Editing Truncated tetrahedron (section)

== Related polyhedrons ==
The truncated tetrahedron can be found in the construction of polyhedrons. For example, the [[augmented truncated tetrahedron]] is a [[Johnson solid]] constructed from a truncated tetrahedron by attaching [[triangular cupola]] onto its hexagonal face.{{r|rajwade}} The [[triakis truncated tetrahedron]] is a polyhedron constructed from a truncated tetrahedron by adding three tetrahedrons onto its triangular faces, as interpreted by the name "[[Conway kis operator|triakis]]". It is classified as [[plesiohedron]], meaning it can [[Tessellation|tessellate]] in three-dimensional space known as [[Honeycomb (geometry)|honeycomb]]; an example is [[triakis truncated tetrahedral honeycomb]].{{r|grunbaum}}

[[File:Truncated triakis tetrahedron.gif|thumb|upright=0.7|Truncated triakis tetrahedron]]
A [[truncated triakis tetrahedron]] is known for its usage in chemistry as a [[fullerene]]. This solid is represented as an [[allotrope]] of carbon (C<sub>28</sub>), forming the smallest stable fullerene,<ref>{{cite journal
 | last = Martin | first = Jan M.L.
 | date = June 1996
 | doi = 10.1016/0009-2614(96)00354-5
 | issue = 1–3
 | journal = Chemical Physics Letters
 | pages = 1–6
 | title = C<sub>28</sub>: the smallest stable fullerene?
 | volume = 255| bibcode = 1996CPL...255....1M
 }}</ref> and experiments have found it to be stabilized by encapsulating a metal atom.<ref name=dkm>{{cite journal
 | last1 = Dunk | first1 = Paul W.
 | last2 = Kaiser | first2 = Nathan K.
 | last3 = Mulet-Gas | first3 = Marc
 | last4 = Rodríguez-Fortea | first4 = Antonio
 | last5 = Poblet | first5 = Josep M.
 | last6 = Shinohara | first6 = Hisanori
 | last7 = Hendrickson | first7 = Christopher L.
 | last8 = Marshall | first8 = Alan G.
 | last9 = Kroto | first9 = Harold W.
 | date = May 2012
 | doi = 10.1021/ja302398h
 | issue = 22
 | journal = Journal of the American Chemical Society
 | pages = 9380–9389
 | publisher = American Chemical Society (ACS)
 | title = The Smallest Stable Fullerene, M@C<sub>28</sub> (M = Ti, Zr, U): Stabilization and Growth from Carbon Vapor
 | volume = 134| pmid = 22519801
 | bibcode = 2012JAChS.134.9380D
 }}</ref> Geometrically, this polyhedron was studied in 1935 by Michael Goldberg as a possible solution to the [[isoperimetric problem]] of maximizing the volume for a given number of faces (16 in this case) and a given surface area.<ref>{{cite journal
 | last = Goldberg | first = Michael
 | journal = [[Tohoku Mathematical Journal]]
 | pages = 226–236
 | title = The isoperimetric problem for polyhedra
 | url = https://www.jstage.jst.go.jp/article/tmj1911/40/0/40_0_226/_pdf
 | volume = 40}}</ref> For this optimization problem, the optimal geometric form for the polyhedron is one in which the faces are all tangent to an [[inscribed sphere]].<ref>{{cite journal
 | last = Fejes Tóth | first = László | author-link = László Fejes Tóth
 | doi = 10.2307/2371944
 | journal = [[American Journal of Mathematics]]
 | jstor = 2371944
 | mr = 24157
 | pages = 174–180
 | title = The isepiphan problem for {{mvar|n}}-hedra
 | volume = 70
 | year = 1948| issue = 1 }}</ref>

{{anchor|Friauf polyhedron}}The ''Friauf polyhedron'' is named after [[J. B. Friauf]] in which he described it as a [[intermetallic]] structure formed by a compound of metallic elements.{{r|friauf}} It can be found in crystals such as complex metallic alloys, an example is dizinc magnesium MgZn<sub>2</sub>.{{r|lcd}} It is a lower symmetry version of the truncated tetrahedron, interpreted as a truncated [[tetragonal disphenoid]] with its three-dimensional symmetry group as the [[dihedral group]] <math> D_{2\mathrm{d}} </math> of order 8.{{cn|date=July 2024}}

Truncating a truncated tetrahedron gives the resulting polyhedron 54 edges, 32 vertices, and 20 faces&mdash;4 hexagons, 4 [[nonagon]]s, and 12 [[trapezoid|trapeziums]]. This polyhedron was used by [[Adidas]] as the underlying geometry of the [[Adidas Jabulani|Jabulani ball]] designed for the [[FIFA World Cup 2010|2010 World Cup]].{{r|kuchel}}