View source for Tetrakis hexahedron

{{Short description|Catalan solid with 24 faces}}
{{Semireg dual polyhedra db|Semireg dual polyhedron stat table|dtO}}
{{multiple image
 | align = right  | total_width = 300
 | image1 = Perspectiva Corporum Regularium 15d.jpg
 | image2 = Dual compound truncated 8 max.png
 | footer = [[Dual compound]] of [[truncated octahedron]] and tetrakis hexahedron. The woodcut on the left is from ''[[Perspectiva Corporum Regularium]]'' (1568) by [[Wenzel Jamnitzer]].
}}
{{multiple image
 | align = right  | total_width = 300
 | image1 = D24 tetrakis hexahedron.JPG
 | image2 = Modell, Kristallform Pyramidenwürfel bzw Tetrakishexaeder -Krantz 412-.jpg
 | footer = [[Dice|Die]] and [[crystal model]]
}}
{{multiple image
 | align = right  | total_width = 300
 | image1 = Meyers b10 s0232 b7.png
 | image2 = Modell, Kristallform Hexakistetraeders -Krantz- (2), crop.jpg
 | footer = Drawing and crystal model of variant with tetrahedral symmetry called ''hexakis tetrahedron'' <ref>''Hexakistetraeder'' in German, see e.g. [[:File:Meyers b10 s0232.jpg|''Meyers'' page]] and [[:File:Brockhaus chrystal shapes (Krystalle. I.).jpg|''Brockhaus'' page]]. The [[:File:Brockhaus and Efron Encyclopedic Dictionary b48 863-4.jpg|same drawing]] appears in ''Brockhaus and Efron'' as ''преломленный пирамидальный тетраэдр'' (''refracted [[triakis tetrahedron|pyramidal tetrahedron]]'').</ref>
}}
In [[geometry]], a '''[[Conway kis operator|tetrakis]] hexahedron''' (also known as a '''tetrahexahedron''', '''hextetrahedron''', '''tetrakis cube''', and '''kiscube'''<ref>Conway, ''Symmetries of Things'', p.284</ref>) is a [[Catalan solid]]. Its dual is the [[truncated octahedron]], an [[Archimedean solid]].

It can be called a '''disdyakis hexahedron''' or '''hexakis tetrahedron''' as the [[Dual (polyhedron)|dual]] of an [[omnitruncated]] [[tetrahedron]], and as the [[barycentric subdivision]] of a tetrahedron.<ref>{{citation
 | last1 = Langer | first1 = Joel C.
 | last2 = Singer | first2 = David A.
 | doi = 10.1007/s00032-010-0124-5
 | issue = 2
 | journal = Milan Journal of Mathematics
 | mr = 2781856
 | pages = 643–682
 | title = Reflections on the lemniscate of Bernoulli: the forty-eight faces of a mathematical gem
 | volume = 78
 | year = 2010}}</ref>

== Cartesian coordinates ==

[[Cartesian coordinates]] for the 14 vertices of a tetrakis hexahedron centered at the origin, are the points 
<math display=block>
  \bigl( {\pm\tfrac32}, 0, 0 \bigr),\  
  \bigl( 0, \pm\tfrac32, 0 \bigr),\ 
  \bigl( 0, 0, \pm\tfrac32 \bigr),\ 
  \bigl({\pm1}, \pm1, \pm1 \bigr).
</math>

The length of the shorter edges of this tetrakis hexahedron equals 3/2 and that of the longer edges equals 2. The faces are acute isosceles triangles. The larger angle of these equals <math>\arccos\tfrac19 \approx 83.62^{\circ}</math> and the two smaller ones equal <math>\arccos\tfrac23 \approx 48.19^{\circ}</math>.

==Orthogonal projections==
The ''tetrakis hexahedron'', dual of the [[truncated octahedron]] has 3 symmetry positions, two located on vertices and one mid-edge.
{|class=wikitable
|+ Orthogonal projections
|- align=center
!Projective<BR>symmetry
|[2]
|[4]
|[6]
|-
!Tetrakis<BR>hexahedron
|[[File:Dual cube t12 e66.png|120px]]
|[[File:Dual cube t12 B2.png|120px]]
|[[File:Dual cube t12.png|120px]]
|-
!Truncated<BR>octahedron
|[[File:Cube t12 e66.png|120px]]
|[[File:3-cube t12 B2.svg|120px]]
|[[File:3-cube t12.svg|120px]]
|}

==Uses==
Naturally occurring ([[crystal]]) formations of tetrahexahedra are observed in [[copper]] and [[fluorite]] systems.

[[Polyhedral dice]] shaped like the tetrakis hexahedron are occasionally used by [[gamer]]s.

A [[24-cell#Projections|24-cell]] viewed under a vertex-first [[perspective projection]] has a surface topology of a tetrakis hexahedron and the geometric proportions of the [[rhombic dodecahedron]], with the rhombic faces divided into two triangles.

The tetrakis hexahedron appears as one of the simplest examples in [[Building_(mathematics)|building]] theory. Consider the [[Riemannian symmetric space]] associated to the [[Group_(mathematics)|group]] [[Special_linear_group|''SL''<sub>4</sub>('''R''')]]. Its [[Tits metric|Tits boundary]] has the structure of a [[Building_(mathematics)#Spherical_building|spherical building]] whose apartments are 2-dimensional spheres. The partition of this sphere into spherical [[simplex|simplices]] (chambers) can be obtained by taking the radial projection of a tetrakis hexahedron.

==Symmetry==
With T<sub>d</sub>, [3,3] (*332) [[tetrahedral symmetry]], the triangular faces represent the 24 fundamental domains of tetrahedral symmetry. This polyhedron can be constructed from 6 [[great circle]]s on a sphere. It can also be seen by a cube with its square faces triangulated by their vertices and face centers and a tetrahedron with its faces divided by vertices, mid-edges, and a central point.

{|class="wikitable" style="text-align: center;"
|- style="vertical-align: center;"
| [[File:Polyhedron great rhombi 4-4 max.png|100px]]
| [[File:Disdyakis 6 max.png|x100px]]
| [[File:Disdyakis 6 in deltoidal 12.png|x120px]]
|-
| [[Truncated octahedron|Truncated<br>octahedron]]
| '''Disdyakis<br>hexahedron'''
| [[Rhombic dodecahedron|Deltoidal<br>dodecahedron]]
|- style="vertical-align: center;"
| [[File:Disdyakis 6 in rhombic 6 max.png|x100px]]
| [[File:Disdyakis 6 in Platonic 4a max.png|x100px]]
| [[File:Disdyakis 6 in Platonic 4b max.png|x100px]]
|-
| [[Cube|Rhombic<br>hexahedron]]
|colspan="2"| [[Tetrahedron]]
|}

{| class="wikitable collapsible collapsed" style="text-align: center;"
!colspan="4"| [[Spherical polyhedron]]
|- valign=top
| [[File:Disdyakis 6 spherical.png|170px]]
| [[File:Disdyakis 6 spherical from redyellow.png|170px]]
| [[File:Disdyakis 6 spherical from red.png|170px]]
| [[File:Disdyakis 6 spherical from blue.png|170px]]
|-
| (see [[:File:Disdyakis 6 spherical.gif|rotating model]])
|colspan="3"| [[Orthographic projection]]s from 2-, 3- and 4-fold axes
|}

The edges of the spherical tetrakis hexahedron belong to six great circles, which correspond to [[Reflection (mathematics)|mirror planes]] in [[tetrahedral symmetry]]. They can be grouped into three pairs of orthogonal circles (which typically intersect on one coordinate axis each). In the images below these square [[hosohedron|hosohedra]] are colored red, green and blue.

{| class="wikitable collapsible collapsed" style="text-align: center;"
!colspan="4"| [[Stereographic projection]]s
|-
|rowspan="2"| [[File:Spherical tetrakis hexahedron RGB.png|230px]]
! 2-fold
! 3-fold
! 4-fold
|-
| [[File:Tetrakis hexahedron stereographic d2.svg|x200px]]
| [[File:Tetrakis hexahedron stereographic d3.svg|x200px]]
| [[File:Tetrakis hexahedron stereographic d4.svg|x200px]]
|}

==Dimensions==

If we denote the edge length of the base cube by {{mvar|a}}, the height of each pyramid summit above the cube is {{tmath|\tfrac{a}{4}.}} The inclination of each triangular face of the pyramid versus the cube face is <math>\arctan\tfrac{1}{2} \approx 26.565^\circ</math> {{OEIS|A073000}}. One edge of the [[Triangle|isosceles triangles]] has length {{mvar|a}}, the other two have length {{tmath|\tfrac{3a}{4},}} which follows by applying the [[Pythagorean theorem]] to height and base length. This yields an altitude of <math>\tfrac{\sqrt 5 a}{4}</math> in the triangle ({{OEIS2C|A204188}}). Its [[Triangle#Computing the area of a triangle|area]] is <math>\tfrac{\sqrt 5 a^2}{8},</math> and the internal angles are <math>\arccos\tfrac{2}{3} \approx 48.1897^\circ</math> and the complementary <math>180^\circ - 2\arccos\tfrac{2}{3} \approx 83.6206^\circ.</math>

The [[Pyramid (geometry)#Mensuration|volume of the pyramid]] is {{tmath|\tfrac{a^3}{12};}} so the total volume of the six pyramids and the cube in the hexahedron is {{tmath|\tfrac{3a^3}{2}.}}

== Kleetope ==
[[File:Pyramid_augmented_cube.png|thumb|Non-convex tetrakis hexahedron with equilateral triangle faces]]
It can be seen as a [[cube]] with [[square pyramid]]s covering each square face; that is, it is the [[Kleetope]] of the cube. A non-convex form of this shape, with [[equilateral triangle]] faces, has the same surface geometry as the [[regular octahedron]], and a paper octahedron model can be re-folded into this shape.<ref>{{citation
 | last = Rus | first = Jacob
 | editor1-last = Swart | editor1-first = David
 | editor2-last = Séquin | editor2-first = Carlo H.
 | editor3-last = Fenyvesi | editor3-first = Kristóf
 | contribution = Flowsnake Earth
 | contribution-url = https://archive.bridgesmathart.org/2017/bridges2017-237.html
 | isbn = 978-1-938664-22-9
 | location = Phoenix, Arizona
 | pages = 237–244
 | publisher = Tessellations Publishing
 | title = Proceedings of Bridges 2017: Mathematics, Art, Music, Architecture, Education, Culture
 | year = 2017}}</ref> This form of the tetrakis hexahedron was illustrated by [[Leonardo da Vinci]] in [[Luca Pacioli]]'s ''[[Divina proportione]]'' (1509).<ref>{{citation|title=Divina proportione|title-link=Divina proportione|first=Luca|last=Pacioli|author-link=Luca Pacioli|year=1509|contribution=Plates 11 and 12|contribution-url=https://archive.org/details/divinaproportion00paci/page/n205}}</ref>

This non-convex form of the tetrakis hexahedron can be folded along the square faces of the inner cube as a [[net (geometry)|net]] for a four-dimensional [[cubic pyramid]].

== Related polyhedra and tilings==
{{Octahedral truncations}}

{{Truncated figure2 table}}

It is a polyhedra in a sequence defined by the [[face configuration]] V4.6.2''n''. This group is special for having all even number of edges per vertex and form bisecting planes through the polyhedra and infinite lines in the plane, and continuing into the hyperbolic plane for any ''n''&nbsp;≥&nbsp;7.

With an even number of faces at every vertex, these polyhedra and tilings can be shown by alternating two colors so all adjacent faces have different colors.

Each face on these domains also corresponds to the fundamental domain of a [[symmetry group]] with order 2,3,''n'' mirrors at each triangle face vertex.
{{Omnitruncated table}}

==See also==
*[[Disdyakis triacontahedron]]
*[[Disdyakis dodecahedron]]
*[[Kisrhombille tiling]]
*[[Compound of three octahedra]]
* [[Deltoidal icositetrahedron]], another 24-face Catalan solid.

==References==
{{reflist}}
*{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
*{{Citation |last1=Wenninger |first1=Magnus |author1-link=Magnus Wenninger |title=Dual Models |publisher=[[Cambridge University Press]] |isbn=978-0-521-54325-5 |mr=730208 |year=1983 |doi=10.1017/CBO9780511569371}} (The thirteen semiregular convex polyhedra and their duals, Page 14, Tetrakishexahedron)
*''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, {{isbn|978-1-56881-220-5}} [https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Tetrakis hexahedron)

==External links==
*{{Mathworld2 |urlname=TetrakisHexahedron |title=Tetrakis hexahedron |urlname2=CatalanSolid |title2=Catalan solid}}
*[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] www.georgehart.com: The Encyclopedia of Polyhedra
**[[VRML]] [http://www.georgehart.com/virtual-polyhedra/vrml/tetrakishexahedron.wrl model]
**[http://www.georgehart.com/virtual-polyhedra/conway_notation.html Conway Notation for Polyhedra] Try: "dtO" or "kC"
*[https://web.archive.org/web/20080828230230/http://polyhedra.org/poly/show/35/tetrakis_hexahedron Tetrakis Hexahedron] – Interactive Polyhedron model
*[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]

{{Catalan solids}}
{{Polyhedron navigator}}

[[Category:Catalan solids]]