View source for Compound of five tetrahedra

{{short description|Compound polyhedron}}
{{more sources needed|date=August 2024}}
{|class="wikitable" style="float:right; margin-left:8px; width:340px"
!bgcolor=#e7dcc3 colspan=2|Compound of five tetrahedra
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|align=center colspan=2|[[Image:Compound of five tetrahedra.png|240px]]
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|bgcolor=#e7dcc3 width=50%|Type||[[Regular polyhedral compound|Regular compound]]
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|bgcolor=#e7dcc3|Coxeter symbol|| {5,3}[5{3,3}] {3,5}{{Sfn|Coxeter|1973|p=98}}
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|bgcolor=#e7dcc3|Index||UC<sub>5</sub>, [[List of Wenninger polyhedron models#Stellations of icosahedron|W<sub>24</sub>]]
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|bgcolor=#e7dcc3|[[Euler characteristic|Elements]]<BR>(As a compound)||5 [[Tetrahedron|tetrahedra]]:<BR>''F'' = 20, ''E'' = 30, ''V'' = 20
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|bgcolor=#e7dcc3|[[Dual polyhedron|Dual compound]]||Self-dual
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|bgcolor=#e7dcc3|[[Symmetry group]]||[[Chirality (mathematics)|chiral]] [[Icosahedral symmetry|icosahedral]] (''I'')
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|bgcolor=#e7dcc3|[[Subgroup]] restricting to one constituent||[[Chirality (mathematics)|chiral]] [[Tetrahedral symmetry|tetrahedral]] (''T'')
|}
[[File:Compound of five tetrahedra (full).stl|thumb|3D model of a compound of five tetrahedra]]
The '''[[polyhedral compound|compound]] of five [[tetrahedron|tetrahedra]]''', also known as the chiricosahedron, is one of the five regular polyhedral compounds. This [[Polyhedral compound|compound]] [[polyhedron]] is also a [[stellation]] of the regular [[icosahedron]]. It was first described by [[Edmund Hess]] in 1876.

It can be seen as a [[#As a faceting|faceting]] of a [[regular dodecahedron]].

==As a compound==
It can be constructed by arranging five [[tetrahedron|tetrahedra]] in [[rotational icosahedral symmetry]] ('''I'''), as colored in the upper right model. It is one of [[Polytope compound#Regular compounds|five regular compounds]] which can be constructed from identical [[Platonic solid]]s.{{Sfn|Coxeter|1973|loc=§3.6 The five regular compounds|pp=47-50}}

It shares the same [[vertex arrangement]] as a [[regular dodecahedron]].

There are two [[Chirality (mathematics)|enantiomorphous]] forms (the same figure but having opposite chirality) of this compound polyhedron. Both forms together create the reflection symmetric [[compound of ten tetrahedra]].

It has a [[Density (polyhedra)|density]] of higher than 1.

{| class=wikitable
|- align=center
|[[Image:Spherical compound of five tetrahedra.png|160px]]<BR>As a [[spherical tiling]]
|[[image:CompoundOfFiveTetrahedra.png|160px]]<BR>Transparent Models<BR>[[:image:CompoundOfFiveTetrahedra.gif|(Animation)]]
|[[File:Five tetrahedra.png|200px]]<br/>Five interlocked tetrahedra
|}

==As a stellation==

It can also be obtained by [[stellation|stellating]] the [[icosahedron]], and is given as [[List of Wenninger polyhedron models#Stellations of icosahedron|Wenninger model index 24]].{{Sfn|Coxeter|1973|loc=§6.2 Stellating the Platonic solids|pp=96-104}}

{| class=wikitable
![[Stellation diagram]]!![[Stellation]] core||[[Convex hull]]
|- valign=top align=center
|[[File:Compound of five tetrahedra stellation facets.svg|200px]]
|[[File:Icosahedron.png|200px]]<BR>[[Stellations of icosahedron|Icosahedron]]
|[[File:Dodecahedron.png|200px]]<BR>[[Dodecahedron]]
|}

==As a faceting==

[[File:Chiroicosahedron-in-dodecahedron.png|thumb|150px|left|Five tetrahedra in a dodecahedron.]]

It is a [[faceting]] of a dodecahedron, as shown at left.
{{-}}

== Group theory ==
The compound of five tetrahedra is a geometric illustration of the notion of [[Group action (mathematics)#orbstab|orbits and stabilizers]], as follows.

The symmetry group of the compound is the (rotational) [[icosahedral group]] ''I'' of order 60, while the stabilizer of a single chosen tetrahedron is the (rotational) [[tetrahedral group]] ''T'' of order 12, and the orbit space ''I''/''T'' (of order 60/12&nbsp;=&nbsp;5) is naturally identified with the 5 tetrahedra – the coset ''gT'' corresponds to which tetrahedron ''g'' sends the chosen tetrahedron to.

==An unusual dual property==
[[File:Second compound stellation of icosahedron.svg|thumb|150px|Compound of five tetrahedra]]
This compound is unusual, in that the [[dual polyhedron|dual]] figure is the [[enantiomorph]] of the original. If the faces are twisted to the right, then the vertices are twisted to the left. When we [[dual polyhedron|dualise]], the faces dualise to right-twisted vertices and the vertices dualise to left-twisted faces, giving the chiral twin. Figures with this property are extremely rare.

==In 4-dimensional space==
The compound of five tetrahedra is related to the regular [[5-cell]], the 4-simplex [[regular 4-polytope]], which is also composed of 5 regular tetrahedra. In the 5-cell the tetrahedra are bonded face-to-face such that each triangular face is shared by two tetrahedral cells.

The compound of five tetrahedra occurs embedded in [[4-dimensional space]], inscribed in the 120 dodecahedral cells of the [[120-cell]]. The 120-cell is the [[120-cell#Geometry|largest and most comprehensive regular 4-polytope]]; the regular 5-cell is the smallest and most elemental. The 120-cell contains inscribed within itself instances of every other regular 4-polytope.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; "It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions."}} In each of the 120-cell's dodecahedral cells, there are two inscribed instances of the compound of 5 tetrahedra (in other words, one instance of the [[compound of ten tetrahedra]]). The 5 tetrahedra of each compound of five occur as cells of another regular 4-polytope inscribed within the 120-cell, the [[600-cell]], which has 600 regular tetrahedra as its cells. The 120-cell is a compound of 5 disjoint 600-cells, and each of its dodecahedral cells is a [[120-cell#Cell rotations of inscribed duals|compound of 5 tetrahedral cells]], one cell from each of the 5 disjoint 600-cells.

== See also ==
* [[Compound of ten tetrahedra]]
* [[Compound of five cubes]]
* [[Compound of five truncated tetrahedra]]

==Citations==
{{reflist}}

==References==
* {{cite book | first=Magnus | last=Wenninger | authorlink=Magnus Wenninger | title=Polyhedron Models | publisher=Cambridge University Press | year=1974 | isbn=0-521-09859-9 }}
* {{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 | last1=Coxeter | first1=Harold Scott MacDonald | author1-link=Harold Scott MacDonald Coxeter | last2=Du Val | first2=P. | last3=Flather | first3=H. T. | last4=Petrie | first4=J. F. | title=[[The Fifty-Nine Icosahedra]] | publisher=Tarquin | edition=3rd | isbn=978-1-899618-32-3 | mr=676126 | year=1999 | orig-year=1938}}
* {{cite web |last=McCooey |first=Robert |title=Uniform Polyhedron Compounds |url=http://www.polytope.net/hedrondude/regcomp3.htm |website=Hedron Dude |access-date=24 June 2025}}

== External links ==
* {{MathWorld|urlname=Tetrahedron5-Compound|title=Tetrahedron 5-Compound}}
* [http://bulatov.org/metal/five_tetrahedra.html Metal Sculpture of Five Tetrahedra Compound]
* [[VRML]] model: [https://web.archive.org/web/20081119051953/http://www.interocitors.com/polyhedra/UCs/05__5_Tetrahedra.wrl]
* [http://demonstrations.wolfram.com/CompoundsOf5And10Tetrahedra/ Compounds of 5 and 10 Tetrahedra] by Sándor Kabai, [[The Wolfram Demonstrations Project]].
* {{KlitzingPolytopes|../incmats/ki.htm|3D compound|}}

{{Icosahedron stellations}}

[[Category:Polyhedral stellation]]
[[Category:Polyhedral compounds]]
[[Category:Chiral polyhedra]]