euclidean 3-space based on the structure of the [[Cubic honeycomb|cubic honeycomb]], {4,3,4}.{{sfnp|McMullen|Schulte|2002|p=224}} A &pi; [[Petrie dual|petrie dual]] operator replaces faces with [[Petrie polygon|petrie polygon]]s; &delta; is a dual operator reverses vertices and faces; &phi;<sub>k</sub> is a ''k''th facetting operator; &eta; is a halving operator, and &sigma; skewing halving operator.]]
{| class="wikitable"
!colspan=3| Regular skew polyhedra with planar faces
|-
|align=center|[[File:Mucube external.png|100px]]<BR>{4,6|4}
|align=center|[[File:Muoctahedron external.png|100px]]<BR>{6,4|4}
|align=center|[[File:Mutetrahedron external.png|100px]]<BR>{6,6|3}
|}
Allowing for skew faces, there are 24 regular apeirohedra in Euclidean 3-space.{{sfnp|McMullen|Schulte|2002|loc=Section 7E}} These include 12 apeirhedra created by blends with the Euclidean apeirohedra, and 12 pure apeirohedra, including the 3 above, which cannot be expressed as a non-trivial blend.

Those pure apeirohedra are:
*{{math|{4,6{{!}}4{{)}}}}, the mucube
*{{math|{&infin;,6}{{sub|4,4}}}}, the Petrial of the mucube
*{{math|{6,6{{!}}3{{)}}}}, the mutetrahedron
*{{math|{&infin;,6}{{sub|6,3}}}}, the Petrial of the mutetrahedron
*{{math|{6,4{{!}}4{{)}}}}, the muoctahedron
*{{math|{&infin;,4}{{sub|6,4}}}}, the Petrial of the muoctahedron
*{{math|{6,6}{{sub|4}}}}, the halving of the mucube
*{{math|{4,6}{{sub|6}}}}, the Petrial of {{math|{6,6}{{sub|4}}}}
*{{math|{&infin;,4}{{sub|·,*3}}}}, the skewing of the muoctahedron
*{{math|{6,4}{{sub|6}}}}, the Petrial of {{math|{&infin;,4}{{sub|·,*3}}}}
*{{math|{∞,3}{{sup|({{mvar|a}})}}}}
*{{math|{∞,3}{{sup|({{mvar|b}})}}}}

====Skew apeirohedra in hyperbolic 3-space====
There are 31 [[Regular skew apeirohedron#Regular skew apeirohedra in hyperbolic 3-space|regular skew apeirohedra]] with convex faces in hyperbolic 3-space with compact or paracompact symmetry:<ref>{{cite journal |last=Garner |first=C.W.L. |title=Regular Skew Polyhedra in Hyperbolic Three-Space |journal=Can. J. Math. |volume=19 |pages=1179–1186 |date=1967 |doi=10.4153/CJM-1967-106-9 |s2cid=124086497 |url=http://cms.math.ca/cjm/a145822|doi-access=free }} Note: His paper says there are 32, but one is self-dual, leaving 31.</ref>
* 14 are compact: {8,10|3}, {10,8|3}, {10,4|3}, {4,10|3}, {6,4|5}, {4,6|5}, {10,6|3}, {6,10|3}, {8,8|3}, {6,6|4}, {10,10|3},{6,6|5}, {8,6|3}, and {6,8|3}.
* 17 are paracompact: {12,10|3}, {10,12|3}, {12,4|3}, {4,12|3}, {6,4|6}, {4,6|6}, {8,4|4}, {4,8|4}, {12,6|3}, {6,12|3}, {12,12|3}, {6,6|6}, {8,6|4}, {6,8|4}, {12,8|3}, {8,12|3}, and {8,8|4}.

===4-apeirotopes===

====Tessellations of Euclidean 3-space====
[[Image:Cubic honeycomb.png|upright=0.8|thumb|Edge framework of cubic honeycomb, {4,3,4}]]

There is only one non-degenerate regular tessellation of 3-space (''[[Honeycomb (geometry)|honeycombs]]''),  {4, 3, 4}:{{sfnp|Coxeter|1973|loc=Table II: Regular honeycombs, p.&nbsp;296}}

{| class="wikitable"
|-
!Name
![[Schläfli symbol|Schläfli]]<BR>{p,q,r}
!Coxeter<BR>{{CDD|node|p|node|q|node|r|node}}
!Cell<BR>type<BR>{p,q}
!Face<BR>type<BR>{p}
!Edge<BR>figure<BR>{r}
![[Vertex figure|Vertex<BR>figure]]<BR>{q,r}
![[Euler characteristic|χ]]
!Dual
|- BGCOLOR="#e0e0e0" align=center
|[[Cubic honeycomb]]||{4,3,4}||{{CDD|node_1|4|node|3|node|4|node}}||{4,3}||{4}||{4}||{3,4}||0||Self-dual
|}

====Improper tessellations of Euclidean 3-space====
[[File:Order-4 square hosohedral honeycomb-sphere.png|upright=0.8|thumb|Regular {2,4,4} honeycomb, seen projected into a sphere.]]
There are six improper regular tessellations, pairs based on the three regular Euclidean tilings. Their cells and vertex figures are all regular [[Hosohedron|hosohedra]] {2,n}, [[Dihedron|dihedra]], {n,2}, and Euclidean tilings. These improper regular tilings are constructionally related to prismatic uniform honeycombs by truncation operations. They are higher-dimensional analogues of the [[Order-2 apeirogonal tiling|order-2 apeirogonal tiling]] and [[Apeirogonal hosohedron|apeirogonal hosohedron]].
{| class=wikitable
|-
![[Schläfli symbol|Schläfli]]<BR>{p,q,r}
!Coxeter<BR>diagram
!Cell<BR>type<BR>{p,q}
!Face<BR>type<BR>{p}
!Edge<BR>figure<BR>{r}
![[Vertex figure|Vertex<BR>figure]]<BR>{q,r}
|- align=center
|[[Order-4 square hosohedral honeycomb|{2,4,4}]]||{{CDD|node_1|2|node|4|node|4|node}}||{2,4}||{2}||{4}||{4,4}
|- align=center
|[[Order-6 triangular hosohedral honeycomb|{2,3,6}]]||{{CDD|node_1|2|node|3|node|6|node}}||{2,3}||{2}||{6}||{3,6}
|- align=center
|{2,6,3}||{{CDD|node_1|2|node|6|node|3|node}}||{2,6}||{2}||{3}||{6,3}
|- align=center
|{4,4,2}||{{CDD|node_1|4|node|4|node|2|node}}||{4,4}||{4}||{2}||{4,2}
|- align=center
|{3,6,2}||{{CDD|node_1|3|node|6|node|2|node}}||{3,6}||{3}||{2}||{6,2}
|- align=center
|{6,3,2}||{{CDD|node_1|6|node|3|node|2|node}}||{6,3}||{6}||{2}||{3,2}
|}

====Tessellations of hyperbolic 3-space====
There are ten flat regular honeycombs of hyperbolic 3-space:{{sfnp|Coxeter|1999|loc="Chapter 10"}} (previously listed above as tessellations)
*4 are compact: {3,5,3}, {4,3,5}, {5,3,4}, and {5,3,5}
*while 6 are paracompact: {3,3,6}, {6,3,3}, {3,4,4}, {4,4,3}, {3,6,3}, {4,3,6}, {6,3,4}, {4,4,4}, {5,3,6}, {6,3,5}, and {6,3,6}.

{| align=right
|
{| class="wikitable"
|+ 4 compact regular honeycombs
|- bgcolor="#d0e0ff" align=center
|[[File:H3 534 CC center.png|150px]]<BR>[[Order-4 dodecahedral honeycomb|{5,3,4}]]
|[[File:H3 535 CC center.png|150px]]<BR>[[Order-5 dodecahedral honeycomb|{5,3,5}]]
|- bgcolor="#d0e0ff" align=center
|[[File:H3 435 CC center.png|150px]]<BR>[[Order-5 cubic honeycomb|{4,3,5}]]
|[[File:H3 353 CC center.png|150px]]<BR>{3,5,3}
|}
|-
|
{| class="wikitable"
|+ 4 of 11 paracompact regular honeycombs
|- bgcolor="#e0d0ff" align=center
|[[File:H3 344 CC center.png|150px]]<BR>[[Order-4 octahedral honeycomb|{3,4,4}]]
|[[File:H3 363 FC boundary.png|150px]]<BR>[[Triangular tiling honeycomb|{3,6,3}]]
|- bgcolor="#e0d0ff" align=center
|[[File:H3 443 FC boundary.png|150px]]<BR>[[Square tiling honeycomb|{4,4,3}]]
|[[File:H3 444 FC boundary.png|150px]]<BR>[[Order-4 square tiling honeycomb|{4,4,4}]]
|}
|}
Tessellations of [[Hyperbolic space|hyperbolic 3-space]] can be called ''[[Honeycomb (geometry)|hyperbolic honeycombs]]''. There are 15 hyperbolic honeycombs in H<sup>3</sup>, 4 compact and 11 paracompact.

{| class="wikitable"
|+ 4 compact regular honeycombs
|-
!Name
![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p,q,r}
!Coxeter<BR>{{CDD|node|p|node|q|node|r|node}}
!Cell<BR>type<BR>{p,q}
!Face<BR>type<BR>{p}
!Edge<BR>figure<BR>{r}
![[Vertex figure|Vertex<BR>figure]]<BR>{q,r}
![[Euler characteristic|χ]]
!Dual
|- BGCOLOR="#e0e0e0" align=center
|[[Icosahedral honeycomb]]||{3,5,3}||{{CDD|node_1|3|node|5|node|3|node}}||[[Icosahedron|{3,5}]]||{3}||{3}||[[Dodecahedron|{5,3}]]||0||Self-dual
|- BGCOLOR="#ffe0e0" align=center
|[[Order-5 cubic honeycomb]]||{4,3,5}||{{CDD|node_1|4|node|3|node|5|node}}||[[Cube|{4,3}]]||{4}||{5}||[[Icosahedron|{3,5}]]||0||{5,3,4}
|- BGCOLOR="#e0e0ff" align=center
|[[Order-4 dodecahedral honeycomb]]||{5,3,4}||{{CDD|node_1|5|node|3|node|4|node}}||[[Dodecahedron|{5,3}]]||{5}||{4}||[[Octahedron|{3,4}]]||0||{4,3,5}
|- BGCOLOR="#e0e0e0" align=center
|[[Order-5 dodecahedral honeycomb]]||{5,3,5}||{{CDD|node_1|5|node|3|node|5|node}}||[[Dodecahedron|{5,3}]]||{5}||{5}||[[Icosahedron|{3,5}]]||0||Self-dual
|}

There are also 11 paracompact H<sup>3</sup> honeycombs (those with infinite (Euclidean) cells and/or vertex figures): {3,3,6}, {6,3,3}, {3,4,4}, {4,4,3}, {3,6,3}, {4,3,6}, {6,3,4}, {4,4,4}, {5,3,6}, {6,3,5}, and {6,3,6}.

{| class="wikitable"
|+ 11 paracompact regular honeycombs
|-
!Name
![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p,q,r}
!Coxeter<BR>{{CDD|node|p|node|q|node|r|node}}
!Cell<BR>type<BR>{p,q}
!Face<BR>type<BR>{p}
!Edge<BR>figure<BR>{r}
![[Vertex figure|Vertex<BR>figure]]<BR>{q,r}
![[Euler characteristic|χ]]
!Dual
|- BGCOLOR="#ffe0e0" align=center
|[[Order-6 tetrahedral honeycomb]]||{3,3,6}||{{CDD|node_1|3|node|3|node|6|node}}||[[Tetrahedron|{3,3}]]||{3}||{6}||[[Triangular tiling|{3,6}]]||0||{6,3,3}
|- BGCOLOR="#e0e0ff" align=center
|[[Hexagonal tiling honeycomb]]||{6,3,3}||{{CDD|node_1|6|node|3|node|3|node}}||[[Hexagonal tiling|{6,3}]]||{6}||{3}||[[Tetrahedron|{3,3}]]||0||{3,3,6}
|- BGCOLOR="#ffe0e0" align=center
|[[Order-4 octahedral honeycomb]]||{3,4,4}||{{CDD|node_1|3|node|4|node|4|node}}||[[Octahedron|{3,4}]]||{3}||{4}||[[Square tiling|{4,4}]]||0||{4,4,3}
|- BGCOLOR="#e0e0ff" align=center
|[[Square tiling honeycomb]]||{4,4,3}||{{CDD|node_1|4|node|4|node|3|node}}||[[Square tiling|{4,4}]]||{4}||{3}||[[Cube|{4,3}]]||0||{3,3,4}
|- BGCOLOR="#e0e0e0" align=center
|[[Triangular tiling honeycomb]]||{3,6,3}||{{CDD|node_1|3|node|6|node|3|node}}||[[Triangular tiling|{3,6}]]||{3}||{3}||[[Hexagonal tiling|{6,3}]]||0||Self-dual
|- BGCOLOR="#ffe0e0" align=center
|[[Order-6 cubic honeycomb]]||{4,3,6}||{{CDD|node_1|4|node|3|node|6|node}}||[[Cube|{4,3}]]||{4}||{4}||[[Triangular tiling|{3,6}]]||0||{6,3,4}
|- BGCOLOR="#e0e0ff" align=center
|[[Order-4 hexagonal tiling honeycomb]]||{6,3,4}||{{CDD|node_1|6|node|3|node|4|node}}||[[Hexagonal tiling|{6,3}]]||{6}||{4}||[[Octahedron|{3,4}]]||0||{4,3,6}
|- BGCOLOR="#e0e0e0" align=center
|[[Order-4 square tiling honeycomb]]||{4,4,4}||{{CDD|node_1|4|node|4|node|4|node}}||[[Square tiling|{4,4}]]||{4}||{4}||[[Square tiling|{4,4}]]||0||Self-dual
|- BGCOLOR="#ffe0e0" align=center
|[[Order-6 dodecahedral honeycomb]]||{5,3,6}||{{CDD|node_1|5|node|3|node|6|node}}||[[Dodecahedron|{5,3}]]||{5}||{5}||[[Triangular tiling|{3,6}]]||0||{6,3,5}
|- BGCOLOR="#e0e0ff" align=center
|[[Order-5 hexagonal tiling honeycomb]]||{6,3,5}||{{CDD|node_1|6|node|3|node|5|node}}||[[Hexagonal tiling|{6,3}]]||{6}||{5}||[[Icosahedron|{3,5}]]||0||{5,3,6}
|- BGCOLOR="#e0e0e0" align=center
|[[Order-6 hexagonal tiling honeycomb]]||{6,3,6}||{{CDD|node_1|6|node|3|node|6|node}}||[[Hexagonal tiling|{6,3}]]||{6}||{6}||[[Triangular tiling|{3,6}]]||0||Self-dual
|}

Noncompact solutions exist as Lorentzian Coxeter groups, and can be visualized with open domains in hyperbolic space (the fundamental tetrahedron having ultra-ideal vertices). All honeycombs with hyperbolic cells or vertex figures and do not have 2 in their Schläfli symbol are noncompact.

There are no regular hyperbolic star-honeycombs in H<sup>3</sup>: all forms with a regular star polyhedron as cell, vertex figure or both end up being spherical.

Ideal vertices now appear when the vertex figure is a Euclidean tiling, becoming inscribable in a horosphere rather than a sphere. They are dual to ideal cells (Euclidean tilings rather than finite polyhedra). As the last number in the Schläfli symbol rises further, the vertex figure becomes hyperbolic, and vertices become ultra-ideal (so the edges do not meet within hyperbolic space). In honeycombs {p, q, ∞} the edges intersect the Poincaré ball only in one ideal point; the rest of the edge has become ultra-ideal. Continuing further would lead to edges that are completely ultra-ideal, both for the honeycomb and for the fundamental simplex (though still infinitely many {p, q} would meet at such edges). In general, when the last number of the Schläfli symbol becomes ∞, faces of codimension two intersect the Poincaré hyperball only in one ideal point.<ref name=hyphon/>

===5-apeirotopes===

====Tessellations of Euclidean 4-space====
There are three kinds of infinite regular tessellations ([[Honeycomb (geometry)|honeycombs]]) that can tessellate Euclidean four-dimensional space:

{| class="wikitable"
|+ 3 regular Euclidean honeycombs
|-
!Name
![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p,q,r,s}
!Facet<BR>type<BR>{p,q,r}
!Cell<BR>type<BR>{p,q}
!Face<BR>type<BR>{p}
!Face<BR>figure<BR>{s}
!Edge<BR>figure<BR>{r,s}
![[Vertex figure|Vertex<BR>figure]]<BR>{q,r,s}
!Dual
|- BGCOLOR="#e0e0e0" align=center
|[[Tesseractic honeycomb]]||{4,3,3,4}||{4,3,3}||{4,3}||{4}||{4}||{3,4}||{3,3,4}||Self-dual
|- BGCOLOR="#ffe0e0" align=center
|[[16-cell honeycomb|16-cell honeycomb]]||{3,3,4,3}||{3,3,4}||{3,3}||{3}||{3}||{4,3}||{3,4,3}||{3,4,3,3}
|- BGCOLOR="#e0e0ff" align=center
|[[24-cell honeycomb|24-cell honeycomb]]||{3,4,3,3}||{3,4,3}||{3,4}||{3}||{3}||{3,3}||{4,3,3}||{3,3,4,3}
|}

{| class="wikitable"
|[[Image:Tesseractic tetracomb.png|175px]]<BR>Projected portion of {4,3,3,4}<BR>(Tesseractic honeycomb)
|[[Image:Demitesseractic tetra hc.png|175px]]<BR>Projected portion of {3,3,4,3}<BR>(16-cell honeycomb)
|[[Image:Icositetrachoronic tetracomb.png|175px]]<BR>Projected portion of {3,4,3,3}<BR>(24-cell honeycomb)
|}

There are also the two improper cases {4,3,4,2} and {2,4,3,4}.

There are three flat regular honeycombs of Euclidean 4-space:{{sfnp|Coxeter|1973|loc=Table II: Regular honeycombs, p.&nbsp;296}} 
* {4,3,3,4}, {3,3,4,3}, and {3,4,3,3}.

There are seven flat regular convex honeycombs of hyperbolic 4-space:{{sfnp|Coxeter|1999|loc="Chapter 10"}}
*5 are compact: {3,3,3,5}, {5,3,3,3}, {4,3,3,5}, {5,3,3,4}, {5,3,3,5} 
*2 are paracompact: {3,4,3,4}, and {4,3,4,3}.

There are four flat regular star honeycombs of hyperbolic 4-space:{{sfnp|Coxeter|1999|loc="Chapter 10"}}
* {5/2,5,3,3}, {3,3,5,5/2}, {3,5,5/2,5}, and {5,5/2,5,3}.

====Tessellations of hyperbolic 4-space====
There are seven convex regular [[Honeycomb (geometry)|honeycombs]] and four star-honeycombs in H<sup>4</sup> space.{{sfnp|Coxeter|1999|loc="Chapter 10" Table IV, p. 213}} Five convex ones are compact, and two are paracompact.

Five compact regular honeycombs in H<sup>4</sup>:
{| class="wikitable"
|+ 5 compact regular honeycombs
|-
!Name
![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p,q,r,s}
!Facet<BR>type<BR>{p,q,r}
!Cell<BR>type<BR>{p,q}
!Face<BR>type<BR>{p}
!Face<BR>figure<BR>{s}
!Edge<BR>figure<BR>{r,s}
![[Vertex figure|Vertex<BR>figure]]<BR>{q,r,s}
!Dual
|- BGCOLOR="#ffe0e0" align=center
|[[Order-5 5-cell honeycomb]]||{3,3,3,5}||[[5-cell|{3,3,3}]]||[[Tetrahedron|{3,3}]]||{3}||{5}||[[Icosahedron|{3,5}]]||[[600-cell|{3,3,5}]]||{5,3,3,3}
|- BGCOLOR="#e0e0ff" align=center
|[[120-cell honeycomb|120-cell honeycomb]]||{5,3,3,3}||[[120-cell|{5,3,3}]]||[[Dodecahedron|{5,3}]]||{5}||{3}||[[Tetrahedron|{3,3}]]||[[5-cell|{3,3,3}]]||{3,3,3,5}
|- BGCOLOR="#ffe0e0" align=center
|[[Order-5 tesseractic honeycomb]]||{4,3,3,5}||[[Tesseract|{4,3,3}]]||[[Cube|{4,3}]]||{4}||{5}||[[Icosahedron|{3,5}]]||[[600-cell|{3,3,5}]]||{5,3,3,4}
|- BGCOLOR="#e0e0ff" align=center
|[[Order-4 120-cell honeycomb]]||{5,3,3,4}||[[120-cell|{5,3,3}]]||[[Dodecahedron|{5,3}]]||{5}||{4}||[[Octahedron|{3,4}]]||[[16-cell|{3,3,4}]]||{4,3,3,5}
|- BGCOLOR="#e0e0e0" align=center
|[[Order-5 120-cell honeycomb]]||{5,3,3,5}||[[120-cell|{5,3,3}]]||[[Dodecahedron|{5,3}]]||{5}||{5}||[[Icosahedron|{3,5}]]||[[600-cell|{3,3,5}]]||Self-dual
|}

The two paracompact regular H<sup>4</sup> honeycombs are: {3,4,3,4}, {4,3,4,3}.

{| class="wikitable"
|+ 2 paracompact regular honeycombs
|-
!Name
![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p,q,r,s}
!Facet<BR>type<BR>{p,q,r}
!Cell<BR>type<BR>{p,q}
!Face<BR>type<BR>{p}
!Face<BR>figure<BR>{s}
!Edge<BR>figure<BR>{r,s}
![[Vertex figure|Vertex<BR>figure]]<BR>{q,r,s}
!Dual
|- BGCOLOR="#ffe0e0" align=center
|[[Order-4 24-cell honeycomb]]||{3,4,3,4}||[[24-cell|{3,4,3}]]||[[Octahedron|{3,4}]]||{3}||{4}||[[Octahedron|{3,4}]]||[[Cubic honeycomb|{4,3,4}]]||{4,3,4,3}
|- BGCOLOR="#e0e0ff" align=center
|[[Cubic honeycomb honeycomb]]||{4,3,4,3}||[[Cubic honeycomb|{4,3,4}]]||[[Cube|{4,3}]]||{4}||{3}||[[Cube|{4,3}]]||[[24-cell|{3,4,3}]]||{3,4,3,4}
|}

Noncompact solutions exist as Lorentzian Coxeter groups, and can be visualized with open domains in hyperbolic space (the fundamental 5-cell having some parts inaccessible beyond infinity). All honeycombs which are not shown in the set of tables below and do not have 2 in their Schläfli symbol are noncompact.

====Star tessellations of hyperbolic 4-space====
There are four regular star-honeycombs in H<sup>4</sup> space, all compact:
{| class="wikitable"
|+ 4 compact regular star-honeycombs
|-
!Name
![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p,q,r,s}
!Facet<BR>type<BR>{p,q,r}
!Cell<BR>type<BR>{p,q}
!Face<BR>type<BR>{p}
!Face<BR>figure<BR>{s}
!Edge<BR>figure<BR>{r,s}
![[Vertex figure|Vertex<BR>figure]]<BR>{q,r,s}
!Dual
!Density
|- BGCOLOR="#ffe0e0" align=center
|[[Small stellated 120-cell honeycomb]]||{5/2,5,3,3}||[[Small stellated 120-cell|{5/2,5,3}]]||{5/2,5}||{5/2}||{3}||{3,3}||[[120-cell|{5,3,3}]]||{3,3,5,5/2}||5
|- BGCOLOR="#e0e0ff" align=center
|[[Pentagrammic-order 600-cell honeycomb]]||{3,3,5,5/2}||[[600-cell|{3,3,5}]]||[[Tetrahedron|{3,3}]]||{3}||{5/2}||{5,5/2}||{3,5,5/2}||{5/2,5,3,3}||5
|- BGCOLOR="#ffe0e0" align=center
|[[Order-5 icosahedral 120-cell honeycomb]]||{3,5,5/2,5}||{3,5,5/2}||[[Icosahedron|{3,5}]]||{3}||{5}||{5/2,5}||{5,5/2,5}||{5,5/2,5,3}||10
|- BGCOLOR="#e0e0ff" align=center
|[[Great 120-cell honeycomb]]||{5,5/2,5,3}||{5,5/2,5}||{5,5/2}||{5}||{3}||[[Dodecahedron|{5,3}]]||{5/2,5,3}||{3,5,5/2,5}||10
|}

===6-apeirotopes===
There is only one flat regular honeycomb of Euclidean 5-space: (previously listed above as tessellations){{sfnp|Coxeter|1973|loc=Table II: Regular honeycombs, p.&nbsp;296}}
* {4,3,3,3,4}

There are five flat regular regular honeycombs of hyperbolic 5-space, all paracompact: (previously listed above as tessellations){{sfnp|Coxeter|1999|loc="Chapter 10"}}
* {3,3,3,4,3}, {3,4,3,3,3}, {3,3,4,3,3}, {3,4,3,3,4}, and {4,3,3,4,3}

====Tessellations of Euclidean 5-space====
The [[Hypercubic honeycomb|hypercubic honeycomb]] is the only family of regular honeycombs that can tessellate each dimension, five or higher, formed by [[Hypercube|hypercube]] facets, four around every ridge.

{| class="wikitable"
|-
!Name
![[Schläfli symbol|Schläfli]]<BR>{''p''<sub>1</sub>, ''p''<sub>2</sub>, ..., ''p''<sub>''n''&minus;1</sub>}
!Facet<BR>type
![[Vertex figure|Vertex<BR>figure]]
!Dual
|- BGCOLOR="#e0e0e0" align=center
|[[Square tiling]]||{4,4}||{4}||{4}||Self-dual
|- BGCOLOR="#e0e0e0" align=center
|[[Cubic honeycomb]]||{4,3,4}||{4,3}||{3,4}||Self-dual
|- BGCOLOR="#e0e0e0" align=center
|[[Tesseractic honeycomb]]||{4,3<sup>2</sup>,4}||{4,3<sup>2</sup>}||{3<sup>2</sup>,4}||Self-dual
|- BGCOLOR="#e0e0e0" align=center
|5-cube honeycomb||{4,3<sup>3</sup>,4}||{4,3<sup>3</sup>}||{3<sup>3</sup>,4}||Self-dual
|- BGCOLOR="#e0e0e0" align=center
|6-cube honeycomb||{4,3<sup>4</sup>,4}||{4,3<sup>4</sup>}||{3<sup>4</sup>,4}||Self-dual
|- BGCOLOR="#e0e0e0" align=center
|7-cube honeycomb||{4,3<sup>5</sup>,4}||{4,3<sup>5</sup>}||{3<sup>5</sup>,4}||Self-dual
|- BGCOLOR="#e0e0e0" align=center
|8-cube honeycomb||{4,3<sup>6</sup>,4}||{4,3<sup>6</sup>}||{3<sup>6</sup>,4}||Self-dual
|- BGCOLOR="#e0e0e0" align=center
|n-[[Hypercubic honeycomb|hypercubic honeycomb]]||{4,3<sup>n−2</sup>,4}||{4,3<sup>n−2</sup>}||{3<sup>n−2</sup>,4}||Self-dual
|}

In E<sup>5</sup>, there are also the improper cases {4,3,3,4,2}, {2,4,3,3,4}, {3,3,4,3,2}, {2,3,3,4,3}, {3,4,3,3,2}, and {2,3,4,3,3}. In E<sup>''n''</sup>, {4,3<sup>n−3</sup>,4,2} and {2,4,3<sup>n−3</sup>,4} are always improper Euclidean tessellations.

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

There are 5 regular honeycombs in H<sup>5</sup>, all paracompact, which include infinite (Euclidean) facets or vertex figures: {3,4,3,3,3}, {3,3,4,3,3}, {3,3,3,4,3}, {3,4,3,3,4}, and {4,3,3,4,3}.

There are no compact regular tessellations of hyperbolic space of dimension 5 or higher and no paracompact regular tessellations in hyperbolic space of dimension 6 or higher.

{| class="wikitable"
|+ 5 paracompact regular honeycombs
|-
!Name
![[Schläfli symbol|Schläfli]]<BR>Symbol<BR>{p,q,r,s,t}
!Facet<BR>type<BR>{p,q,r,s}
!4-face<BR>type<BR>{p,q,r}
!Cell<BR>type<BR>{p,q}
!Face<BR>type<BR>{p}
!Cell<BR>figure<BR>{t}
!Face<BR>figure<BR>{s,t}
!Edge<BR>figure<BR>{r,s,t}
![[Vertex figure|Vertex<BR>figure]]<BR>{q,r,s,t}
!Dual
|- BGCOLOR="#ffe0e0" align=center
|[[5-orthoplex honeycomb|5-orthoplex honeycomb]]||{3,3,3,4,3}||[[5-orthoplex|{3,3,3,4}]]||[[5-cell|{3,3,3}]]||[[Tetrahedron|{3,3}]]||{3}||{3}||[[Cube|{4,3}]]||[[24-cell|{3,4,3}]]||[[16-cell honeycomb|{3,3,4,3}]]||{3,4,3,3,3}

|- BGCOLOR="#e0e0ff" align=center
|[[24-cell honeycomb honeycomb|24-cell honeycomb honeycomb]]||{3,4,3,3,3}||[[24-cell honeycomb|{3,4,3,3}]]||[[24-cell|{3,4,3}]]||[[Octahedron|{3,4}]]||{3}||{3}||[[Tetrahedron|{3,3}]]||[[5-cell|{3,3,3}]]||[[5-cube|{4,3,3,3}]]||{3,3,3,4,3}

|- BGCOLOR="#e0ffe0" align=center
|[[16-cell honeycomb honeycomb|16-cell honeycomb honeycomb]]||{3,3,4,3,3}||[[16-cell honeycomb|{3,3,4,3}]]||[[16-cell|{3,3,4}]]||[[Tetrahedron|{3,3}]]||{3}||{3}||[[Tetrahedron|{3,3}]]||[[Tesseract|{4,3,3}]]||[[24-cell honeycomb|{3,4,3,3}]]||self-dual

|- BGCOLOR="#ffe0e0" align=center
|[[Order-4 24-cell honeycomb honeycomb]]||{3,4,3,3,4}||[[16-cell honeycomb|{3,4,3,3}]]||[[24-cell|{3,4,3}]]||[[Octahedron|{3,4}]]||{3}||{4}||[[Octahedron|{3,4}]]||[[16-cell|{3,3,4}]]||[[Tesseractic honeycomb|{4,3,3,4}]]||{4,3,3,4,3}

|- BGCOLOR="#e0e0ff" align=center
|[[Tesseractic honeycomb honeycomb]]||{4,3,3,4,3}||[[Tesseractic honeycomb|{4,3,3,4}]]||[[Tesseract|{4,3,3}]]||[[Cube|{4,3}]]||{4}||{3}||[[Cube|{4,3}]]||[[24-cell|{3,4,3}]]||[[16-cell honeycomb|{3,3,4,3}]]||{3,4,3,3,4}

|}

Since there are no regular star ''n''-polytopes for ''n''&nbsp;≥&nbsp;5, that could be potential cells or vertex figures, there are no more hyperbolic star honeycombs in H<sup>''n''</sup> for ''n''&nbsp;≥&nbsp;5.

===Apeirotopes of rank 7 or more===

====Tessellations of hyperbolic 6-space and higher====
There are no regular compact or paracompact tessellations of hyperbolic space of dimension 6 or higher. However, any Schläfli symbol of the form {p,q,r,s,...} not covered above (p,q,r,s,... natural numbers above 2, or infinity) will form a noncompact tessellation of hyperbolic ''n''-space.<ref name=hyphon/>

== Compound polytopes ==
{{main article|Polytope compound}}

=== Two dimensional compounds ===
For any natural number n, there are n-pointed star regular polygonal stars with Schläfli symbols {n/m} for all m such that m < n/2 (strictly speaking {n/m}={n/(n−m)}) and m and n are coprime. When m and n are not coprime, the star polygon obtained will be a regular polygon with ''n''/''m'' sides. A new figure is obtained by rotating these regular ''n''/''m''-gons one vertex to the left on the original polygon until the number of vertices rotated equals ''n''/''m'' minus one, and combining these figures. An extreme case of this is where ''n''/''m'' is 2, producing a figure consisting of ''n''/2 straight line segments; this is called a '''[[Degeneracy (mathematics)|degenerate]] star polygon'''.

In other cases where ''n'' and ''m'' have a common factor, a star polygon for a lower ''n'' is obtained, and rotated versions can be combined. These figures are called '''star figures''', '''improper star polygons''' or '''compound polygons'''. The same notation {''n''/''m''} is often used for them, although authorities such as Grünbaum (1994) regard (with some justification) the form ''k''{''n''} as being more correct, where usually ''k'' = ''m''.

A further complication comes when we compound two or more star polygons, as for example two pentagrams, differing by a rotation of 36°, inscribed in a decagon. This is correctly written in the form ''k''{''n''/''m''}, as 2{5/2}, rather than the commonly used {10/4}.

Coxeter's extended notation for compounds is of the form ''c''{''m'',''n'',...}[''d''{''p'',''q'',...}]''e''{''s'',''t'',...}, indicating that ''d'' distinct {''p'',''q'',...}'s together  cover the vertices of {''m'',''n'',...} ''c'' times and the facets of {''s'',''t'',...} ''e'' times. If no regular {''m'',''n'',...} exists, the first part of the notation is removed, leaving [''d''{''p'',''q'',...}]''e''{''s'',''t'',...}; the opposite holds if no regular {''s'',''t'',...} exists. The dual of ''c''{''m'',''n'',...}[''d''{''p'',''q'',...}]''e''{''s'',''t'',...} is ''e''{''t'',''s'',...}[''d''{''q'',''p'',...}]''c''{''n'',''m'',...}. If ''c'' or ''e'' are 1, they may be omitted. For compound polygons, this notation reduces to {''nk''}[''k''{''n''/''m''}]{''nk''}: for example, the [[Hexagram|hexagram]] may be written thus as {6}[2{3}]{6}.

{| class=wikitable
|+ Examples for ''n''=2..10, ''nk''≤30
|- align=center
|[[File:regular_star_figure_2(2,1).svg|50px]]<BR>2{2}
|[[File:regular_star_figure_3(2,1).svg|50px]]<BR>3{2}
|[[File:regular_star_figure_4(2,1).svg|50px]]<BR>4{2}
|[[File:regular_star_figure_5(2,1).svg|50px]]<BR>5{2}
|[[File:regular_star_figure_6(2,1).svg|50px]]<BR>6{2}
|[[File:regular_star_figure_7(2,1).svg|50px]]<BR>7{2}
|[[File:regular_star_figure_8(2,1).svg|50px]]<BR>8{2}
|[[File:regular_star_figure_9(2,1).svg|50px]]<BR>9{2}
|[[File:regular_star_figure_10(2,1).svg|50px]]<BR>10{2}
|[[File:regular_star_figure_11(2,1).svg|50px]]<BR>11{2}
|[[File:regular_star_figure_12(2,1).svg|50px]]<BR>12{2}
|[[File:regular_star_figure_13(2,1).svg|50px]]<BR>13{2}
|[[File:regular_star_figure_14(2,1).svg|50px]]<BR>14{2}
|[[File:regular_star_figure_15(2,1).svg|50px]]<BR>15{2}
|- align=center
|[[File:regular_star_figure_2(3,1).svg|50px]]<BR>[[Hexagram|2{3}]]
|[[File:regular_star_figure_3(3,1).svg|50px]]<BR>[[Enneagram (geometry)|3{3}]]
|[[File:regular_star_figure_4(3,1).svg|50px]]<BR>[[Dodecagram|4{3}]]
|[[File:regular_star_figure_5(3,1).svg|50px]]<BR>[[Pentadecagram|5{3}]]
|[[File:regular_star_figure_6(3,1).svg|50px]]<BR>[[Octadecagram|6{3}]]
|[[File:regular_star_figure_7(3,1).svg|50px]]<BR>7{3}
|[[File:regular_star_figure_8(3,1).svg|50px]]<BR>8{3}
|[[File:regular_star_figure_9(3,1).svg|50px]]<BR>9{3}
|[[File:regular_star_figure_10(3,1).svg|50px]]<BR>10{3}
|[[File:regular_star_figure_2(4,1).svg|50px]]<BR>[[Octagram|2{4}]]
|[[File:regular_star_figure_3(4,1).svg|50px]]<BR>[[Dodecagram|3{4}]]
|[[File:regular_star_figure_4(4,1).svg|50px]]<BR>[[Hexadecagram|4{4}]]
|[[File:regular_star_figure_5(4,1).svg|50px]]<BR>[[Icosagram|5{4}]]
|[[File:regular_star_figure_6(4,1).svg|50px]]<BR>6{4}
|[[File:regular_star_figure_7(4,1).svg|50px]]<BR>7{4}
|- align=center
|[[File:regular_star_figure_2(5,1).svg|50px]]<BR>[[Decagram (geometry)|2{5}]]
|[[File:regular_star_figure_3(5,1).svg|50px]]<BR>[[Pentadecagram|3{5}]]
|[[File:regular_star_figure_4(5,1).svg|50px]]<BR>[[Icosagram|4{5}]]
|[[File:regular_star_figure_5(5,1).svg|50px]]<BR>5{5}
|[[File:regular_star_figure_6(5,1).svg|50px]]<BR>6{5}
|[[File:regular_star_figure_2(5,2).svg|50px]]<BR>2{5/2}
|[[File:regular_star_figure_3(5,2).svg|50px]]<BR>3{5/2}
|[[File:regular_star_figure_4(5,2).svg|50px]]<BR>4{5/2}
|[[File:regular_star_figure_5(5,2).svg|50px]]<BR>5{5/2}
|[[File:regular_star_figure_6(5,2).svg|50px]]<BR>6{5/2}
|[[File:regular_star_figure_2(6,1).svg|50px]]<BR>[[Dodecagram|2{6}]]
|[[File:regular_star_figure_3(6,1).svg|50px]]<BR>[[Octadecagram|3{6}]]
|[[File:regular_star_figure_4(6,1).svg|50px]]<BR>4{6}
|[[File:regular_star_figure_5(6,1).svg|50px]]<BR>5{6}
|- align=center
|[[File:regular_star_figure_2(7,1).svg|50px]]<BR>[[Tetradecagram|2{7}]]
|[[File:regular_star_figure_3(7,1).svg|50px]]<BR>3{7}
|[[File:regular_star_figure_4(7,1).svg|50px]]<BR>4{7}
|[[File:regular_star_figure_2(7,2).svg|50px]]<BR>2{7/2}
|[[File:regular_star_figure_3(7,2).svg|50px]]<BR>3{7/2}
|[[File:regular_star_figure_4(7,2).svg|50px]]<BR>4{7/2}
|[[File:regular_star_figure_2(7,3).svg|50px]]<BR>2{7/3}
|[[File:regular_star_figure_3(7,3).svg|50px]]<BR>3{7/3}
|[[File:regular_star_figure_4(7,3).svg|50px]]<BR>4{7/3}
|[[File:regular_star_figure_2(8,1).svg|50px]]<BR>[[Hexadecagram|2{8}]]
|[[File:regular_star_figure_3(8,1).svg|50px]]<BR>3{8}
|[[File:regular_star_figure_2(8,3).svg|50px]]<BR>2{8/3}
|[[File:regular_star_figure_3(8,3).svg|50px]]<BR>3{8/3}
|- align=center
|[[File:regular_star_figure_2(9,1).svg|50px]]<BR>[[Octadecagram|2{9}]]
|[[File:regular_star_figure_3(9,1).svg|50px]]<BR>3{9}
|[[File:regular_star_figure_2(9,2).svg|50px]]<BR>2{9/2}
|[[File:regular_star_figure_3(9,2).svg|50px]]<BR>3{9/2}
|[[File:regular_star_figure_2(9,4).svg|50px]]<BR>2{9/4}
|[[File:regular_star_figure_3(9,4).svg|50px]]<BR>3{9/4}
|[[File:regular_star_figure_2(10,1).svg|50px]]<BR>[[Icosagram|2{10}]]
|[[File:regular_star_figure_3(10,1).svg|50px]]<BR>3{10}
|[[File:regular_star_figure_2(10,3).svg|50px]]<BR>2{10/3}
|[[File:regular_star_figure_3(10,3).svg|50px]]<BR>3{10/3}
|- align=center
|[[File:regular_star_figure_2(11,1).svg|50px]]<BR>2{11}
|[[File:regular_star_figure_2(11,2).svg|50px]]<BR>2{11/2}
|[[File:regular_star_figure_2(11,3).svg|50px]]<BR>2{11/3}
|[[File:regular_star_figure_2(11,4).svg|50px]]<BR>2{11/4}
|[[File:regular_star_figure_2(11,5).svg|50px]]<BR>2{11/5}
|[[File:regular_star_figure_2(12,1).svg|50px]]<BR>[[Icositetragon|2{12}]]
|[[File:regular_star_figure_2(12,5).svg|50px]]<BR>2{12/5}
|[[File:regular_star_figure_2(13,1).svg|50px]]<BR>2{13}
|[[File:regular_star_figure_2(13,2).svg|50px]]<BR>2{13/2}
|[[File:regular_star_figure_2(13,3).svg|50px]]<BR>2{13/3}
|[[File:regular_star_figure_2(13,4).svg|50px]]<BR>2{13/4}
|[[File:regular_star_figure_2(13,5).svg|50px]]<BR>2{13/5}
|[[File:regular_star_figure_2(13,6).svg|50px]]<BR>2{13/6}
|- align=center
|[[File:regular_star_figure_2(14,1).svg|50px]]<BR>2{14}
|[[File:regular_star_figure_2(14,3).svg|50px]]<BR>2{14/3}
|[[File:regular_star_figure_2(14,5).svg|50px]]<BR>2{14/5}
|[[File:regular_star_figure_2(15,1).svg|50px]]<BR>[[Triacontagon|2{15}]]
|[[File:regular_star_figure_2(15,2).svg|50px]]<BR>2{15/2}
|[[File:regular_star_figure_2(15,4).svg|50px]]<BR>2{15/4}
|[[File:regular_star_figure_2(15,7).svg|50px]]<BR>2{15/7}
|}

Regular skew polygons also create compounds, seen in the edges of [[Prismatic compound of antiprisms|prismatic compound of antiprisms]], for instance:
{| class=wikitable
|+ Regular compound skew polygon
|- align=center
|colspan=2|Compound<BR>skew squares
|Compound<BR>skew hexagons
|Compound<BR>skew decagons
|- align=center
|Two {2}#{ }
|Three {2}#{ }
|Two {3}#{ }
|Two {5/3}#{ }
|- align=center
|[[File:Compound_skew_square_in_cube.png|150px]]
|[[File:Skew_tetragons_in_compound_of_three_digonal_antiprisms.png|150px]]
|[[File:Compound_skew_hexagon_in_hexagonal_prism.png|150px]]
|[[File:Compound_skew_hexagon_in_pentagonal_crossed_antiprism.png|150px]]
|}

===Three dimensional compounds===
A regular polyhedron compound can be defined as a compound which, like a regular polyhedron, is vertex-transitive, edge-transitive, and face-transitive. With this definition there are 5 regular compounds.

{| class="wikitable"
|- align=center
!Symmetry
|[4,3], O<sub>h</sub>
|[5,3]<sup>+</sup>, I
|colspan=3|[5,3], I<sub>h</sub>
|- align=center
!Duality
!colspan=3|Self-dual
!colspan=2|Dual pairs
|- align=center
!Image
| [[Image:Compound of two tetrahedra.png|100px]]
| [[Image:Compound of five tetrahedra.png|100px]]
| [[Image:Compound of ten tetrahedra.png|100px]]
| [[Image:Compound of five cubes.png|100px]]
| [[Image:Compound of five octahedra.png|100px]]
|- align=center
!Spherical
| [[Image:Spherical compound of two tetrahedra.png|100px]]
| [[Image:Spherical compound of five tetrahedra.png|100px]]
| [[Image:Spherical compound of ten tetrahedra.png|100px]]
| [[Image:Spherical compound of five cubes.png|100px]]
| [[Image:Spherical compound of five octahedra.png|100px]]
|- align=center
!Polyhedra
| [[Compound of two tetrahedra|2 {3,3}]]
| [[Compound of five tetrahedra|5 {3,3}]]
| [[Compound of ten tetrahedra|10 {3,3}]]
| [[Compound of five cubes|5 {4,3}]]
| [[Compound of five octahedra|5 {3,4}]]
|- align=center
!Coxeter
| [[Cube|{4,3}]][2[[Tetrahedron|{3,3}]]][[Octahedron|{3,4}]]
| [[Dodecahedron|{5,3}]][5[[Tetrahedron|{3,3}]]][[Icosahedron|{3,5}]]
| 2[[Dodecahedron|{5,3}]][10[[Tetrahedron|{3,3}]]]2[[Icosahedron|{3,5}]]
| 2[[Dodecahedron|{5,3}]][5[[Cube|{4,3}]]]
| [5[[Octahedron|{3,4}]]]2[[Icosahedron|{3,5}]]
|}

Coxeter's notation for regular compounds is given in the table above, incorporating [[Schläfli symbol]]s. The material inside the square brackets, [''d''{''p'',''q''}], denotes the components of the compound: ''d'' separate {''p'',''q''}'s. The material ''before'' the square brackets denotes the vertex arrangement of the compound: ''c''{''m'',''n''}[''d''{''p'',''q''}] is a compound of ''d'' {''p'',''q''}'s sharing the vertices of an {''m'',''n''} counted ''c'' times. The material ''after'' the square brackets denotes the facet arrangement of the compound: [''d''{''p'',''q''}]''e''{''s'',''t''} is a compound of ''d'' {''p'',''q''}'s sharing the faces of {''s'',''t''} counted ''e'' times. These may be combined: thus ''c''{''m'',''n''}[''d''{''p'',''q''}]''e''{''s'',''t''}  is a compound of ''d'' {''p'',''q''}'s sharing the vertices of {''m'',''n''} counted ''c'' times ''and'' the faces of {''s'',''t''} counted ''e'' times. This notation can be generalised to compounds in any number of dimensions.{{sfnp|Coxeter|1973|p=48}}

==== Euclidean and hyperbolic plane compounds ====
There are eighteen two-parameter families of regular compound tessellations of the Euclidean plane. In the hyperbolic plane, five one-parameter families and seventeen isolated cases are known, but the completeness of this listing has not yet been proven.

The Euclidean and hyperbolic compound families 2 {''p'',''p''} (4 ≤ ''p'' ≤ &infin;, ''p'' an integer) are analogous to the spherical stella octangula, 2 {3,3}.

{| class="wikitable"
|+ A few examples of Euclidean and hyperbolic regular compounds
!Self-dual
!colspan=2|Duals
!Self-dual
|-
!2 [[Square tiling|{4,4}]]
!2 [[Hexagonal tiling|{6,3}]]
!2 [[Triangular tiling|{3,6}]]
!2 [[Infinite-order apeirogonal tiling|{&infin;,&infin;}]]
|- align=center
|[[File:Kah 4 4.png|160px]] 
|[[File:Compound_2_hexagonal_tilings.png|160px]]
|[[File:Compound_2_triangular_tilings.png|160px]]
| [[File:Infinite-order apeirogonal tiling and dual.png|160px]]
|- align=center
|<nowiki>{{4,4}}</nowiki> or a{4,4} or {4,4}[2{4,4}]{4,4}<BR>{{CDD|nodes_10ru|split2-44|node}} +  {{CDD|nodes_01rd|split2-44|node}} or {{CDD|node_h3|4|node|4|node}}
| [2{6,3}]{3,6}
| a{6,3} or {6,3}[2{3,6}]<BR> {{CDD|branch_10ru|split2|node}} + {{CDD|branch_01rd|split2|node}} or {{CDD|node_h3|6|node|3|node}}
|<nowiki>{{&infin;,&infin;}}</nowiki> or a{&infin;,&infin;} or {4,&infin;}[2{&infin;,&infin;}]{&infin;,4}<BR> {{CDD|labelinfin|branch_10ru|split2-ii|node}} + {{CDD|labelinfin|branch_01rd|split2-ii|node}} or {{CDD|node_h3|infin|node|infin|node}}
|-
!
!3 {6,3}
!3 {3,6}
!3 [[Infinite-order apeirogonal tiling|{&infin;,&infin;}]]
|- align=center
|
|[[File:Compound_3_hexagonal_tilings.png|160px]]
|[[File:Compound_3_triangular_tilings.png|160px]]
|[[File:Iii symmetry 000.png|160px]]
|- align=center
|
|2{3,6}[3{6,3}]{6,3}
|{3,6}[3{3,6}]2{6,3}<br>{{CDD|branch_10ru|split2|node}} + {{CDD|branch_01rd|split2|node}} + {{CDD|branch|split2|node_1}} 
|<br>{{CDD|labelinfin|branch_10ru|split2-ii|node}} + {{CDD|labelinfin|branch_01rd|split2-ii|node}} + {{CDD|labelinfin|branch|split2-ii|node_1}}
|}

=== Four dimensional compounds ===
{| class=wikitable align=right
|+ Orthogonal projections
|[[File:Regular compound 75 tesseracts.png|120px]]
|[[File:Regular compound 75 16-cells.png|120px]]
|-
!75 [[Tesseract|{4,3,3}]]
!75 [[16-cell|{3,3,4}]]
|}

Coxeter lists 32 regular compounds of regular 4-polytopes in his book ''Regular Polytopes''.<ref name="cox">{{harvtxt|Coxeter|1973}}. Table VII, p. 305</ref> [[Biography:Peter McMullen|McMullen]] adds six in his paper ''New Regular Compounds of 4-Polytopes'', in which he also proves that the list is now complete.{{sfnp|McMullen|2018}} In the following tables, the superscript (var) indicates that the labeled compounds are distinct from the other compounds with the same symbols.

{| class=wikitable
|+ Self-dual regular compounds
!Compound 
!Constituent
!Symmetry
! Vertex arrangement
! Cell arrangement
|-
| 120 {3,3,3} || [[5-cell|5-cell]]  ||[5,3,3], order 14400<ref name="cox"/> || {5,3,3} || {3,3,5}
|-
| 120 {3,3,3}<sup>(var)</sup> || [[5-cell|5-cell]]  ||order 1200{{sfnp|McMullen|2018}} || {5,3,3} || {3,3,5}
|-
| 720 {3,3,3} || [[5-cell|5-cell]]  ||[5,3,3], order 14400{{sfnp|McMullen|2018}} || 6{5,3,3} || 6{3,3,5}
|-
| 5 {3,4,3} || [[24-cell|24-cell]]  || [5,3,3], order 14400<ref name="cox"/> || {3,3,5} || {5,3,3}
|}

{| class=wikitable
|+ Regular compounds as dual pairs
!Compound 1
!Compound 2
!Symmetry
!Vertex arrangement (1)
!Cell arrangement (1)
!Vertex arrangement (2)
!Cell arrangement (2)
|-
|3 [[16-cell|{3,3,4}]]<ref>{{KlitzingPolytopes|..//incmats/stico.htm|Uniform compound|stellated icositetrachoron}}</ref>||3 [[Tesseract|{4,3,3}]]||[3,4,3], order 1152<ref name="cox"/> || {3,4,3} || 2{3,4,3} || 2{3,4,3} || {3,4,3}
|-
|15 [[16-cell|{3,3,4}]]||15 [[Tesseract|{4,3,3}]]||[5,3,3], order 14400<ref name="cox"/> || {3,3,5} || 2{5,3,3} || 2{3,3,5} || {5,3,3}
|-
|75 [[16-cell|{3,3,4}]]||75 [[Tesseract|{4,3,3}]]||[5,3,3], order 14400<ref name="cox"/> || 5{3,3,5} || 10{5,3,3} || 10{3,3,5} || 5{5,3,3}
|-
|75 [[16-cell|{3,3,4}]]||75 [[Tesseract|{4,3,3}]]||[5,3,3], order 14400<ref name="cox"/> || {5,3,3} || 2{3,3,5} || 2{5,3,3} || {3,3,5}
|-
|75 [[16-cell|{3,3,4}]]||75 [[Tesseract|{4,3,3}]]||order 600{{sfnp|McMullen|2018}} || {5,3,3} || 2{3,3,5} || 2{5,3,3} || {3,3,5}
|-
|300 [[16-cell|{3,3,4}]]||300 [[Tesseract|{4,3,3}]]||[5,3,3]<sup>+</sup>, order 7200<ref name="cox"/> || 4{5,3,3} || 8{3,3,5} || 8{5,3,3} || 4{3,3,5}
|-
|600 [[16-cell|{3,3,4}]]||600 [[Tesseract|{4,3,3}]]||[5,3,3], order 14400<ref name="cox"/> || 8{5,3,3} || 16{3,3,5} || 16{5,3,3} || 8{3,3,5}
|-
|25 [[24-cell|{3,4,3}]]||25 [[24-cell|{3,4,3}]]||[5,3,3], order 14400<ref name="cox"/> || {5,3,3} || 5{5,3,3} || 5{3,3,5} || {3,3,5}
|}

There are two different compounds of 75 tesseracts: one shares the vertices of a 120-cell, while the other shares the vertices of a 600-cell. It immediately follows therefore that the corresponding dual compounds of 75 16-cells are also different.

{| class=wikitable
|+ Self-dual star compounds
!Compound 
!Symmetry
!Vertex arrangement
!Cell arrangement
|-
| 5 [[Great 120-cell|{5,5/2,5}]] || [5,3,3]<sup>+</sup>, order 7200<ref name="cox"/> || {5,3,3} || {3,3,5}
|-
| 10 [[Great 120-cell|{5,5/2,5}]] || [5,3,3], order 14400<ref name="cox"/> || 2{5,3,3} || 2{3,3,5}
|-
| 5 [[Grand stellated 120-cell|{5/2,5,5/2}]] || [5,3,3]<sup>+</sup>, order 7200<ref name="cox"/> || {5,3,3} || {3,3,5}
|-
| 10 [[Grand stellated 120-cell|{5/2,5,5/2}]] || [5,3,3], order 14400<ref name="cox"/> || 2{5,3,3} || 2{3,3,5}
|}

{| class=wikitable
|+ Regular star compounds as dual pairs
!Compound 1
!Compound 2
!Symmetry
!Vertex arrangement (1)
!Cell arrangement (1)
!Vertex arrangement (2)
!Cell arrangement (2)
|-
|5 [[Icosahedral 120-cell|{3,5,5/2}]]||5 [[Small stellated 120-cell|{5/2,5,3}]]||[5,3,3]<sup>+</sup>, order 7200<ref name="cox"/> || {5,3,3} || {3,3,5} || {5,3,3} || {3,3,5}
|-
|10 [[Icosahedral 120-cell|{3,5,5/2}]]||10 [[Small stellated 120-cell|{5/2,5,3}]]||[5,3,3], order 14400<ref name="cox"/> || 2{5,3,3} || 2{3,3,5} || 2{5,3,3} || 2{3,3,5}
|-
|5 [[Great grand 120-cell|{5,5/2,3}]]||5 [[Great icosahedral 120-cell|{3,5/2,5}]]||[5,3,3]<sup>+</sup>, order 7200<ref name="cox"/> || {5,3,3} || {3,3,5} || {5,3,3} || {3,3,5}
|-
|10 [[Great grand 120-cell|{5,5/2,3}]]||10 [[Great icosahedral 120-cell|{3,5/2,5}]]||[5,3,3], order 14400<ref name="cox"/> || 2{5,3,3} || 2{3,3,5} || 2{5,3,3} || 2{3,3,5}
|-
|5 [[Great stellated 120-cell|{5/2,3,5}]]||5 [[Grand 120-cell|{5,3,5/2}]]||[5,3,3]<sup>+</sup>, order 7200<ref name="cox"/> || {5,3,3} || {3,3,5} || {5,3,3} || {3,3,5}
|-
|10 [[Great stellated 120-cell|{5/2,3,5}]]||10 [[Grand 120-cell|{5,3,5/2}]]||[5,3,3], order 14400<ref name="cox"/> || 2{5,3,3} || 2{3,3,5} || 2{5,3,3} || 2{3,3,5}
|}

There are also fourteen ''partially regular'' compounds, that are either vertex-transitive or cell-transitive but not both. The seven vertex-transitive partially regular compounds are the duals of the seven cell-transitive partially regular compounds.
{| class=wikitable
|+ Partially regular compounds as dual pairs
!Compound 1<BR>Vertex-transitive
!Compound 2<BR>Cell-transitive
!Symmetry
|-
|2 [[16-cell|16-cell]]s<ref>{{KlitzingPolytopes|..//incmats/haddet.htm|Uniform compound|demidistesseract}}</ref>||2 [[Tesseract|tesseract]]s||[4,3,3], order 384<ref name="cox"/>
|-
|25 [[24-cell|24-cell]]<sup>(var)</sup>||25 [[24-cell|24-cell]]<sup>(var)</sup>||order 600{{sfnp|McMullen|2018}}
|-
|100 [[24-cell|24-cell]]||100 [[24-cell|24-cell]]||[5,3,3]<sup>+</sup>, order 7200<ref name="cox"/>
|-
|200 [[24-cell|24-cell]]||200 [[24-cell|24-cell]]||[5,3,3], order 14400<ref name="cox"/>
|-
|5 [[600-cell|600-cell]]||5 [[120-cell|120-cell]]||[5,3,3]<sup>+</sup>, order 7200<ref name="cox"/>
|-
|10 [[600-cell|600-cell]]||10 [[120-cell|120-cell]]||[5,3,3], order 14400<ref name="cox"/>
|}
{| class=wikitable
|+ Partially regular star compounds as dual pairs
!Compound 1<BR>Vertex-transitive
!Compound 2<BR>Cell-transitive
!Symmetry
|-
|5 [[Grand 600-cell|{3,3,5/2}]]||5 [[Great grand stellated 120-cell|{5/2,3,3}]]||[5,3,3]<sup>+</sup>, order 7200<ref name="cox"/>
|-
|10 [[Grand 600-cell|{3,3,5/2}]]||10 [[Great grand stellated 120-cell|{5/2,3,3}]]||[5,3,3], order 14400<ref name="cox"/>
|}

Although the 5-cell and 24-cell are both self-dual, their dual compounds (the compound of two 5-cells and compound of two 24-cells) are not considered to be regular, unlike the compound of two tetrahedra and the various dual polygon compounds, because they are neither vertex-regular nor cell-regular: they are not facetings or stellations of any regular 4-polytope. However, they are vertex-, edge-, face-, and cell-transitive.

==== Euclidean 3-space compounds ====
The only regular Euclidean compound honeycombs are an infinite family of compounds of [[Cubic honeycomb|cubic honeycomb]]s, all sharing vertices and faces with another cubic honeycomb. This compound can have any number of cubic honeycombs. The Coxeter notation is {4,3,4}[''d''{4,3,4}]{4,3,4}.

=== Five dimensions and higher compounds===
There are no regular compounds in five or six dimensions. There are three known seven-dimensional compounds (16, 240, or 480 [[7-simplex|7-simplices]]), and six known eight-dimensional ones (16, 240, or 480 8-cubes or 8-orthoplexes). There is also one compound of ''n''-simplices in ''n''-dimensional space provided that ''n'' is one less than a power of two, and also two compounds (one of ''n''-cubes and a dual one of ''n''-orthoplexes) in ''n''-dimensional space if ''n'' is a power of two.

The Coxeter notation for these compounds are (using α<sup>''n''</sup> = {3<sup>''n''−1</sup>}, β<sup>''n''</sup> = {3<sup>''n''−2</sup>,4}, γ<sub>''n''</sub> = {4,3<sup>''n''−2</sup>}):

* 7-simplexes: ''c''γ<sub>7</sub>[16''c''α<sub>7</sub>]''c''β<sub>7</sub>, where ''c'' = 1, 15, or 30
* 8-orthoplexes: ''c''γ<sub>8</sub>[16''c''β<sub>8</sub>]
* 8-cubes: [16''c''γ<sub>8</sub>]''c''β<sub>8</sub>

The general cases (where ''n'' = 2<sup>''k''</sup> and ''d'' = 2<sup>2<sup>''k''</sup> − ''k'' − 1</sup>, ''k'' = 2, 3, 4, ...):

* Simplexes: γ<sub>''n''−1</sub>[''d''α<sub>''n''−1</sub>]β<sub>''n''−1</sub>
* Orthoplexes: γ<sub>''n''</sub>[''d''β<sub>''n''</sub>]
* Hypercubes: [''d''γ<sub>''n''</sub>]β<sub>''n''</sub>

==== Euclidean honeycomb compounds====
A known family of regular Euclidean compound honeycombs in five or more dimensions is an infinite family of compounds of [[Hypercubic honeycomb|hypercubic honeycomb]]s, all sharing vertices and faces with another hypercubic honeycomb. This compound can have any number of hypercubic honeycombs. The Coxeter notation is δ<sub>''n''</sub>[''d''δ<sub>''n''</sub>]δ<sub>''n''</sub> where δ<sub>''n''</sub> = {∞} when ''n'' = 2 and {4,3<sup>''n''−3</sup>,4} when ''n'' ≥ 3.

== Abstract polytopes ==

The [[Abstract polytope|abstract polytope]]s arose out of an attempt to study polytopes apart from the geometrical space they are embedded in. They include the tessellations of spherical, Euclidean and hyperbolic space, and of other [[Manifold|manifold]]s. There are infinitely many of every rank greater than 1. See [http://www.abstract-polytopes.com/atlas/ this atlas] for a sample. Some notable examples of abstract regular polytopes that do not appear elsewhere in this list are the [[11-cell|11-cell]], {3,5,3}, and the [[57-cell|57-cell]], {5,3,5}, which have regular projective polyhedra as cells and vertex figures.

The elements of an abstract polyhedron are its body (the maximal element), its faces, edges, vertices and the ''null polytope'' or empty set. These abstract elements can be mapped into ordinary space or ''realised'' as geometrical figures. Some abstract polyhedra have well-formed or ''faithful'' realisations, others do not. A ''flag'' is a connected set of elements of each rank - for a polyhedron that is the body, a face, an edge of the face, a vertex of the edge, and the null polytope. An abstract polytope is said to be ''regular'' if its combinatorial symmetries are transitive on its flags - that is to say, that any flag can be mapped onto any other under a symmetry of the polyhedron. Abstract regular polytopes remain an active area of research.

Five such regular abstract polyhedra, which can not be realised faithfully and symmetrically, were identified by H. S. M. Coxeter in his book ''Regular Polytopes'' (1977) and again by J. M. Wills in his paper "The combinatorially regular polyhedra of index 2" (1987).<ref>{{cite web |url=http://homepages.wmich.edu/~drichter/regularpolyhedra.htm |title=The Regular Polyhedra (of index two) |author=David A. Richter}}</ref> They are all topologically equivalent to [[Toroid|toroid]]s. Their construction, by arranging ''n'' faces around each vertex, can be repeated indefinitely as tilings of the [[Hyperbolic geometry#Models of the hyperbolic plane|hyperbolic plane]]. In the diagrams below, the hyperbolic tiling images have colors corresponding to those of the polyhedra images.

:{| class="wikitable" width=400
|- align=center 
! Polyhedron
|[[Image:DU36 medial rhombic triacontahedron.png|100px]]<br>Medial rhombic triacontahedron
|[[Image:Dodecadodecahedron.png|100px]]<br>[[Dodecadodecahedron]]
|[[Image:DU41 medial triambic icosahedron.png|100px]]<br>Medial triambic icosahedron
|[[Image:Ditrigonal dodecadodecahedron.png|100px]]<br>[[Ditrigonal dodecadodecahedron]]
|[[Image:Excavated dodecahedron.png|100px]]<br>Excavated dodecahedron
|- align=center
![[Vertex figure]]
|{5}, {5/2}<br>[[File:Regular polygon 5.svg|40px]][[File:Pentagram green.svg|40px]]
|(5.5/2)<sup>2</sup><br>[[File:Dodecadodecahedron vertfig.png|60px]]
|{5}, {5/2}<br>[[File:Regular polygon 5.svg|40px]][[File:Pentagram green.svg|40px]]
|(5.5/3)<sup>3</sup><br>[[File:Ditrigonal dodecadodecahedron vertfig.png|60px]]
|[[File:Medial triambic icosahedron face.svg|60px]]
|- align=center valign=top
!Faces
|30 rhombi<br>[[File:Rhombus definition2.svg|60px]]
|12 pentagons<br>12 pentagrams<br>[[File:Regular polygon 5.svg|40px]][[File:Pentagram green.svg|40px]]
|20 hexagons<br>[[File:Medial triambic icosahedron face.svg|60px]]
|12 pentagons<br>12 pentagrams<br>[[File:Regular polygon 5.svg|40px]][[File:Pentagram green.svg|40px]]
|20 hexagrams<br>[[File:Star hexagon face.png|60px]]
|- align=center
! Tiling
|[[Image:Uniform tiling 45-t0.png|100px]]<br>[[Order-5 square tiling|{4, 5}]]
|[[Image:Uniform tiling 552-t1.png|100px]]<br>[[Order-4 pentagonal tiling|{5, 4}]]
|[[Image:Uniform tiling 65-t0.png|100px]]<br>[[Order-5 hexagonal tiling|{6, 5}]]
|[[Image:Uniform tiling 553-t1.png|100px]]<br>[[Order-6 pentagonal tiling|{5, 6}]]
|[[Image:Uniform tiling 66-t2.png|100px]]<br>[[Order-6 hexagonal tiling|{6, 6}]]
|- align=center
! [[Euler characteristic|χ]]
| −6
| −6
| −16
| −16
| −20
|}

These occur as dual pairs as follows:
*The medial rhombic triacontahedron and [[Dodecadodecahedron|dodecadodecahedron]] are dual to each other.
*The medial triambic icosahedron and [[Ditrigonal dodecadodecahedron|ditrigonal dodecadodecahedron]] are dual to each other.
*The excavated dodecahedron is self-dual.

== See also ==
* [[Polygon]]
** [[Regular polygon]]
** [[Star polygon]]
* [[Polyhedron]]
** [[Regular polyhedron]] (5 regular [[Platonic solid]]s and 4 Kepler–Poinsot solids)
*** [[Uniform polyhedron]]
** Petrial
* [[4-polytope|4-polytope]]
** [[Regular 4-polytope]] (16 regular 4-polytopes, 4 convex and 10 star (Schläfli–Hess))
** Uniform 4-polytope
* [[Tessellation]]
** Tilings of regular polygons
** [[Convex uniform honeycomb]]
* [[Regular polytope]]
** Uniform polytope
* [[Regular map (graph theory)]]

==Notes==
{{reflist|30em}}

== References ==
{{refbegin}}
*{{citation
 | chapter = Chapter 10: Regular Honeycombs in Hyperbolic Space |pages = 199–214
 | isbn = 0-486-40919-8
 | mr = 1717154
 | lccn = 99035678
 | publisher = Dover Publications, Inc. | location = Mineola, NY
 | title = The Beauty of Geometry: Twelve Essays
 | year = 1999}}. See in particular Summary Tables II, III, IV, V, pp.&nbsp;212&ndash;213.
**Originally published in {{citation |ref=none
 | contribution = Regular honeycombs in hyperbolic space
 | mr = 0087114
 | pages = 155–169
 | publisher = North-Holland Publishing Co. | location = Amsterdam
 | title = Proceedings of the International Congress of Mathematicians, 1954, Amsterdam |volume=III
 | contribution-url = http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf
 | archive-url = https://web.archive.org/web/20150402180536/http://www.mathunion.org/ICM/ICM1954.3/Main/icm1954.3.0155.0169.ocr.pdf |archive-date=2015-04-02 |url-status=dead
 | year = 1956}}.
*{{cite book |last=Coxeter |first=H. S. M. |date=1973 |orig-year=1948 |title=Regular Polytopes |location=New York |publisher=Dover Publications |edition=Third |oclc=798003 |isbn=0-486-61480-8 |mr=0370327}} See in particular Tables I and II: Regular polytopes and honeycombs, pp.&nbsp;294&ndash;296.
*{{citation
 | last = Johnson | first = Norman W.
 | contribution = Regular inversive polytopes 
 | pages = 85–95 [http://www.foibg.com/conf/ITA2012/MDA2012_Abstracts_Book.pdf Paper 27]
 | title = International Conference on Mathematics of Distances and Applications (July 2–5, 2012, Varna, Bulgaria)
 | contribution-url = http://www.foibg.com/ibs_isc/ibs-25/ibs-25-p07.pdf
 | year = 2012}}
*{{citation
 | last1 = McMullen
 | first1 = Peter
 | last2 = Schulte
 | first2 = Egon
 | doi = 10.1017/CBO9780511546686
 | isbn = 0-521-81496-0
 | mr = 1965665
 | publisher = Cambridge University Press
 | location = Cambridge
 | series = Encyclopedia of Mathematics and its Applications
 | title = Abstract Regular Polytopes
 | volume = 92
 | year = 2002
 | s2cid = 115688843
 | url-access = registration
 | url = https://archive.org/details/abstractregularp0000mcmu
 }}
*{{citation|first=Peter|last=McMullen|title=New Regular Compounds of 4-Polytopes|journal=New Trends in Intuitive Geometry|series=Bolyai Society Mathematical Studies|volume=27|pages=307–320|year=2018|doi=10.1007/978-3-662-57413-3_12|isbn=978-3-662-57412-6}}.
*{{cite arXiv |eprint=1511.02851 |title=Visualizing Hyperbolic Honeycombs |first1=Roice |last1=Nelson |first2=Henry |last2=Segerman |date=2015|class=math.HO }} [http://hyperbolichoneycombs.org hyperbolichoneycombs.org/]
*{{citation
 | mr = 0100239
 | publisher = Dover Publications, Inc. | location = New York
 | title = An Introduction to the Geometry of {{mvar|n}} Dimensions
 | year = 1958}}. Reprint of 1930 ed., published by E. P. Dutton. See in particular Chapter X: The Regular Polytopes.
{{refend}}

== External links ==
*[http://www.math.utah.edu/~alfeld/math/polyhedra/polyhedra.html The Platonic Solids]
*[http://www.georgehart.com/virtual-polyhedra/kepler-poinsot-info.html Kepler-Poinsot Polyhedra]
*[https://web.archive.org/web/20110717184828/http://weimholt.com/andrew/polytope.shtml Regular 4d Polytope Foldouts]
*[https://web.archive.org/web/20070207021813/http://members.aol.com/Polycell/glossary.html Multidimensional Glossary (Look up '''Hexacosichoron''' and '''Hecatonicosachoron''')]
*[http://www.stat.berkeley.edu/~evans/shapiro/TesseractApplet.html Polytope Viewer]
*[https://web.archive.org/web/20060615201915/http://presh.com/hovinga/ Polytopes and optimal packing of p points in n dimensional spheres]
*[http://www.abstract-polytopes.com/atlas/ An atlas of small regular polytopes]
* [https://www.fields.utoronto.ca/programs/scientific/11-12/discretegeom/gradcourses/Course_notes.pdf Regular polyhedra through time] I. Hubard, ''Polytopes, Maps and their Symmetries''
*[http://nan.ma/star/#/ Regular Star Polytopes], Nan Ma

{{Polytopes}}
{{Honeycombs}}
{{Tessellation}}

{{DEFAULTSORT:Polytopes, Regular}}
[[Category:Polytopes| Regular]]
[[Category:Multi-dimensional geometry]]
[[Category:Lists of shapes]]
[[Category:Mathematics-related lists]]

{{Sourceattribution|List of regular polytopes and compounds}}