View source for Truncated tetrahedron

{{short description|Archimedean solid with 8 faces}}
{{infobox polyhedron
| name               = Truncated tetrahedron
| image              = Afgeknotte driezijdige piramide.png
| type               = [[Archimedean solid]],<br>[[Uniform polyhedron]]
| faces              = 4 [[hexagon]]s<br>4 [[triangle]]s
| edges              = 18
| vertices           = 12
| dual               = [[triakis tetrahedron]]
| symmetry           = [[tetrahedral symmetry]] <math> \mathrm{T}_\mathrm{h} </math>
| vertex_figure      = Polyhedron truncated 4a vertfig.svg
| net                = Polyhedron truncated 4a net.svg
}}

In [[geometry]], the '''truncated tetrahedron''' is an [[Archimedean solid]]. It has 4 regular [[hexagon]]al faces, 4 [[equilateral triangle]] faces, 12 vertices and 18 edges (of two types). It can be constructed by [[truncation (geometry)|truncating]] all 4 vertices of a regular [[tetrahedron]].

== Construction ==
The truncated tetrahedron can be constructed from a [[regular tetrahedron]] by cutting all of its vertices off, a process known as [[Truncation (geometry)|truncation]].{{r|kuchel}} The resulting polyhedron has 4 equilateral triangles and 4 regular hexagons, 18 edges, and 12 vertices.{{r|berman}} With edge length 1, the [[Cartesian coordinates]] of the 12 vertices are points

<math display=block>
  \bigl( {\pm\tfrac{3\sqrt{2}}{4} }, \pm\tfrac{\sqrt{2}}{4}, \pm\tfrac{\sqrt{2}}{4} \bigr)
</math>

that have an even number of minus signs.

== Properties ==
Given the edge length <math> a </math>. The surface area of a truncated tetrahedron <math> A </math> is the sum of 4 regular hexagons and 4 equilateral triangles' area, and its [[volume]] <math> V </math> is:{{r|berman}}
<math display="block">\begin{align}
A &= 7\sqrt{3}a^2 &&\approx 12.124a^2, \\
V &= \tfrac{23}{12}\sqrt{2}a^3 &&\approx 2.711a^3.
\end{align}</math>

The dihedral angle of a truncated tetrahedron between triangle-to-hexagon is approximately 109.47°, and that between adjacent hexagonal faces is approximately 70.53°.{{r|johnson}}

The densest packing of the truncated tetrahedron is believed to be <math display="inline"> \Phi = \frac{207}{208} </math>, as reported by two independent groups using [[Monte Carlo methods]] by {{harvtxt|Damasceno|Engel|Glotzer|2012}} and {{harvtxt|Jiao|Torquato|2011}}.{{r|damasceno|jiao}} Although no mathematical proof exists that this is the best possible packing for the truncated tetrahedron, the high proximity to the unity and independence of the findings make it unlikely that an even denser packing is to be found. If the truncation of the corners is slightly smaller than that of a truncated tetrahedron, this new shape can be used to fill space completely.{{r|damasceno}}

[[File:Truncated tetrahedron.stl|thumb|3D model of a truncated tetrahedron]]
The truncated tetrahedron is an [[Archimedean solid]], meaning it is a highly symmetric and semi-regular polyhedron, and two or more different [[regular polygon]]al faces meet in a vertex.{{r|diudea}} The truncated tetrahedron has the same [[Point groups in three dimensions|three-dimensional group symmetry]] as the regular tetrahedron, the [[tetrahedral symmetry]] <math> \mathrm{T}_\mathrm{h} </math>.{{r|kocakoca}} The polygonal faces that meet for every vertex are one equilateral triangle and two regular hexagons, and the [[vertex figure]] is denoted as <math> 3 \cdot 6^2 </math>. Its [[dual polyhedron]] is [[triakis tetrahedron]], a [[Catalan solid]], shares the same symmetry as the truncated tetrahedron.{{r|williams}}

== 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}}

==Truncated tetrahedral graph==
[[File:Tuncated tetrahedral graph.png|thumb|240px|The graph of a truncated tetrahedron]]
In the [[mathematics|mathematical]] field of [[graph theory]], a '''truncated tetrahedral graph''' is an [[Archimedean graph]], the [[1-skeleton|graph of vertices and edges]] of the truncated tetrahedron, one of the [[Archimedean solid]]s. It has 12 [[Vertex (graph theory)|vertices]] and 18 edges.<ref>An Atlas of Graphs, page 267, truncated tetrahedral graph</ref> It is a connected cubic graph,<ref>An Atlas of Graphs, page 130, connected cubic graphs, 12 vertices, C105</ref> and connected cubic transitive graph.<ref>An Atlas of Graphs, page 161, connected cubic transitive graphs, 12 vertices, Ct11</ref>

== Examples ==
<gallery class="center">
File:De divina proportione - Tetraedron Abscisum Vacuum.jpg | drawing in [[De divina proportione]] (1509)
File:Perspectiva Corporum Regularium 09a.jpg | drawing in [[Perspectiva Corporum Regularium]] (1568)
File:Modell, Kristallform (Verzerrungen) Oktaeder (Spinell) -Krantz 4, 6, 7, 391- (8).jpg | [[crystal model]]
File:Tetraedro truncado (Matemateca IME-USP).jpg | photos from different perspectives ([[Matemateca]])
File:D4 truncated tetrahedron.JPG | 4-sided [[Dice#Polyhedral dice|die]]
File:Permutohedron in simplex of order 4, with truncated tetrahedron (0-based).png | 12 [[permutations#Permutations of multisets|permutations]] of <math>(4, 2, 0, 0)</math> (brown)
</gallery>

==See also==
* [[Quarter cubic honeycomb]] – Fills space using truncated tetrahedra and smaller tetrahedra
* [[Truncated 5-cell]] – Similar uniform polytope in 4-dimensions
* [[Truncated triakis tetrahedron]]
* [[Triakis truncated tetrahedron]]
* [[Octahedron]] – a rectified tetrahedron
* [[Truncated Triangular Pyramid Number]]

==References==
{{reflist|refs=

<ref name=berman>{{cite journal
 | last = Berman | first = Martin
 | doi = 10.1016/0016-0032(71)90071-8
 | journal = Journal of the Franklin Institute
 | mr = 290245
 | pages = 329–352
 | title = Regular-faced convex polyhedra
 | volume = 291
 | year = 1971| issue = 5
}}</ref>

<ref name="damasceno">{{cite journal
 | arxiv = 1109.1323
 | title = Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces
 | journal = ACS Nano
 | volume = 6 | issue = 2012 | pages = 609–614
 | year = 2012
 | last1 = Damasceno | first1 = Pablo F.
 | last2 = Engel | first2 = Michael
 | last3 = Glotzer | first3 = Sharon C.
 | doi = 10.1021/nn204012y
 | pmid = 22098586
 | s2cid = 12785227
}}</ref>

<ref name="diudea">{{cite book
 | last = Diudea | first = M. V.
 | year = 2018
 | title = Multi-shell Polyhedral Clusters
 | series = Carbon Materials: Chemistry and Physics
 | volume = 10
 | publisher = [[Springer Science+Business Media|Springer]]
 | isbn = 978-3-319-64123-2
 | doi = 10.1007/978-3-319-64123-2
 | url = https://books.google.com/books?id=p_06DwAAQBAJ&pg=PA39
 | page = 39
}}</ref>

<ref name="friauf">{{cite journal
 | last = Friauf | first = J. B.
 | title = The crystal structure of the intermetallic compounds
 | year = 1927
 | journal = [[Journal of the American Chemical Society]]
 | volume = 49 | issue = 12
 | pages = 3107–3114
 | doi = 10.1021/ja01411a017
| bibcode = 1927JAChS..49.3107F
 }}</ref>

<ref name="grunbaum">{{cite journal
 | last1 = Grünbaum | first1 = Branko | author1-link = Branko Grünbaum
 | last2 = Shephard | first2 = G. C. | author2-link = Geoffrey Colin Shephard
 | doi = 10.1090/S0273-0979-1980-14827-2
 | issue = 3
 | journal = [[Bulletin of the American Mathematical Society]]
 | mr = 585178
 | pages = 951–973
 | series = New Series
 | title = Tilings with congruent tiles
 | volume = 3
 | year = 1980| doi-access = free
}}</ref>

<ref name="kocakoca">{{cite book
 | last1 = Koca | first1 = M.
 | last2 = Koca | first2 = N. O.
 | year = 2013
 | title = Mathematical Physics: Proceedings of the 13th Regional Conference, Antalya, Turkey, 27&ndash;31 October 2010
 | contribution = Coxeter groups, quaternions, symmetries of polyhedra and 4D polytopes
 | contribution-url = https://books.google.com/books?id=ILnBkuSxXGEC&pg=PA46
 | page = 46&ndash;47
 | publisher = World Scientific
}}</ref>

<ref name="kuchel">{{cite journal
 | last = Kuchel | first = Philip W.
 | year = 2012
 | title = 96.45 Can you 'bend' a truncated truncated tetrahedron?
 | journal = [[The Mathematical Gazette]]
 | volume = 96 | issue = 536 | pages = 317–323
 | doi = 10.1017/S0025557200004666
 | jstor = 23248575
}}</ref>

<ref name="jiao">{{citation
 | last1 = Jiao | first1 = Yang
 | last2 = Torquato | first2 = Salvatore
 | arxiv = 1107.2300
 | date = October 2011
 | doi = 10.1063/1.3653938
 | issue = 15
 | journal = The Journal of Chemical Physics
 | article-number = 151101
 | publisher = AIP Publishing
 | title = Communication: A packing of truncated tetrahedra that nearly fills all of space and its melting properties
 | volume = 135| pmid = 22029288
 | bibcode = 2011JChPh.135o1101J
 }}</ref>

<ref name="johnson">{{cite journal
 | last = Johnson | first = Norman W. | author-link = Norman W. Johnson
 | year = 1966
 | title = Convex polyhedra with regular faces
 | journal = [[Canadian Journal of Mathematics]]
 | volume = 18
 | pages = 169–200
 | doi = 10.4153/cjm-1966-021-8
 | mr = 0185507
 | s2cid = 122006114
 | zbl = 0132.14603 | doi-access = free
}} See line II.</ref>

<ref name="lcd">{{cite book
 | last1 = Lalena | first1 = John N.
 | last2 = Cleary | first2 = David A.
 | last3 = Duparc | first3 = Olivier B.
 | title = Principles of Inorganic Materials Design
 | year = 2020
 | publisher = [[John Wiley & Sons]]
 | page = 150
 | isbn = 9781119486916
 | url = https://books.google.com/books?id=7y3fDwAAQBAJ&pg=PA150
}}</ref>

<ref name="rajwade">{{cite book
 | last = Rajwade | first = A. R.
 | title = Convex Polyhedra with Regularity Conditions and Hilbert's Third Problem
 | series = Texts and Readings in Mathematics
 | year = 2001
 | url = https://books.google.com/books?id=afJdDwAAQBAJ&pg=PA84
 | page = 84&ndash;89
 | publisher = Hindustan Book Agency
 | isbn = 978-93-86279-06-4
 | doi = 10.1007/978-93-86279-06-4
}}</ref>

<ref name="williams">{{cite book
 | last = Williams | first = Robert | authorlink = Robert Williams (geometer)
 | year = 1979
 | title = The Geometrical Foundation of Natural Structure: A Source Book of Design
 | publisher = Dover Publications, Inc.
 | url = https://archive.org/details/geometricalfound00will/page/72
 | page = 72
| isbn = 978-0-486-23729-9 }}</ref>

}}
* {{citation|last1=Read|first1=R. C.|last2=Wilson|first2=R. J.|title=An Atlas of Graphs|publisher=Oxford University Press|year= 1998}}
{{Commons category}}

==External links==
* {{mathworld2 |urlname=TruncatedTetrahedron |title=Truncated tetrahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
** {{mathworld | urlname = TruncatedTetrahedralGraph | title = Truncated tetrahedral graph}}
* {{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3x3o - tut}}
* [http://www.dr-mikes-math-games-for-kids.com/polyhedral-nets.html?net=3cqTmfu7gdEZ8I7kRUVvji6qxBATVSp2WpmIWGx7l7pWe7bveylFxv3piHnPNZN&name=Truncated+Tetrahedron#applet Editable printable net of a truncated tetrahedron with interactive 3D view]
* [http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
* [http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra

{{Archimedean solids}}
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[[Category:Archimedean solids]]
[[Category:Truncated tilings]]
[[Category:Individual graphs]]
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