View source for 5-simplex

{{Short description|Regular 5-polytope}}
{{Uniform polyteron db|Uniform polyteron stat table|hix}}
In [[Five-dimensional space|five-dimensional]] [[geometry]], a 5-[[simplex]] is a self-dual [[Regular polytope|regular]] [[5-polytope]]. It has six [[vertex (geometry)|vertices]], 15 [[Edge (geometry)|edge]]s, 20 triangle [[Face (geometry)|faces]], 15 tetrahedral [[Cell (mathematics)|cells]], and 6 [[5-cell]] [[Facet (geometry)|facets]]. It has a [[dihedral angle]] of cos<sup>−1</sup>({{sfrac|1|5}}), or approximately 78.46°.

The 5-simplex is a solution to the problem: ''Make 20 equilateral triangles using 15 matchsticks, where each side of every triangle is exactly one matchstick.''

== Alternate names ==

It can also be called a '''hexateron''', or '''hexa-5-tope''', as a 6-[[facet (geometry)|facetted]] polytope in 5-dimensions. The [[5-polytope#A note on generality of terms for n-polytopes and elements|name]] ''hexateron'' is derived from ''hexa-'' for having six [[Facet (mathematics)|facets]] and ''[[5-polytope|teron]]'' (with ''ter-'' being a corruption of ''[[tetra-]]'') for having four-dimensional facets.

By Jonathan Bowers, a hexateron is given the acronym '''hix'''.<ref>{{KlitzingPolytopes|polytera.htm|5D uniform polytopes (polytera)|x3o3o3o3o — hix}}</ref>

== As a configuration==
This [[Regular 4-polytope#As configurations|configuration matrix]] represents the 5-simplex. The rows and columns correspond to vertices, edges, faces, cells and 4-faces. The diagonal numbers say how many of each element occur in the whole 5-simplex. The nondiagonal numbers say how many of the column's element occur in or at the row's element. This self-dual simplex's matrix is identical to its 180 degree rotation.<ref>{{harvnb|Coxeter|1973|loc=§1.8 Configurations}}</ref><ref>{{cite book |first=H.S.M. |last=Coxeter |title=Regular Complex Polytopes |url=https://books.google.com/books?id=9BY9AAAAIAAJ&pg=PA117 |pages=117 |edition=2nd |publisher=Cambridge University Press |year=1991 |isbn=9780521394901}}</ref>

<math>\begin{bmatrix}\begin{matrix}6 & 5 & 10 & 10 & 5 \\ 2 & 15 & 4 & 6 & 4 \\ 3 & 3 & 20 & 3 & 3 \\ 4 & 6 & 4 & 15 & 2 \\ 5 & 10 & 10 & 5 & 6 \end{matrix}\end{bmatrix}</math>

== Regular hexateron cartesian coordinates ==
The ''hexateron'' can be constructed from a [[5-cell]] by adding a 6th vertex such that it is equidistant from all the other vertices of the 5-cell.

The  [[Cartesian coordinates]] for the vertices of an origin-centered regular hexateron having edge length&nbsp;2 are:

:<math>\begin{align}
&\left(\tfrac{1}\sqrt{15},\ \tfrac{1}\sqrt{10},\ \tfrac{1}\sqrt{6},\ \tfrac{1}\sqrt{3},\ \pm1\right)\\[5pt]
&\left(\tfrac{1}\sqrt{15},\ \tfrac{1}\sqrt{10},\ \tfrac{1}\sqrt{6},\ -\tfrac{2}\sqrt{3},\ 0\right)\\[5pt]
&\left(\tfrac{1}\sqrt{15},\ \tfrac{1}\sqrt{10},\ -\tfrac\sqrt{3}\sqrt{2},\ 0,\ 0\right)\\[5pt]
&\left(\tfrac{1}\sqrt{15},\ -\tfrac{2\sqrt 2}\sqrt{5},\ 0,\ 0,\ 0\right)\\[5pt]
&\left(-\tfrac\sqrt{5}\sqrt{3},\ 0,\ 0,\ 0,\ 0\right)
\end{align}</math>

The vertices of the ''5-simplex'' can be more simply positioned on a [[hyperplane]] in 6-space as permutations of (0,0,0,0,0,1) ''or'' (0,1,1,1,1,1).  These constructions can be seen as facets of the [[6-orthoplex]] or [[rectified 6-cube]] respectively.

== Projected images ==
{{5-simplex Coxeter plane graphs|t0|100}}

{| class=wikitable width=320
|[[Image:hexateron.png|320px]]<BR>[[Stereographic projection]] 4D to 3D of [[Schlegel diagram]] 5D to 4D of hexateron.
|}

== Lower symmetry forms ==

A lower symmetry form is a ''5-cell pyramid'' {3,3,3}∨(&nbsp;), with [3,3,3] symmetry order 120, constructed as a [[5-cell]] base in a 4-space [[hyperplane]], and an [[Apex (geometry)|apex]] point ''above'' the hyperplane. The five ''sides'' of the pyramid are made of 5-cell cells. These are seen as [[vertex figure]]s of truncated regular [[6-polytope]]s, like a [[truncated 6-cube]].

Another form is {3,3}∨{&nbsp;}, with [3,3,2,1] symmetry order 48, the joining of an orthogonal digon and a tetrahedron, orthogonally offset, with all pairs of vertices connected between. Another form is {3}∨{3}, with [3,2,3,1] symmetry order 36, and extended symmetry <nowiki>[[</nowiki>3,2,3],1], order 72. It represents joining of 2 orthogonal triangles, orthogonally offset, with all pairs of vertices connected between.

The form {&nbsp;}∨{&nbsp;}∨{&nbsp;} has symmetry [2,2,1,1], order 8, extended by permuting 3 segments as [3[2,2],1] or [4,3,1,1], order 48.

These are seen in the [[vertex figure]]s of [[Bitruncation|bitruncated]] and tritruncated regular 6-polytopes, like a [[bitruncated 6-cube]] and a [[tritruncated 6-simplex]]. The edge labels here represent the types of face along that direction, and thus represent different edge lengths.

The vertex figure of the [[omnitruncated 5-simplex honeycomb]], {{CDD|node_1|split1|nodes_11|3ab|nodes_11|split2|node_1}}, is a 5-simplex with a [[petrie polygon]] cycle of 5 long edges. Its symmetry is isomorphic to dihedral group Dih<sub>6</sub> or simple rotation group [6,2]<sup>+</sup>, order 12.
{| class=wikitable
|+ Vertex figures for [[uniform 6-polytope]]s
!Join||{3,3,3}∨(&nbsp;)||{3,3}∨{&nbsp;}||{3}∨{3}||{&nbsp;}∨{&nbsp;}∨{&nbsp;}||
|-
![[Coxeter notation|Symmetry]]||[3,3,3,1]<BR>Order 120||[3,3,2,1]<BR>Order 48||<nowiki>[[</nowiki>3,2,3],1]<BR>Order 72||[3[2,2],1,1]=[4,3,1,1]<BR>Order 48||~[6] or ~[6,2]<sup>+</sup><BR>Order 12
|-
!Diagram
|[[File:Truncated 6-simplex_verf.png|160px]]
|[[File:Bitruncated 6-simplex_verf.png|160px]]
|[[File:Tritruncated 6-simplex_verf.png|160px]]
|[[File:3-3-3-prism-verf.png|160px]]
|[[File:Omnitruncated 5-simplex honeycomb verf.png|160px]]
|- align=center
!Polytope
|[[truncated 6-simplex]]<BR>{{CDD|branch_11|3b|nodeb|3b|nodeb|3b|nodeb|3b|nodeb}}
|[[bitruncated 6-simplex]]<BR>{{CDD|branch_11|3ab|nodes|3b|nodeb|3b|nodeb}}
|[[tritruncated 6-simplex]]<BR>{{CDD|branch_11|3ab|nodes|3ab|nodes}}
|3-3-3 prism<BR> {{CDD|branch_10|2|branch_10|2|branch_10}}
|[[Omnitruncated 5-simplex honeycomb]]<BR>{{CDD|node_1|split1|nodes_11|3ab|nodes_11|split2|node_1}}
|}

== Compound==
The compound of two 5-simplexes in dual configurations can be seen in this A6 [[Coxeter plane]] projection, with a red and blue 5-simplex vertices and edges. This compound has <nowiki>[[3,3,3,3]]</nowiki> symmetry, order 1440. The intersection of these two 5-simplexes is a uniform [[Rectified 5-simplexes#Intersection of two 5-simplices|birectified 5-simplex]]. {{CDD|node_1|split1|nodes|3ab|nodes}} = {{CDD|node|split1|nodes|3ab|nodes_10l}} ∩ {{CDD|node|split1|nodes|3ab|nodes_01l}}.
:[[File:Compound two 5-simplexes.png|240px]]

== Related uniform 5-polytopes ==

It is first in a dimensional series of uniform polytopes and honeycombs, expressed by [[Coxeter]] as 1<sub>3k</sub> series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral [[hosohedron]].
{{1 3k polytopes}}

It is first in a dimensional series of uniform polytopes and honeycombs, expressed by [[Coxeter]] as 3<sub>k1</sub> series. A degenerate 4-dimensional case exists as 3-sphere tiling, a tetrahedral [[dihedron]].
{{3_k1_polytopes}}

The 5-simplex, as 2<sub>20</sub> polytope is first in dimensional series 2<sub>2k</sub>.
{{2 2k polytopes}}

The regular 5-simplex is one of 19 [[Uniform 5-polytope#The A5 .5B3.2C3.2C3.2C3.5D family .285-simplex.29|uniform polytera]] based on the [3,3,3,3] [[Coxeter group]], all shown here in A<sub>5</sub> [[Coxeter plane]] [[orthographic projection]]s. (Vertices are colored by projection overlap order, red, orange, yellow, green, cyan, blue, purple having progressively more vertices)

{{Hexateron family}}

==See also==
* [[11-cell]]

== Notes ==
{{reflist}}

== References ==
* {{cite book |author-link=Thorold Gosset |first=T. |last=Gosset |chapter=On the Regular and Semi-Regular Figures in Space of n Dimensions |title=Messenger of Mathematics |publisher=Macmillan |year=1900 |pages=43– |url=https://books.google.com/books?id=BZo_AQAAIAAJ}}
* [[Harold Scott MacDonald Coxeter|Coxeter, H.S.M.]]: 
** {{cite book |title-link=Regular Polytopes (book) |author-mask=1 |first=H.S.M. |last=Coxeter |chapter=Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5) |title=Regular Polytopes |publisher=Dover |edition=3rd |year=1973 |isbn=0-486-61480-8 |pages=[https://archive.org/details/regularpolytopes00coxe_869/page/n319 296] }}
** {{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://books.google.com/books?id=fUm5Mwfx8rAC&pg=PP1}}
*** (Paper 22) {{cite journal |author-mask=1 |first=H.S.M. |last=Coxeter |title=Regular and Semi Regular Polytopes I |journal=Math. Zeit. |volume=46 |pages=380–407 |year=1940 |doi=10.1007/BF01181449 |s2cid=186237114 |url=https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PA251 }}
*** (Paper 23) {{cite journal |author-mask=1 |first=H.S.M. |last=Coxeter |title=Regular and Semi-Regular Polytopes II |journal=Math. Zeit. |volume=188 |pages=559–591 |year=1985 |issue=4 |doi=10.1007/BF01161657 |s2cid=120429557 |url=https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PA279}}
*** (Paper 24) {{cite journal |author-mask=1 |first=H.S.M. |last=Coxeter |title=Regular and Semi-Regular Polytopes III |journal=Math. Zeit. |volume=200 |pages=3–45 |year=1988 |doi=10.1007/BF01161745 |s2cid=186237142 |url=https://books.google.com/books?id=fUm5Mwfx8rAC&pg=PA313}}
* {{cite book |author-link=John Horton Conway |first1=John H. |last1=Conway |first2=Heidi |last2=Burgiel |first3=Chaim |last3=Goodman-Strauss |chapter=26. Hemicubes: 1<sub>n1</sub> |title=The Symmetries of Things |year=2008 |isbn=978-1-56881-220-5 |pages=409 }}
* {{cite document |author-link=Norman Johnson (mathematician) |first=Norman |last=Johnson |title=Uniform Polytopes |date=1991 |type=Manuscript|publisher=Norman Johnson }}
**  {{cite thesis |first=N.W. |last=Johnson |title=The Theory of Uniform Polytopes and Honeycombs |date=1966 |type=PhD |publisher=University of Toronto |url=https://search.library.utoronto.ca/details?402790}}

== External links ==
*{{GlossaryForHyperspace | anchor=Simplex | title=Simplex }}
* [http://www.polytope.net/hedrondude/topes.htm Polytopes of Various Dimensions], Jonathan Bowers
* [http://tetraspace.alkaline.org/glossary.htm Multi-dimensional Glossary]

{{Polytopes}}

[[Category:5-polytopes]]