Editing 5-cell (section)

== Irregular 5-cells ==
In the case of [[Simplex|simplexes]] such as the 5-cell, certain irregular forms are in some sense more fundamental than the regular form. Although regular 5-cells cannot fill 4-space or the regular 4-polytopes, there are irregular 5-cells which do. These '''characteristic 5-cells''' are the [[fundamental domain]]s of the different [[Coxeter group|symmetry groups]] which give rise to the various 4-polytopes.

===Orthoschemes===
A '''4-orthoscheme''' is a 5-cell where all 10 faces are [[Triangle#By_internal_angles|right triangles]]. (The 5 vertices form 5 tetrahedral [[Cell (geometry)|cells]] face-bonded to each other, with a total of 10 edges and 10 triangular faces.) An [[Schläfli orthoscheme|orthoscheme]] is an irregular [[simplex]] that is the [[convex hull]] of a [[Tree (graph theory)|tree]] in which all edges are mutually perpendicular. In a 4-dimensional orthoscheme, the tree consists of four perpendicular edges connecting all five vertices in a linear path that makes three right-angled turns. The elements of an orthoscheme are also orthoschemes (just as the elements of a regular simplex are also regular simplexes). Each tetrahedral cell of a 4-orthoscheme is a [[Tetrahedron#Orthoschemes|3-orthoscheme]], and each triangular face is a 2-orthoscheme (a right triangle).

Orthoschemes are the [[Orthoscheme#Characteristic simplex of the general regular polytope|characteristic simplexes]] of the regular polytopes, because each regular polytope is [[Coxeter group|generated by reflections]] in the bounding facets of its particular characteristic orthoscheme.{{Sfn|Coxeter|1973|loc=§11.7 Regular figures and their truncations|pp=198-202}} For example, the special case of the 4-orthoscheme with equal-length perpendicular edges is the characteristic orthoscheme of the [[4-cube]] (also called the ''tesseract'' or ''8-cell''), the 4-dimensional analogue of the 3-dimensional cube. If the three perpendicular edges of the 4-orthoscheme are of unit length, then all its edges are of length {{radic|1}}, {{radic|2}}, {{radic|3}}, or {{radic|4}}, precisely the [[Tesseract#Radial equilateral symmetry|chord lengths of the unit 4-cube]] (the lengths of the 4-cube's edges and its various diagonals). Therefore this 4-orthoscheme fits within the 4-cube, and the 4-cube (like every regular convex polytope) can be [[Dissection into orthoschemes|dissected into instances of its characteristic orthoscheme]].

[[File:Triangulated cube.svg|thumb|400px|A 3-cube dissected into six [[Tetrahedron#Orthoschemes|3-orthoschemes]]. Three are left-handed and three are right handed. A left and a right meet at each square face.]]A 3-orthoscheme is easily illustrated, but a 4-orthoscheme is more difficult to visualize. A 4-orthoscheme is a [[Hyperpyramid|tetrahedral pyramid]] with a 3-orthoscheme as its base. It has four more edges than the 3-orthoscheme, joining the four vertices of the base to its apex (the fifth vertex of the 5-cell). Pick out any one of the 3-orthoschemes of the six shown in the 3-cube illustration. Notice that it touches four of the cube's eight vertices, and those four vertices are linked by a 3-edge path that makes two right-angled turns. Imagine that this 3-orthoscheme is the base of a 4-orthoscheme, so that from each of those four vertices, an unseen 4-orthoscheme edge connects to a fifth apex vertex (which is outside the 3-cube and does not appear in the illustration at all). Although the four additional edges all reach the same apex vertex, they will all be of different lengths. The first of them, at one end of the 3-edge orthogonal path, extends that path with a fourth orthogonal {{radic|1}} edge by making a third 90 degree turn and reaching perpendicularly into the fourth dimension to the apex. The second of the four additional edges is a {{radic|2}} diagonal of a cube face (not of the illustrated 3-cube, but of another of the tesseract's eight 3-cubes). The third additional edge is a {{radic|3}} diagonal of a 3-cube (again, not the original illustrated 3-cube). The fourth additional edge (at the other end of the orthogonal path) is a [[Tesseract#Radial equilateral symmetry|long diameter of the tesseract]] itself, of length {{radic|4}}. It reaches through the exact center of the tesseract to the [[Antipodal point|antipodal]] vertex (a vertex of the opposing 3-cube), which is the apex. Thus the '''characteristic 5-cell of the 4-cube''' has four {{radic|1}} edges, three {{radic|2}} edges, two {{radic|3}} edges, and one {{radic|4}} edge.

The 4-cube {{CDD|node_1|4|node|3|node|3|node}} can be [[Schläfli orthoscheme#Properties|dissected into 24 such 4-orthoschemes]] {{CDD|node|4|node|3|node|3|node}} eight different ways, with six 4-orthoschemes surrounding each of four orthogonal {{radic|4}} tesseract long diameters. The 4-cube can also be dissected into 384 ''smaller'' instances of this same characteristic 4-orthoscheme, just one way, by all of its symmetry hyperplanes at once, which divide it into 384 4-orthoschemes that all meet at the center of the 4-cube.

More generally, any regular polytope can be dissected into ''g'' instances of its characteristic orthoscheme that all meet at the regular polytope's center.{{Sfn|Kim|Rote|2016|pp=17-20|loc=§10 The Coxeter Classification of Four-Dimensional Point Groups}} The number ''g'' is the ''order'' of the polytope, the number of reflected instances of its characteristic orthoscheme that comprise the polytope when a ''single'' mirror-surfaced orthoscheme instance is reflected in its own facets. More generally still, characteristic simplexes are able to fill uniform polytopes because they possess all the requisite elements of the polytope. They also possess all the requisite angles between elements (from 90 degrees on down). The characteristic simplexes are the [[genetic code]]s of polytopes: like a [[Swiss Army knife]], they contain one of everything needed to construct the polytope by replication.

Every regular polytope, including the regular 5-cell, has its characteristic orthoscheme. There is a 4-orthoscheme which is the '''characteristic 5-cell of the regular 5-cell'''. It is a [[Hyperpyramid|tetrahedral pyramid]] based on the [[Tetrahedron#Orthoschemes|characteristic tetrahedron of the regular tetrahedron]]. The regular 5-cell {{CDD|node_1|3|node|3|node|3|node}} can be dissected into 120 instances of this characteristic 4-orthoscheme {{CDD|node|3|node|3|node|3|node}} just one way, by all of its symmetry hyperplanes at once, which divide it into 120 4-orthoschemes that all meet at the center of the regular 5-cell.

{| class="wikitable floatright"
!colspan=6|Characteristics of the regular 5-cell{{Sfn|Coxeter|1973|pp=292-293|loc=Table I(ii); "5-cell, 𝛼<sub>4</sub>"}}
|-
!align=right|
!align=center|edge{{Sfn|Coxeter|1973|p=139|loc=§7.9 The characteristic simplex}}
!colspan=2 align=center|arc
!colspan=2 align=center|dihedral{{Sfn|Coxeter|1973|p=290|loc=Table I(ii); "dihedral angles"}}
|-
!align=right|𝒍
|align=center|<small><math>\sqrt{\tfrac{5}{2}} \approx 1.581</math></small>
|align=center|<small>104°30′40″</small>
|align=center|<small><math>\pi - 2\text{𝜂}</math></small>
|align=center|<small>75°29′20″</small>
|align=center|<small><math>\pi - 2\text{𝟁}</math></small>
|-
|
|
|
|
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|-
!align=right|𝟀
|align=center|<small><math>\sqrt{\tfrac{1}{10}} \approx 0.316</math></small>
|align=center|<small>75°29′20″</small>
|align=center|<small><math>2\text{𝜂}</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
|-
!align=right|𝝉{{Efn|{{Harv|Coxeter|1973}} uses the greek letter 𝝓 (phi) to represent one of the three ''characteristic angles'' 𝟀, 𝝓, 𝟁 of a regular polytope. Because 𝝓 is commonly used to represent the [[golden ratio]] constant ≈ 1.618, for which Coxeter uses 𝝉 (tau), we reverse Coxeter's conventions, and use 𝝉 to represent the characteristic angle.|name=reversed greek symbols}}
|align=center|<small><math>\sqrt{\tfrac{1}{30}} \approx 0.183</math></small>
|align=center|<small>52°15′20″</small>
|align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
|-
!align=right|𝟁
|align=center|<small><math>\sqrt{\tfrac{2}{15}} \approx 0.103</math></small>
|align=center|<small>52°15′20″</small>
|align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small>
|align=center|<small>60°</small>
|align=center|<small><math>\tfrac{\pi}{3}</math></small>
|-
|
|
|
|
|
|-
!align=right|<small><math>_0R^3/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{3}{20}} \approx 0.387</math></small>
|align=center|<small>75°29′20″</small>
|align=center|<small><math>2\text{𝜂}</math></small>
|align=center|<small>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</math></small>
|-
!align=right|<small><math>_1R^3/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{1}{20}} \approx 0.224</math></small>
|align=center|<small>52°15′20″</small>
|align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small>
|align=center|<small>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</math></small>
|-
!align=right|<small><math>_2R^3/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{1}{60}} \approx 0.129</math></small>
|align=center|<small>52°15′20″</small>
|align=center|<small><math>\tfrac{\pi}{2}-\text{𝜂}</math></small>
|align=center|<small>90°</small>
|align=center|<small><math>\tfrac{\pi}{2}</math></small>
|-
|
|
|
|
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|-
!align=right|<small><math>_0R^4/l</math></small>
|align=center|<small><math>\sqrt{1} = 1.0</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|<small><math>_1R^4/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{3}{8}} \approx 0.612</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|<small><math>_2R^4/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{1}{6}} \approx 0.408</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
!align=right|<small><math>_3R^4/l</math></small>
|align=center|<small><math>\sqrt{\tfrac{1}{16}} = 0.25</math></small>
|align=center|
|align=center|
|align=center|
|align=center|
|-
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|
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|-
!align=right|<small><math>\text{𝜼}</math></small>
|align=center|
|align=center|<small>37°44′40″</small>
|align=center|<small><math>\tfrac{\text{arc sec }4}{2}</math></small>
|align=center|
|align=center|
|}
The characteristic 5-cell (4-orthoscheme) of the regular 5-cell has four more edges than its base characteristic tetrahedron (3-orthoscheme), which join the four vertices of the base to its apex (the fifth vertex of the 4-orthoscheme, at the center of the regular 5-cell). The four edges of each 4-orthoscheme which meet at the center of a regular 4-polytope are of unequal length, because they are the four characteristic radii of the regular 4-polytope: a vertex radius, an edge center radius, a face center radius, and a cell center radius. If the regular 5-cell has unit radius and edge length <small><math>\sqrt{\tfrac{5}{2}}</math></small>, its characteristic 5-cell's ten edges have lengths <small><math>\sqrt{\tfrac{1}{10}}</math></small>, <small><math>\sqrt{\tfrac{1}{30}}</math></small>, <small><math>\sqrt{\tfrac{2}{15}}</math></small> around its exterior right-triangle face (the edges opposite the ''characteristic angles'' 𝟀, 𝝉, 𝟁),{{Efn|name=reversed greek symbols}} plus <small><math>\sqrt{\tfrac{3}{20}}</math></small>, <small><math>\sqrt{\tfrac{1}{20}}</math></small>, <small><math>\sqrt{\tfrac{1}{60}}</math></small> (the other three edges of the exterior 3-orthoscheme facet the characteristic tetrahedron, which are the ''characteristic radii'' of the regular tetrahedron), plus <small><math>\sqrt{1}</math></small>, <small><math>\sqrt{\tfrac{3}{8}}</math></small>, <small><math>\sqrt{\tfrac{1}{6}}</math></small>, <small><math>\sqrt{\tfrac{1}{16}}</math></small> (edges which are the characteristic radii of the regular 5-cell). The 4-edge path along orthogonal edges of the orthoscheme is <small><math>\sqrt{\tfrac{1}{30}}</math></small>, <small><math>\sqrt{\tfrac{2}{15}}</math></small>, <small><math>\sqrt{\tfrac{1}{60}}</math></small>, <small><math>\sqrt{\tfrac{1}{16}}</math></small>, first from a regular 5-cell vertex to a regular 5-cell edge center, then turning 90° to a regular 5-cell face center, then turning 90° to a regular 5-cell tetrahedral cell center, then turning 90° to the regular 5-cell center.

===Isometries===
There are many lower symmetry forms of the 5-cell, including these found as uniform polytope [[vertex figure]]s:
{| class=wikitable width=600
!Symmetry
![3,3,3]<BR>Order 120
![3,3,1]<BR>Order 24
![3,2,1]<BR>Order 12
![3,1,1]<BR>Order 6
!~[5,2]<sup>+</sup><BR>Order 10
|- align=center
!Name
| Regular 5-cell
| Tetrahedral [[polyhedral pyramid|pyramid]]
| 
| Triangular pyramidal pyramid
|
|- align=center
![[Schläfli symbol|Schläfli]]
| {3,3,3}
| {3,3}∨(&nbsp;)
| {3}∨{&nbsp;}
| {3}∨(&nbsp;)∨(&nbsp;)
| 
|- align=center valign=top
!valign=center|Example<BR>Vertex<BR>figure
|[[File:5-simplex verf.png|120px]]<BR>[[5-simplex]]
|[[File:Truncated 5-simplex verf.png|120px]]<BR>[[Truncated 5-simplex]]
|[[File:Bitruncated 5-simplex verf.png|120px]]<BR>[[Bitruncated 5-simplex]]
|[[File:Canitruncated 5-simplex verf.png|120px]]<BR>[[Cantitruncated 5-simplex]]
|[[File:Omnitruncated 4-simplex honeycomb verf.png|120px]]<BR>[[Omnitruncated 4-simplex honeycomb]]
|}

The '''tetrahedral pyramid''' is a special case of a '''5-cell''', a [[polyhedral pyramid]], constructed as a regular [[tetrahedron]] base in a 3-space [[hyperplane]], and an [[Apex (geometry)|apex]] point ''above'' the hyperplane. The four ''sides'' of the pyramid are made of [[triangular pyramid]] cells.

Many [[uniform 5-polytope]]s have '''tetrahedral pyramid''' [[vertex figure]]s with [[Schläfli symbol]]s ( )∨{3,3}.
{| class=wikitable
|+ Symmetry [3,3,1], order 24
|-
![[Schlegel diagram|Schlegel<BR>diagram]]
|[[File:5-cell prism verf.png|100px]]
|[[File:Tesseractic prism verf.png|100px]]
|[[File:120-cell prism verf.png|100px]]
|[[File:Truncated 5-simplex verf.png|100px]]
|[[File:Truncated 5-cube verf.png|100px]]
|[[File:Truncated 24-cell honeycomb verf.png|100px]]
|-
!Name<BR>[[Coxeter diagram|Coxeter]]
![[5-cell prism|{ }×{3,3,3}]]<BR>{{CDD|node_1|2|node_1|3|node|3|node|3|node}}
![[Tesseractic prism|{ }×{4,3,3}]]<BR>{{CDD|node_1|2|node_1|4|node|3|node|3|node}}
![[120-cell prism|{ }×{5,3,3}]]<BR>{{CDD|node_1|2|node_1|5|node|3|node|3|node}}
![[Truncated 5-simplex|t{3,3,3,3}]]<BR>{{CDD|node_1|3|node_1|3|node|3|node|3|node}}
![[Truncated 5-cube|t{4,3,3,3}]]<BR>{{CDD|node_1|4|node_1|3|node|3|node|3|node}}
![[Truncated 24-cell honeycomb|t{3,4,3,3}]]<BR>{{CDD|node_1|3|node_1|4|node|3|node|3|node}}
|}

Other uniform 5-polytopes have irregular 5-cell vertex figures. The symmetry of a vertex figure of a [[uniform polytope]] is represented by removing the ringed nodes of the Coxeter diagram.
{| class=wikitable
!Symmetry
!colspan=2|[3,2,1], order 12
!colspan=2|[3,1,1], order 6
![2<sup>+</sup>,4,1], order 8
![2,1,1], order 4
|- align=center
![[Schläfli symbol|Schläfli]]
|colspan=2|{3}∨{ &nbsp;}||colspan=2|{3}∨(&nbsp;)∨(&nbsp;)||colspan=2|{&nbsp;}∨{&nbsp;}∨(&nbsp;)
|-
![[Schlegel diagram|Schlegel<BR>diagram]]
|[[File:Bitruncated 5-simplex verf.png|100px]]
|[[File:Bitruncated penteract verf.png|100px]]
|[[File:Canitruncated 5-simplex verf.png|100px]]
|[[File:Canitruncated 5-cube verf.png|100px]]
|[[File:Bicanitruncated 5-simplex verf.png|100px]]
|[[File:Bicanitruncated 5-cube verf.png|100px]]
|-
!Name<BR>[[Coxeter diagram|Coxeter]]
![[bitruncated 5-simplex|t<sub>12</sub>α<sub>5</sub>]]<BR>{{CDD|node|3|node_1|3|node_1|3|node|3|node}}
![[bitruncated 5-cube|t<sub>12</sub>γ<sub>5</sub>]]<BR>{{CDD|node|4|node_1|3|node_1|3|node|3|node}}
![[Cantitruncated 5-simplex|t<sub>012</sub>α<sub>5</sub>]]<BR>{{CDD|node_1|3|node_1|3|node_1|3|node|3|node}}
![[Cantitruncated 5-cube|t<sub>012</sub>γ<sub>5</sub>]]<BR>{{CDD|node_1|4|node_1|3|node_1|3|node|3|node}}
![[Bicantitruncated 5-simplex|t<sub>123</sub>α<sub>5</sub>]]<BR>{{CDD|node|3|node_1|3|node_1|3|node_1|3|node}}
![[Bicantitruncated 5-cube|t<sub>123</sub>γ<sub>5</sub>]]<BR>{{CDD|node|4|node_1|3|node_1|3|node_1|3|node}}
|}

{| class=wikitable
!Symmetry
!colspan=3|[2,1,1], order 2
![2<sup>+</sup>,1,1], order 2
![ ]<sup>+</sup>, order 1
|- align=center
![[Schläfli symbol|Schläfli]]
|colspan=3|{&nbsp;}∨(&nbsp;)∨(&nbsp;)∨(&nbsp;)||colspan=2|(&nbsp;)∨(&nbsp;)∨(&nbsp;)∨(&nbsp;)∨(&nbsp;)
|-
![[Schlegel diagram|Schlegel<BR>diagram]]
|[[File:Runcicantitruncated 5-simplex verf.png|100px]]
|[[File:Runcicantitruncated 5-cube verf.png|100px]]
|[[File:Runcicantitruncated 5-orthoplex verf.png|100px]]
|[[File:Omnitruncated 5-simplex verf.png|100px]]
|[[File:Omnitruncated 5-cube verf.png|100px]]
|-
!Name<BR>[[Coxeter diagram|Coxeter]]
![[Runcicantitruncated 5-simplex|t<sub>0123</sub>α<sub>5</sub>]]<BR>{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node}}
![[Runcicantitruncated 5-cube|t<sub>0123</sub>γ<sub>5</sub>]]<BR>{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node}}
![[Runcicantitruncated 5-orthoplex|t<sub>0123</sub>β<sub>5</sub>]]<BR>{{CDD|node_1|3|node_1|3|node_1|3|node_1|4|node}}
![[Omnitruncated 5-simplex|t<sub>01234</sub>α<sub>5</sub>]]<BR>{{CDD|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}
![[Omnitruncated 5-cube|t<sub>01234</sub>γ<sub>5</sub>]]<BR>{{CDD|node_1|4|node_1|3|node_1|3|node_1|3|node_1}}
|}