View source for Regular 4-polytope

{{Short description|Four-dimensional analogues of the regular polyhedra in three dimensions}}
[[File:Hypercube.svg|thumb|The [[tesseract]] is one of 6 convex regular 4-polytopes]]
In [[mathematics]], a '''regular 4-polytope''' or '''regular polychoron''' is a [[regular polytope|regular]] [[4-polytope|four-dimensional polytope]]. They are the four-dimensional analogues of the [[Regular polyhedron|regular polyhedra]] in three dimensions and the [[regular polygon]]s in two dimensions.

There are six [[Convex polytope|convex]] and ten [[star polytope|star]] regular 4-polytopes, giving a total of sixteen.

== History ==
The convex regular 4-polytopes were first described by the Swiss [[mathematician]] [[Ludwig Schläfli]] in the mid-19th century.{{Sfn|Coxeter|1973|p=141|loc=§7-x. Historical remarks}} He discovered that there are precisely six such figures.

Schläfli also found four of the regular star 4-polytopes: the [[grand 120-cell]], [[great stellated 120-cell]], [[grand 600-cell]], and [[great grand stellated 120-cell]]. He skipped the remaining six because he would not allow forms that failed the [[Euler characteristic]] on cells or vertex figures (for zero-hole tori: ''F''&nbsp;&minus;&nbsp;''E''&nbsp;+&nbsp;''V''&nbsp;{{=}}&nbsp;2). That excludes cells and vertex figures such as the [[great dodecahedron]] {5,{{sfrac|5|2}}} and [[small stellated dodecahedron]] {{{sfrac|5|2}},5}.

[[Edmund Hess]] (1843&ndash;1903) published the complete list in his 1883 German book ''Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder''.

== Construction ==
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> which form its cells and a [[dihedral angle]] constraint
:<math>\sin\frac{\pi}p \sin\frac{\pi}r > \cos\frac{\pi}q</math>

to ensure that the cells meet to form a closed 3-surface.

The six convex and ten star polytopes described are the only solutions to these constraints.

There are four nonconvex [[Schläfli symbol]]s {p,q,r} that have valid cells {p,q} and vertex figures {q,r}, and pass the dihedral test, but fail to produce finite figures: {3,{{sfrac|5|2}},3}, {4,3,{{sfrac|5|2}}}, {{{sfrac|5|2}},3,4}, {{{sfrac|5|2}},3,{{sfrac|5|2}}}.

==Regular convex 4-polytopes==
The regular convex 4-polytopes are the four-dimensional analogues of the [[Platonic solid]]s in three dimensions and the convex [[regular polygon]]s in two dimensions.

Each convex regular 4-polytope is bounded by a set of 3-dimensional ''[[cell (mathematics)|cells]]'' which are all Platonic solids of the same type and size. These are fitted together along their respective faces (face-to-face) in a regular fashion, forming the ''surface'' of the 4-polytope which is a closed, curved 3-dimensional space (analogous to the way the surface of the earth is a closed, curved 2-dimensional space).

=== Properties ===

Like their 3-dimensional analogues, the convex regular 4-polytopes can be naturally ordered by size as a measure of 4-dimensional content (hypervolume) for the same radius. Each greater polytope in the sequence is ''rounder'' than its predecessor, enclosing more content within the same radius.{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii): The sixteen regular polytopes {''p,q,r''} in four dimensions}} The 4-simplex (5-cell) has the smallest content, and the 120-cell has the largest.

{{Regular convex 4-polytopes}}

The following table lists some properties of the six convex regular 4-polytopes. The symmetry groups of these 4-polytopes are all [[Coxeter group]]s and given in the notation described in that article. The number following the name of the group is the [[order (group theory)|order]] of the group.

{| class="wikitable sortable" style="text-align:left;"
! Names ||Image|| Family || [[Schläfli symbol|Schläfli]]<BR>[[Coxeter diagram|Coxeter]] || [[Vertex (geometry)|V]] || [[Edge (geometry)|E]] || [[Face (geometry)|F]] || [[Cell (geometry)|C]] || [[Vertex figure|Vert.<BR>fig.]] || [[Dual polytope|Dual]]
!colspan=2 | [[Coxeter group|Symmetry group]]
|- BGCOLOR="#e8ffe8" style="text-align:center;"
| [[5-cell|<u>5-cell</u><BR>pentachoron<BR>pentatope<BR>4-simplex]] || [[Image:4-simplex t0.svg|125px]] || [[simplex|''n''-simplex]]<br>(A<sub>n</sub> family) || {3,3,3}<BR>{{CDD|node_1|3|node|3|node|3|node}}|| 5 || 10 || 10<br>[[triangle|{3}]] || 5<br>[[tetrahedron|{3,3}]] || [[tetrahedron|{3,3}]] || ''self-dual'' || ''A''<sub>4</sub><BR>[3,3,3]|| 120
|- style="text-align:center; background: linear-gradient(0deg, rgb(240,224,240) 0%, rgb(224,224,255) 10%, rgb(224,224,255) 100%);"
| [[16-cell|<u>16-cell</u><BR>hexadecachoron<BR>4-orthoplex]] || [[Image:4-cube t3.svg|125px]]|| [[cross-polytope|''n''-orthoplex]]<br>(B<sub>n</sub> family) || {3,3,4}<BR>{{CDD|node_1|3|node|3|node|4|node}} || 8 || 24 || 32<br>[[triangle|{3}]] || 16<br>[[tetrahedron|{3,3}]] || [[octahedron|{3,4}]] || 8-cell || rowspan="2" style="background: linear-gradient(0deg, rgb(255,224,224) 0%, rgb(224,224,255) 100%);" | ''B''<sub>4</sub><BR>[4,3,3] || rowspan="2" style="background: linear-gradient(0deg, rgb(255,224,224) 0%, rgb(224,224,255) 100%);" | 384
|- style="text-align:center; background: linear-gradient(0deg, rgb(255,224,224) 0%, rgb(255,224,224) 90%, rgb(240,224,240) 100%);"
| [[8-cell|<u>8-cell</u><BR>octachoron<BR>tesseract<BR>4-cube]] || [[Image:4-cube t0.svg|125px]] || hypercube<BR>[[hypercube|''n''-cube]]<br>(B<sub>n</sub> family) || {4,3,3}<BR>{{CDD|node_1|4|node|3|node|3|node}} || 16 || 32 || 24<br>[[Square|{4}]] || 8<br>[[cube|{4,3}]] || [[tetrahedron|{3,3}]] || 16-cell
|- BGCOLOR="#e8ffe8" style="text-align:center;"
| [[24-cell|<u>24-cell</u><BR>icositetrachoron<BR>octaplex<BR>polyoctahedron<br>(pO)]] || [[Image:24-cell t0 F4.svg|125px]] || F<sub>n</sub> family || {3,4,3}<BR>{{CDD|node_1|3|node|4|node|3|node}} || 24 || 96 || 96<br>[[triangle|{3}]] || 24<br>[[octahedron|{3,4}]] || [[cube|{4,3}]] || ''self-dual'' || ''F''<sub>4</sub><BR>[3,4,3] || 1152
|- style="text-align:center; background: linear-gradient(0deg, rgb(240,224,240) 0%, rgb(224,224,255) 10%, rgb(224,224,255) 100%);"
| [[600-cell|<u>600-cell</u><BR>hexacosichoron<BR>tetraplex<BR>polytetrahedron<br>(pT)]] || [[Image:600-cell graph H4.svg|125px]] || [[pentagonal polytope|n-pentagonal<br>polytope]]<br>(H<sub>n</sub> family) || {3,3,5}<BR>{{CDD|node_1|3|node|3|node|5|node}} || 120 || 720 || 1200<br>[[triangle|{3}]] || 600<br>[[tetrahedron|{3,3}]] || [[regular icosahedron|{3,5}]] || 120-cell || rowspan="2" style="background: linear-gradient(0deg, rgb(255,224,224) 0%, rgb(224,224,255) 100%);" | ''H''<sub>4</sub><BR>[5,3,3] || rowspan="2" style="background: linear-gradient(0deg, rgb(255,224,224) 0%, rgb(224,224,255) 100%);" | 14400
|- style="text-align:center; background: linear-gradient(0deg, rgb(255,224,224) 0%, rgb(255,224,224) 90%, rgb(240,224,240) 100%);"
| [[120-cell|<u>120-cell</u><BR>hecatonicosachoron<BR>dodecacontachoron<BR>dodecaplex<BR>polydodecahedron<br>(pD)]] || [[Image:120-cell graph H4.svg|125px]]|| [[pentagonal polytope|n-pentagonal<br>polytope]]<br>(H<sub>n</sub> family) || {5,3,3}<BR>{{CDD|node_1|5|node|3|node|3|node}} || 600 || 1200 || 720<br>[[pentagon|{5}]] || 120<br>[[dodecahedron|{5,3}]] || [[tetrahedron|{3,3}]] || 600-cell
|}

[[John Horton Conway|John Conway]] advocated the names simplex, orthoplex, tesseract, octaplex or polyoctahedron (pO), tetraplex or polytetrahedron (pT), and dodecaplex or polydodecahedron (pD).<ref>{{harvnb|Conway|Burgiel|Goodman-Strauss|2008|loc=Ch. 26. Higher Still}}</ref>

[[Norman Johnson (mathematician)|Norman Johnson]] advocated the names n-cell, or pentachoron, hexadecachoron, tesseract or octachoron, icositetrachoron, hexacosichoron, and hecatonicosachoron (or dodecacontachoron), coining the term ''polychoron'' being a 4D analogy to the 3D polyhedron, and 2D polygon, expressed from the [[Greek language|Greek]] roots ''poly'' ("many") and ''choros'' ("room" or "space").<ref>[https://www.mit.edu/~hlb/Associahedron/program.pdf "Convex and abstract polytopes", Programme and abstracts, MIT, 2005]</ref><ref>{{cite book |first=Norman W. |last=Johnson |title=Geometries and Transformations |chapter-url=https://books.google.com/books?id=adBVDwAAQBAJ&pg=PA246 |date=2018 |publisher=Cambridge University Press |isbn=978-1-107-10340-5 |pages=246– |chapter=§ 11.5 Spherical Coxeter groups}}</ref>

The [[Euler characteristic]] for all 4-polytopes is zero, we have the 4-dimensional analogue of Euler's polyhedral formula:
:<math>N_0 - N_1 + N_2 - N_3 = 0\,</math>
where ''N''<sub>''k''</sub> denotes the number of ''k''-faces in the polytope (a vertex is a 0-face, an edge is a 1-face, etc.).

The topology of any given 4-polytope is defined by its [[Betti number]]s and [[torsion coefficient (topology)|torsion coefficient]]s.<ref name="richeson">{{cite book |first=David S. |last=Richeson |title=Euler's Gem: The Polyhedron Formula and the Birth of Topology |url=https://books.google.com/books?id=zyIRIcRSNwsC |date=2012 |publisher=Princeton University Press |isbn=978-0-691-15457-2 |chapter-url=https://books.google.com/books?id=zyIRIcRSNwsC&pg=PA253 |chapter=23. Henri Poincaré and the Ascendancy of Topology |pages=256–}}</ref>

=== As configurations===
A regular 4-polytope can be completely described as a [[Configuration_(polytope)|configuration matrix]] containing counts of its component elements. The rows and columns correspond to vertices, edges, faces, and cells. The diagonal numbers (upper left to lower right) say how many of each element occur in the whole 4-polytope. The non-diagonal numbers say how many of the column's element occur in or at the row's element. For example, there are 2 vertices ''in'' each edge (each edge ''has'' 2 vertices), and 2 cells meet ''at'' each face (each face ''belongs to'' 2 cells), in any regular 4-polytope. The configuration for the dual polytope can be obtained by rotating the matrix by 180 degrees.<ref>{{harvnb|Coxeter|1973|loc=§ 1.8 Configurations}}</ref><ref>Coxeter, Complex Regular Polytopes, p.117</ref>
{| class=wikitable
![[5-cell]]<BR>{3,3,3}
![[16-cell]]<BR>{3,3,4}
![[8-cell]]<BR>{4,3,3}
![[24-cell]]<BR>{3,4,3}
![[600-cell]]<BR>{3,3,5}
![[120-cell]]<BR>{5,3,3}
|- style="font-size:85%;"
|bgcolor="#e8ffe8"|<math>\begin{bmatrix}\begin{matrix}5 & 4 & 6 & 4 \\ 2 & 10 & 3 & 3 \\ 3 & 3 & 10 & 2 \\ 4 & 6 & 4 & 5 \end{matrix}\end{bmatrix}</math>
|bgcolor="#FFe0e0"|<math>\begin{bmatrix}\begin{matrix}8 & 6 & 12 & 8 \\ 2 & 24 & 4 & 4 \\ 3 & 3 & 32 & 2 \\ 4 & 6 & 4 & 16 \end{matrix}\end{bmatrix}</math>
|bgcolor="#e0e0ff"|<math>\begin{bmatrix}\begin{matrix}16 & 4 & 6 & 4 \\ 2 & 32 & 3 & 3 \\ 4 & 4 & 24 & 2 \\ 8 & 12 & 6 & 8 \end{matrix}\end{bmatrix}</math>
|bgcolor="#e8ffe8"|<math>\begin{bmatrix}\begin{matrix}24 & 8 & 12 & 6 \\ 2 & 96 & 3 & 3 \\ 3 & 3 & 96 & 2 \\ 6 & 12 & 8 & 24 \end{matrix}\end{bmatrix}</math>
|bgcolor="#ffe0e0"|<math>\begin{bmatrix}\begin{matrix}120 & 12 & 30 & 20 \\ 2 & 720 & 5 & 5 \\ 3 & 3 & 1200 & 2 \\ 4 & 6 & 4 & 600 \end{matrix}\end{bmatrix}</math>
|bgcolor="#e0e0ff"|<math>\begin{bmatrix}\begin{matrix}600 & 4 & 6 & 4 \\ 2 & 1200 & 3 & 3 \\ 5 & 5 & 720 & 2 \\ 20 & 30 & 12 & 120 \end{matrix}\end{bmatrix}</math>
|}

===Visualization===

The following table shows some 2-dimensional projections of these 4-polytopes. Various other visualizations can be found in the external links below. The [[Coxeter-Dynkin diagram]] graphs are also given below the [[Schläfli symbol]].

{| class="wikitable" style="text-align:center;"
|-
! A<sub>4</sub> = [3,3,3] ||colspan=2| B<sub>4</sub> = [4,3,3] || F<sub>4</sub> = [3,4,3] ||colspan=2| H<sub>4</sub> = [5,3,3]
|-
! [[5-cell]] || [[16-cell]] || [[Tesseract|8-cell]] || [[24-cell]] || [[600-cell]] || [[120-cell]]
|-
! {3,3,3} || {3,3,4} || {4,3,3} || {3,4,3} || {3,3,5} || {5,3,3}
|-
!{{CDD|node_1|3|node|3|node|3|node}}
!{{CDD|node_1|3|node|3|node|4|node}}
!{{CDD|node_1|4|node|3|node|3|node}}
!{{CDD|node_1|3|node|4|node|3|node}}
!{{CDD|node_1|3|node|3|node|5|node}}
!{{CDD|node_1|5|node|3|node|3|node}}
|-
|colspan=6|Solid 3D [[orthographic projection]]s
|- valign=top style="line-height:120%"
| [[Image:Tetrahedron.png|125px]]<BR>[[Tetrahedron|Tetrahedral<BR>envelope]]<br />(cell/vertex-centered)
| [[File:16-cell ortho cell-centered.png|125px]]<BR>[[Cube|Cubic envelope]]<br />(cell-centered)
| [[Image:Hexahedron.png|125px]]<BR>[[Cube|Cubic envelope]]<br />(cell-centered)
| [[Image:Ortho solid 24-cell.png|125px]]<BR>[[Cuboctahedron|Cuboctahedral<BR>envelope]]<br />(cell-centered)
| [[Image:Ortho solid 600-cell.png|125px]]<BR>[[Pentakis icosidodecahedron|Pentakis icosidodecahedral<BR>envelope]]<br />(vertex-centered)
| [[Image:Ortho solid 120-cell.png|125px]]<BR>[[Truncated rhombic triacontahedron|Truncated rhombic<BR>triacontahedron<BR>envelope]]<br />(cell-centered)
|-
|colspan=6|Wireframe [[Schlegel diagram]]s ([[Perspective projection]])
|- valign=bottom
| [[Image:Schlegel wireframe 5-cell.png|125px]]<BR>Cell-centered
| [[Image:Schlegel wireframe 16-cell.png|125px]]<BR>Cell-centered
| [[Image:Schlegel wireframe 8-cell.png|125px]]<BR>Cell-centered
| [[Image:Schlegel wireframe 24-cell.png|125px]]<BR>Cell-centered
| [[Image:Schlegel wireframe 600-cell vertex-centered.png|125px]]<BR>Vertex-centered
| [[Image:Schlegel wireframe 120-cell.png|125px]]<BR>Cell-centered
|-
|colspan=6|Wireframe [[stereographic projection]]s ([[3-sphere]])
|- valign=top
| [[Image:Stereographic polytope 5cell.png|125px]]
| [[Image:Stereographic polytope 16cell.png|125px]]
| [[Image:Stereographic polytope 8cell.png|125px]]
| [[Image:Stereographic polytope 24cell.png|125px]]
| [[Image:Stereographic polytope 600cell.png|125px]]
| [[Image:Stereographic polytope 120cell.png|125px]]
|}

==Regular star (Schläfli–Hess) 4-polytopes==
[[File:Relationship among regular star polychora.png|320px|thumb|This shows the relationships among the four-dimensional starry polytopes. The 2 convex forms and 10 starry forms can be seen in 3D as the vertices of a [[cuboctahedron]].<ref>{{harvnb|Conway|Burgiel|Goodman-Strauss|2008|p=406, Fig 26.2}}</ref>]] 
[[File:Relationship among regular star polychora-8.png|320px|thumb|A subset of relations among 8 forms from the 120-cell, polydodecahedron (pD). The three operations {a,g,s} are commutable, defining a cubic framework. There are 7 [[Density (geometry)|densities]] seen in vertical positioning, with 2 dual forms having the same density.]] 
The '''Schläfli&ndash;Hess 4-polytopes''' are the complete set of 10 [[Regular polytope|regular]] self-intersecting [[Star polytope|'''star polychora''']] ([[4-polytope|four-dimensional polytopes]]).<ref>Coxeter, ''Star polytopes and the Schläfli function f{&alpha;,&beta;,&gamma;)'' p. 122 2. ''The Schläfli-Hess polytopes''</ref> They are named in honor of their discoverers: [[Ludwig Schläfli]] and [[Edmund Hess]]. Each is represented by a [[Schläfli symbol]] {''p'',''q'',''r''} in which one of the numbers is [[pentagram|{{sfrac|5|2}}]]. They are thus analogous to the regular nonconvex [[Kepler&ndash;Poinsot polyhedron|Kepler&ndash;Poinsot polyhedra]], which are in turn analogous to the pentagram.

=== Names ===

Their names given here were given by [[John Horton Conway|John Conway]], extending [[Arthur Cayley|Cayley's]] names for the [[Kepler&ndash;Poinsot polyhedra]]: along with ''stellated'' and ''great'', he adds a ''grand'' modifier.  Conway offered these operational definitions:
#'''[[stellation]]''' – replaces edges with longer edges in same lines. (Example: a [[pentagon]] stellates into a [[pentagram]])
#'''greatening''' – replaces the faces with large ones in same planes. (Example: an [[regular icosahedron|icosahedron]] greatens into a [[great icosahedron]])
#'''aggrandizement''' – replaces the cells with large ones in same 3-spaces. (Example: a [[600-cell]] aggrandizes into a [[grand 600-cell]])

John Conway names the 10 forms from 3 regular celled 4-polytopes: pT=polytetrahedron {3,3,5} (a tetrahedral [[600-cell]]), pI=polyicosahedron {3,5,{{sfrac|5|2}}} (an [[icosahedral 120-cell]]), and pD=polydodecahedron {5,3,3} (a dodecahedral [[120-cell]]), with prefix modifiers: ''g'', ''a'', and ''s'' for great, (ag)grand, and stellated. The final stellation, the ''great grand stellated polydodecahedron'' contains them all as ''gaspD''.

=== Symmetry ===
All ten polychora have [3,3,5] ([[Coxeter group#Finite Coxeter groups|H<sub>4</sub>]]) [[hexacosichoric symmetry]]. They are generated from 6 related [[Goursat tetrahedron|Goursat tetrahedra]] [[Goursat tetrahedron#Rational solutions|rational-order symmetry groups]]: [3,5,5/2], [5,5/2,5], [5,3,5/2], [5/2,5,5/2], [5,5/2,3], and [3,3,5/2].

Each group has 2 regular star-polychora, except for two groups which are self-dual, having only one. So there are 4 dual-pairs and 2 self-dual forms among the ten regular star polychora.

=== Properties ===
Note:
* There are 2 unique [[vertex arrangement]]s, matching those of the [[120-cell]] and [[600-cell]]. 
* There are 4 unique [[edge arrangement]]s, which are shown as ''wireframes'' [[Orthographic projection (geometry)|orthographic projections]]. 
* There are 7 unique [[face arrangement]]s, shown as ''solids'' (face-colored) orthographic projections.

The cells (polyhedra), their faces (polygons), the ''polygonal [[edge figure]]s'' and ''polyhedral [[vertex figure]]s'' are identified by their [[Schläfli symbol]]s.

{| class="wikitable sortable"
! Name<BR>Conway (abbrev.)<!--<BR>(Bowers acronym)-->
! Orthogonal<BR>projection
! [[Schläfli symbol|Schläfli]]<BR>[[Coxeter-Dynkin diagram|Coxeter]]
! [[Cell (geometry)|C]]<BR>{p, q}
! [[Face (geometry)|F]]<BR>{p}
! [[Edge (geometry)|E]]<BR>{r}
! [[Vertex (geometry)|V]]<BR>{q, r}
![[density (polytope)|Dens.]]
! [[Euler characteristic|χ]]
|- align=center BGCOLOR="#e0e0ff" 
| [[Icosahedral 120-cell]]<BR>polyicosahedron (pI)<!--(fix)-->
| [[File: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>[[regular icosahedron|{3,5}]]<BR>[[File:Icosahedron.png|25px]]
| 1200<BR>[[Triangle|{3}]]<BR>[[File:Regular triangle.svg|25px]]
| 720<BR>[[Pentagram|{5/2]]}<BR>[[File:Star polygon 5-2.svg|25px]]
| 120<BR>[[Great dodecahedron|{5,5/2}]]<BR>[[File:Great dodecahedron.png|25px]]
| 4
| 480
|- align=center BGCOLOR="#ffe0e0" 
| [[Small stellated 120-cell]]<BR>stellated polydodecahedron (spD)<!--<BR>(sishi)-->
| [[File: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>[[File:Small stellated dodecahedron.png|25px]]
| 720<BR>[[Pentagram|{5/2}]]<BR>[[File:Star polygon 5-2.svg|25px]]
| 1200<BR>[[Triangle|{3}]]<BR>[[File:Regular triangle.svg|25px]]
| 120<BR>[[Dodecahedron|{5,3}]]<BR>[[File:Dodecahedron.png|25px]]
| 4
| &minus;480
|- align=center BGCOLOR="#e0ffe0" 
| [[Great 120-cell]]<BR>great polydodecahedron (gpD)<!--<BR>(gohi)-->
| [[File: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>[[File:Great dodecahedron.png|25px]]
| 720<BR>[[Pentagon|{5}]]<BR>[[File:Regular pentagon.svg|25px]]
| 720<BR>[[Pentagon|{5}]]<BR>[[File:Regular pentagon.svg|25px]]
| 120<BR>[[Small stellated dodecahedron|{5/2,5}]]<BR>[[File:Small stellated dodecahedron.png|25px]]
| 6
| 0
|- align=center BGCOLOR="#e0e0ff" 
| [[Grand 120-cell]]<BR>grand polydodecahedron (apD)<!--<BR>(gahi)-->
| [[File: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>[[File:Dodecahedron.png|25px]]
| 720<BR>[[Pentagon|{5}]]<BR>[[File:Regular pentagon.svg|25px]]
| 720<BR>[[Pentagram|{5/2}]]<BR>[[File:Star polygon 5-2.svg|25px]]
| 120<BR>[[Great icosahedron|{3,5/2}]]<BR>[[File:Great icosahedron.png|25px]]
| 20
| 0
|- align=center BGCOLOR="#ffe0e0" 
| [[Great stellated 120-cell]]<BR>great stellated polydodecahedron (gspD)<!--<BR>(gishi)-->
| [[File: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>[[File:Great stellated dodecahedron.png|25px]]
| 720<BR>[[Pentagram|{5/2}]]<BR>[[File:Star polygon 5-2.svg|25px]]
| 720<BR>[[Pentagon|{5}]]<BR>[[File:Regular pentagon.svg|25px]]
| 120<BR>[[regular icosahedron|{3,5}]]<BR>[[File:Icosahedron.png|25px]]
| 20
| 0
|- align=center BGCOLOR="#e0ffe0" 
| [[Grand stellated 120-cell]]<BR>grand stellated polydodecahedron (aspD)<!--<BR>(gashi)-->
| [[File: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>[[File:Small stellated dodecahedron.png|25px]]
| 720<BR>[[Pentagram|{5/2}]]<BR>[[File:Star polygon 5-2.svg|25px]]
| 720<BR>[[Pentagram|{5/2}]]<BR>[[File:Star polygon 5-2.svg|25px]]
| 120<BR>[[Great dodecahedron|{5,5/2}]]<BR>[[File:Great dodecahedron.png|25px]]
| 66
| 0
|- align=center BGCOLOR="#e0e0ff" 
| [[Great grand 120-cell]]<BR>great grand polydodecahedron (gapD)<!--<BR>(gaghi)-->
| [[File: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>[[File:Great dodecahedron.png|25px]]
| 720<BR>[[Pentagon|{5}]]<BR>[[File:Regular pentagon.svg|25px]]
| 1200<BR>[[Triangle|{3}]]<BR>[[File:Regular triangle.svg|25px]]
| 120<BR>[[Great stellated dodecahedron|{5/2,3}]]<BR>[[File:Great stellated dodecahedron.png|25px]]
| 76
| &minus;480
|- align=center BGCOLOR="#ffe0e0" 
| [[Great icosahedral 120-cell]]<BR>great polyicosahedron (gpI)<!--<BR>(gofix)-->
| [[File: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>[[File:Great icosahedron.png|25px]]
| 1200<BR>[[Triangle|{3}]]<BR>[[File:Regular triangle.svg|25px]]
| 720<BR>[[Pentagon|{5}]]<BR>[[File:Regular pentagon.svg|25px]]
| 120<BR>[[Small stellated dodecahedron|{5/2,5}]]<BR>[[File:Small stellated dodecahedron.png|25px]]
| 76
| 480
|- align=center BGCOLOR="#e0e0ff" 
| [[Grand 600-cell]]<BR>grand polytetrahedron (apT)<!--<BR>(gax)-->
| [[File: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>[[File:Tetrahedron.png|25px]]
| 1200<BR>[[Triangle|{3}]]<BR>[[File:Regular triangle.svg|25px]]
| 720<BR>[[Pentagram|{5/2}]]<BR>[[File:Star polygon 5-2.svg|25px]]
| 120<BR>[[Great icosahedron|{3,5/2}]]<BR>[[File:Great icosahedron.png|25px]]
| 191
| 0
|- align=center BGCOLOR="#ffe0e0" 
| [[Great grand stellated 120-cell]]<BR>great grand stellated polydodecahedron (gaspD)<!--<BR>(gogishi)-->
| [[File: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>[[File:Great stellated dodecahedron.png|25px]]
| 720<BR>[[Pentagram|{5/2}]]<BR>[[File:Star polygon 5-2.svg|25px]]
| 1200<BR>[[Triangle|{3}]]<BR>[[File:Regular triangle.svg|25px]]
| 600<BR>[[Tetrahedron|{3,3}]]<BR>[[File:Tetrahedron.png|25px]]
| 191
| 0
|}

== See also ==
*[[Regular polytope]]
* [[List of regular polytopes]]
* Infinite regular 4-polytopes:
** [[List of regular polytopes#Tessellations of Euclidean 3-space|One regular Euclidean honeycomb:]]  {4,3,4}
** [[List of regular polytopes#Tessellations of hyperbolic 3-space|Four compact regular hyperbolic honeycombs:]] {3,5,3}, {4,3,5}, {5,3,4}, {5,3,5}
** [[List of regular polytopes#Tessellations of hyperbolic 3-space|Eleven paracompact regular hyperbolic honeycombs:]] {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}.
*[[Abstract polytope|Abstract]] regular 4-polytopes: 
** [[11-cell]] {3,5,3} 
** [[57-cell]] {5,3,5}
*[[Uniform 4-polytope]] [[uniform polytope|uniform]] 4-polytope families constructed from these 6 regular forms.
*[[Platonic solid]]
* [[Kepler-Poinsot polyhedra]] — regular [[star polyhedron]]
* [[Star polygon]] — regular star polygons
* [[4-polytope]]
* [[5-polytope]]
* [[List_of_regular_polytopes_and_compounds#6_dimensions|6-polytope]]

== Notes ==
{{Notelist}}

==References==

===Citations===
{{Reflist}}

===Bibliography===
{{refbegin}}
* {{Cite book | last=Coxeter | first=H.S.M. | author-link=Harold Scott MacDonald Coxeter | year=1973 | orig-year=1948 | title=Regular Polytopes | publisher=Dover | place=New York | edition=3rd | title-link=Regular Polytopes (book) }}
*{{cite book |author-link=H. S. M. Coxeter |first=H.S.M. |last=Coxeter |title=Introduction to Geometry |publisher=Wiley |edition=2nd |year=1969 |isbn=0-471-50458-0 }}
*{{cite book |author-link=Duncan MacLaren Young Sommerville |author=D.M.Y. Sommerville |title=Introduction to the Geometry of '''n''' Dimensions |chapter-url=https://books.google.com/books?id=4vXDDwAAQBAJ&pg=PA161 |date=2020 |publisher=Courier Dover |isbn=978-0-486-84248-6 |pages=159–192 |chapter=X. The Regular Polytopes |orig-year=1930}}
*{{cite book |author-link=John Horton Conway |first1=John H. |last1=Conway |first2=Heidi |last2=Burgiel |first3=Chaim |last3=Goodman-Strauss  |chapter=26. Regular Star-polytopes |title=The Symmetries of Things |year=2008 |isbn=978-1-56881-220-5 |pages=404–8 }}
*{{cite web |author-link=Edmund Hess |first=Edmund  |last=Hess  |title=Einleitung in die Lehre von der Kugelteilung mit besonderer Berücksichtigung ihrer Anwendung auf die Theorie der Gleichflächigen und der gleicheckigen Polyeder |date=1883 |url=http://www.hti.umich.edu/cgi/b/bib/bibperm?q1=ABN8623.0001.001}}
*{{cite journal |author-link=Edmund Hess |first=Edmund |last=Hess |title=Uber die regulären Polytope höherer Art |journal=Sitzungsber Gesells Beförderung Gesammten Naturwiss Marburg |pages=31–57 |year=1885 }}
*{{cite book |editor-first=F. Arthur |editor-last=Sherk |editor2-first=Peter |editor2-last=McMullen |editor3-first=Anthony C. |editor3-last=Thompson |editor4-first=Asia Ivic |editor4-last=Weiss |title=Kaleidoscopes: Selected Writings of H.S.M. Coxeter |publisher=Wiley |year=1995 |isbn=978-0-471-01003-6 |url=https://archive.org/details/kaleidoscopessel0000coxe |url-access=registration }}
** (Paper 10) {{cite journal |first=H.S.M. |last=Coxeter |title=Star Polytopes and the Schlafli Function f(α,β,γ) |journal=[[Elemente der Mathematik]] |volume=44 |issue=2 |pages=25–36 |year=1989 |url=https://eudml.org/doc/141447}}
*{{cite book |author-link=Harold Scott MacDonald Coxeter |first=H.S.M. |last=Coxeter |title=Regular Complex Polytopes |publisher=Cambridge University Press |edition=2nd |year=1991 |isbn=978-0-521-39490-1 }}
*{{cite web |first1=Peter |last1=McMullen |first2=Egon |last2=Schulte |title=Abstract Regular Polytopes |date=2002 |url=http://assets.cambridge.org/052181/4960/sample/0521814960ws.pdf}}
{{refend}}

==External links==
*{{Mathworld | urlname=RegularPolychoron | title=Regular polychoron }} 
*[http://polytope.net/hedrondude/regulars.htm Jonathan Bowers, 16 regular 4-polytopes]
*[http://www.weimholt.com/andrew/polytope.shtml Regular 4D Polytope Foldouts]
*[http://www.math.cmu.edu/~fho/jenn/polytopes/index.html Catalog of Polytope Images] A collection of stereographic projections of 4-polytopes.
*[http://www.math.cmu.edu/~fho/jenn/polytopes A Catalog of Uniform Polytopes]
*[http://www.dimensions-math.org/ Dimensions] 2 hour film about the fourth dimension (contains stereographic projections of all regular 4-polytopes)
* [https://web.archive.org/web/20061107052613/http://www.mathematik.uni-regensburg.de/Goette/sterne/ Reguläre Polytope]
* [https://web.archive.org/web/20070704012333/http://davidf.faricy.net/polyhedra/Star_Polychora.html The Regular Star Polychora]
* [https://pages.uoregon.edu/koch/hypersolids/hypersolids.html Hypersolids]

{{4D regular polytopes}}

{{DEFAULTSORT:Regular 4-polytope}}
[[Category:Regular 4-polytopes| ]]