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| {{Semireg dual polyhedra db|Semireg dual polyhedron stat table|dtO}}
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| In [[geometry]], a '''[[Conway kis operator|tetrakis]] hexahedron''' (also known as a '''tetrahexahedron''') is a [[Catalan solid]]. Its dual is the [[truncated octahedron]], an [[Archimedean solid]]. It can be seen as a [[cube]] with [[square pyramid]]s covering each square face; that is, it is the [[Kleetope]] of the cube.
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| It also can be called a '''disdyakis hexahedron''' as the [[Dual (polyhedron)|dual]] of an [[omnitruncated]] [[tetrahedron]].
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| ==Orthogonal projections==
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| The ''tetrakis hexahedron'', dual of the [[truncated octahedron]] has five special [[orthogonal projection]]s, centered, on a vertex, on two types of edges, and two types of faces: Hexagon, and square. The last two correspond to the B<sub>2</sub> and A<sub>2</sub> [[Coxeter plane]]s.
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| {|class=wikitable width=640
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| |+ Orthogonal projections
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| |-
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| !Centered by
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| !Vertex
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| !Edge<br>4-6
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| !Edge<br>6-6
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| !Face<br>Square
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| !Face<br>Hexagon
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| |-
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| !Truncated<BR>octahedron
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| |[[File:Cube t12 v.png|100px]]
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| |[[File:Cube t12 e46.png|100px]]
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| |[[File:Cube t12 e66.png|100px]]
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| |[[File:3-cube t12 B2.svg|100px]]
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| |[[File:3-cube t12.svg|100px]]
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| |-
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| !Hexakis<BR>hexahedron
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| |[[File:Dual cube t12 v.png|100px]]
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| |[[File:Dual cube t12 e46.png|100px]]
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| |[[File:Dual cube t12 e66.png|100px]]
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| |[[File:Dual cube t12 B2.png|100px]]
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| |[[File:Dual cube t12.png|100px]]
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| |- align=center
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| !Projective<BR>symmetry
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| |[2]
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| |[2]
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| |[2]
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| |[4]
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| |[6]
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| |}
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| ==Uses==
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| Naturally occurring ([[crystal]]) formations of tetrahexahedra are observed in [[copper]] and [[fluorite]] systems.
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| [[Polyhedral dice]] shaped like the tetrakis hexahedron are occasionally used by [[gamer]]s.
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| 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.
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| ==Symmetry==
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| With ''T<sub>d</sub>'', [3,3] (*332) [[tetrahedral symmetry]], the triangular faces represent the 24 fundamental domains of tetrahdral symmetry. This polyhedron can be constructed from 6 [[great circle]]s on a sphere.
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| {|class=wikitable width=360
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| |[[File:Tetrahedral reflection domains.png|180px]]
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| |[[File:Disdyakis cube.png|180px]]
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| |}
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| ==Dimensions==
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| If we denote the edge length of the base cube by {{math|a}},
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| the height of each pyramid summit above the cube is {{math|a/4}}.
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| The inclination of each triangular face of the pyramid versus the cube face | |
| is {{math|arctan(1/2)}}, approximately 26.565 degrees {{OEIS|A073000}}. | |
| One edge of the [[Triangle|isosceles triangles]] has length {{math|a}}, the other two have
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| length {{math|3a/4}}, which follows by applying the [[Pythagorean theorem]] to height and base length. This yields an altitude of {{math|√5 a/4}} in the triangle ({{OEIS2C|A204188}}). Its [[Triangle#Computing the area of a triangle|area]] is {{math|√5a/8}}, and the internal angles are {{math|arccos(2/3)}} (approximately 48.1897 degrees) and the complementary {{math|180-2arccos(2/3)}} (approximately 83.6206 degrees).
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| The [[Pyramid (geometry)#Volume|volume]] of the pyramid
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| is {{math|a<sup>3</sup>/12}}; so the total volume of the six pyramids and the cube in the hexahedron is {{math|3a<sup>3</sup>/2}}.
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| == Related polyhedra and tilings==
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| {{Octahedral truncations}}
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| {{Truncated figure2 table}}
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| It is a polyhedra in a sequence defined by the [[face configuration]] ''V4.6.2n''. 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 <math>n \ge 7.</math>
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| 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.
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| 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.
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| {{Omnitruncated table}}
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| ==See also==
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| *[[Disdyakis triacontahedron]]
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| *[[Disdyakis dodecahedron]]
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| *[[Kisrhombille tiling]]
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| *[[Compound of three octahedra]]
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| ==References==
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| *{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
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| *{{Citation |last1=Wenninger |first1=Magnus |author1-link=Magnus Wenninger |title=Dual Models |publisher=[[Cambridge University Press]] |isbn=978-0-521-54325-5 |id={{MathSciNet|id=730208}} |year=1983}} (The thirteen semiregular convex polyhedra and their duals, Page 14, Tetrakishexahedron)
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| *''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, ISBN 978-1-56881-220-5 [http://www.akpeters.com/product.asp?ProdCode=2205] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 284, Tetrakis hexahedron)
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| ==External links==
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| *{{Mathworld2 |urlname=TetrakisHexahedron |title=Tetrakis hexahedron |urlname2=CatalanSolid |title2=Catalan solid}}
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| *[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] www.georgehart.com: The Encyclopedia of Polyhedra
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| **[[VRML]] [http://www.georgehart.com/virtual-polyhedra/vrml/tetrakishexahedron.wrl model]
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| **[http://www.georgehart.com/virtual-polyhedra/conway_notation.html Conway Notation for Polyhedra] Try: "dtO" or "kC"
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| *[http://polyhedra.org/poly/show/35/tetrakis_hexahedron Tetrakis Hexahedron] – Interactive Polyhedron model
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| *[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
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| {{Polyhedron navigator}}
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| [[Category:Catalan solids]]
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Interweaving social trends form a deep net in which we all have been trapped. When You see, the Tygers of Pan Tang sang 'It's lonely on the top bar. Everyones trying to do we in', these people stole much from clash of clans hack into tool no survey. A society without battle of clans hack solution no survey is as being a society with no knowledge, in that it is quite good.
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