View source for Triakis octahedron

{{Short description|Catalan solid with 24 faces}}
{{Semireg dual polyhedra db|Semireg dual polyhedron stat table|dtC}}
In [[geometry]], a '''triakis octahedron''' (or '''trigonal trisoctahedron'''<ref>{{Cite web | url = https://etc.usf.edu/clipart/keyword/forms | title = Clipart tagged: 'forms'| publisher = etc.usf.edu}}</ref> or '''kisoctahedron'''<ref>Conway, Symmetries of things, p. 284</ref>) is an [[Archimedean solid|Archimedean dual]] solid, or a [[Catalan solid]]. Its dual is the [[truncated cube]].

It can be seen as an [[octahedron]] with [[triangular pyramid]]s added to each face; that is, it is the [[Kleetope]] of the octahedron. It is also sometimes called a ''trisoctahedron'', or, more fully, ''trigonal trisoctahedron''. Both names reflect that it has three triangular faces for every face of an octahedron. The ''tetragonal trisoctahedron'' is another name for the [[deltoidal icositetrahedron]], a different polyhedron with three quadrilateral faces for every face of an octahedron.

This convex polyhedron is topologically similar to the concave [[stellated octahedron]]. They have the same face connectivity, but the vertices are at different relative distances from the center.

If its shorter edges have length of 1, its surface area and volume are:
:<math>\begin{align} A &= 3\sqrt{7+4\sqrt{2}} \\ V &= \frac{3+2\sqrt{2}}{2} \end{align}</math>

== Cartesian coordinates ==

Let {{nowrap|1=''α'' = {{sqrt|2}} − 1}}, then the 14 points {{nowrap|(±''α'', ±''α'', ±''α'')}} and {{nowrap|(±1, 0, 0)}}, {{nowrap|(0, ±1, 0)}} and {{nowrap|(0, 0, ±1)}} are the vertices of a triakis octahedron centered at the origin.

The length of the long edges equals {{sqrt|2}}, and that of the short edges {{nowrap|2{{sqrt|2}} − 2}}.

The faces are isosceles triangles with one obtuse and two acute angles. The obtuse angle equals {{nowrap|arccos({{sfrac|1|4}} − {{sfrac|{{sqrt|2}}|2}})}} ≈ {{val|117.20057038016}}° and the acute ones equal {{nowrap|arccos({{sfrac|1|2}} + {{sfrac|{{sqrt|2}}|4}})}} ≈ {{val|31.39971480992}}°.

==Orthogonal projections==
The ''triakis octahedron'' has three symmetry positions, two located on vertices, and one mid-edge:
{|class=wikitable
|+ Orthogonal projections
|- align=center
!Projective<br>symmetry
|[2]
|[4]
|[6]
|- align=center
!Triakis<br>octahedron
|[[File:Dual truncated cube t01 e88.png|100px]]
|[[File:Dual truncated cube t01 B2.png|120px]]
|[[File:Dual truncated cube t01.png|120px]]
|-
!Truncated<br>cube
|[[File:Cube t01 e88.png|120px]]
|[[File:3-cube t01 B2.svg|120px]]
|[[File:3-cube t01.svg|120px]]
|}

==Cultural references==
* A triakis octahedron is a vital element in the plot of cult author [[Hugh Cook (science fiction author)|Hugh Cook]]'s novel ''[[Chronicles of an Age of Darkness#The Wishstone and the Wonderworkers|The Wishstone and the Wonderworkers]]''.

== Related polyhedra ==
The triakis octahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.

{{Octahedral truncations}}

The triakis octahedron is a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These [[face-transitive]] figures have (*''n''32) reflectional [[Orbifold notation|symmetry]].
[[File:Triakis octahedron.stl|thumb|3D model of a triakis octahedron]]
[[File:Kleetope of octahedron.gif|thumb|right|Animation of triakis octahedron and other related polyhedra]] [[File:Spherical triakis octahedron.png|160px|thumb|Spherical triakis octahedron]]
{{Truncated figure1 table}}

The triakis octahedron is also a part of a sequence of polyhedra and tilings, extending into the hyperbolic plane. These [[face-transitive]] figures have (*''n''42) reflectional [[Orbifold notation|symmetry]].
{{Truncated figure4 table}}

==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 17, Triakisoctahedron)
* ''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, Triakis octahedron)

==External links==
* {{Mathworld2 |urlname=SmallTriakisOctahedron |title=Triakis octahedron |urlname2=CatalanSolid |title2=Catalan solid}}
* [https://web.archive.org/web/20080908050514/http://polyhedra.org/poly/show/34/triakis_octahedron Triakis Octahedron] – Interactive Polyhedron Model
* [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/triakisoctahedron.wrl model] {{Webarchive|url=https://web.archive.org/web/20181011103313/http://www.georgehart.com/virtual-polyhedra/vrml/triakisoctahedron.wrl |date=2018-10-11 }}
** [http://www.georgehart.com/virtual-polyhedra/conway_notation.html Conway Notation for Polyhedra] Try: "dtC"

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[[Category:Catalan solids]]