View source for Truncated tetrahedron

{{short description|Archimedean solid with 8 faces}}
{{Semireg polyhedra db|Semireg polyhedron stat table|tT}}
[[File:Truncated tetrahedron.stl|thumb|3D model of a truncated tetrahedron]]

In [[Geometry|geometry]], the '''truncated tetrahedron''' is an [[Archimedean solid]]. It has 4 regular [[Hexagon|hexagon]]al faces, 4 [[Equilateral triangle|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|tetrahedron]] at one third of the original edge length.

A deeper truncation, removing a tetrahedron of half the original edge length from each vertex, is called [[Rectification (geometry)|rectification]]. The rectification of a tetrahedron produces an [[Octahedron|octahedron]].<ref>{{cite web |url=https://theory.org/geotopo/tt/html/truncatering.html |title=Truncated Trickery: Truncatering |first1=Matt |last1=Chisholm |first2=Jeremy |last2=Avnet |work=theory.org |year=1997 |access-date=2013-09-02}}</ref>

A ''truncated tetrahedron'' is the [[Goldberg polyhedron]] {{math|G{{sub|III}}(1,1),}} containing triangular and hexagonal faces.

A ''truncated tetrahedron'' can be called a '''cantic cube''', with Coxeter diagram, {{CDD|node_h1|4|node|3|node_1}}, having half of the vertices of the cantellated cube ([[Rhombicuboctahedron|rhombicuboctahedron]]), {{CDD|node_1|4|node|3|node_1}}. There are two dual positions of this construction, and combining them creates the uniform [[Compound of two truncated tetrahedra|compound of two truncated tetrahedra]].

==Area and volume==
The area ''A'' and the [[Physics:Volume|volume]] ''V'' of a truncated tetrahedron of edge length ''a'' are:
:<math>\begin{align}
A &= 7\sqrt{3}a^2 &&\approx 12.124\,355\,65a^2 \\
V &= \tfrac{23}{12}\sqrt{2}a^3 &&\approx 2.710\,575\,995a^3. \end{align}</math>

==Densest packing==
The densest packing of the Archimedean truncated tetrahedron is believed to be Φ&nbsp;=&nbsp;{{sfrac|207|208}}, as reported by two independent groups using Monte Carlo methods.<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="Jiao">{{cite arXiv | eprint=1107.2300| title=A Packing of Truncated Tetrahedra that Nearly Fills All of Space | date= Sep 2011| last1=Jiao | first1=Yang | last2=Torquato | first2=Sal | class=cond-mat.soft }}</ref> Although no mathematical proof exists that this is the best possible packing for the truncated tetrahedron, the high proximity to the unity and independency of the findings make it unlikely that an even denser packing is to be found. In fact, if the truncation of the corners is slightly smaller than that of an Archimedean truncated tetrahedron, this new shape can be used to completely fill space.<ref name="Damasceno" />

==Cartesian coordinates==
Cartesian coordinates for the 12 vertices of a [[Truncation (geometry)|truncated]] [[Tetrahedron|tetrahedron]] centered at the origin, with edge length √8, are all permutations of (±1,±1,±3) with an even number of minus signs:
* (+3,+1,+1), (+1,+3,+1), (+1,+1,+3)
* (−3,−1,+1), (−1,−3,+1), (−1,−1,+3)
* (−3,+1,−1), (−1,+3,−1), (−1,+1,−3)
* (+3,−1,−1), (+1,−3,−1), (+1,−1,−3)

{|class=wikitable width=600
|- valign=top
|[[File:Truncated tetrahedron in unit cube.png|200px]]
|[[File:Triangulated truncated tetrahedron.png|200px]]
|[[Image:UC54-2 truncated tetrahedra.png|200px]]
|- valign=top
|Orthogonal projection showing Cartesian coordinates inside its bounding box: (±3,±3,±3).
|The hexagonal faces of the truncated tetrahedra can be divided into six coplanar equilateral triangles. The four new vertices have Cartesian coordinates:<br /> (−1,−1,−1), (−1,+1,+1),<br />(+1,−1,+1), (+1,+1,−1). As a solid, this can represent a 3D [[Dissection problem|dissection]], making four red octahedra and six yellow tetrahedra.
|The set of vertex permutations (±1,±1,±3) with an odd number of minus signs forms a complementary truncated tetrahedron, and combined they form a uniform compound polyhedron.
|}

Another simple construction exists in 4-space as cells of the truncated 16-cell, with vertices as coordinate permutation of:
:(0,0,1,2)

== Orthogonal projection ==
{| class=wikitable
|+ Orthogonal projection
!Centered by
!Edge normal
!Face normal
!Edge
!Face
|- align=center
!Wireframe
|[[File:Polyhedron truncated 4a from redyellow max.png|100px]]
|
|[[File:Polyhedron truncated 4a from blue max.png|100px]]
|[[File:Polyhedron truncated 4a from red max.png|100px]] [[File:Polyhedron truncated 4a from yellow max.png|100px]]
|- align=center
!Wireframe
|[[File:tetrahedron t01 ae.png|100px]]
|[[File:tetrahedron t01 af36.png|100px]]
|[[File:3-simplex t01.svg|100px]]
|[[File:3-simplex t01 A2.svg|100px]]
|- align=center
!Dual
|[[File:Dual tetrahedron t01 ae.png|100px]]
|[[File:Dual tetrahedron t01 af36.png|100px]]
|[[File:Dual tetrahedron t01.png|100px]]
|[[File:Dual tetrahedron t01 A2.png|100px]]
|- align=center
|- align=center
!Projective<br />symmetry
![1]
![1]
![4]
![3]
|}

==Spherical tiling==
The truncated tetrahedron can also be represented as a spherical tiling, and projected onto the plane via a [[Stereographic projection|stereographic projection]]. This projection is [[Conformal map|conformal]], preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.
{|class=wikitable
|- align=center valign=top
|[[File:Uniform tiling 332-t12.png|160px]]
|[[Image:truncated tetrahedron stereographic projection triangle.png|160px]]<br />[[Triangle|triangle]]-centered
|[[Image:truncated tetrahedron stereographic projection hexagon.png|160px]]<br />[[Hexagon|hexagon]]-centered
|-
![[Orthographic projection]]
!colspan=3|[[Stereographic projection]]s
|}

=== Friauf polyhedron ===
A lower symmetry version of the truncated tetrahedron (a truncated tetragonal disphenoid with order 8 D<sub>2d</sub> symmetry) is called a '''Friauf polyhedron''' in crystals such as [[Chemistry:Complex metallic alloys|complex metallic alloys]]. This form fits 5 Friauf polyhedra around an axis, giving a 72-degree [[Dihedral angle|dihedral angle]] on a subset of 6-6 edges.<ref>http://met.iisc.ernet.in/~lord/webfiles/clusters/polyclusters.pdf {{Bare URL PDF|date=March 2022}}</ref> It is named after J. B. Friauf and his 1927 paper "The crystal structure of the intermetallic compound MgCu<sub>2</sub>".<ref>{{cite journal|last=Friauf|first=J. B.|title=The crystal structure of the intermetallic compound MgCu<sub>2</sub>|date=1927|journal=J. Am. Chem. Soc.|volume=49|pages=3107–3114|doi=10.1021/ja01411a017}}</ref>

==Uses==
Giant truncated tetrahedra were used for the "Man the Explorer" and "Man the Producer" theme pavilions in Expo 67. They were made of massive girders of steel bolted together in a geometric lattice. The truncated tetrahedra were interconnected with lattice steel platforms. All of these buildings were demolished after the end of Expo 67, as they had not been built to withstand the severity of the Montreal weather over the years. Their only remnants are in the Montreal city archives, the Public Archives Of Canada and the photo collections of tourists of the times.<ref>{{Cite web|url=http://expo67.ncf.ca/man_the_producer_p1.html|title=Expo 67 - Man the Producer - page 1}}</ref>

The Tetraminx puzzle has a truncated tetrahedral shape. This puzzle shows a [[Dissection problem|dissection]] of a truncated tetrahedron into 4 octahedra and 6 tetrahedra. It contains 4 central planes of rotations.{{cn|date=November 2023}}
:[[File:Tetraminx.jpg|160px]]

==Truncated tetrahedral graph==
{{Infobox graph
 | name = Truncated tetrahedral graph
 | image = [[File:Tuncated tetrahedral graph.png|240px]]
 | image_caption = 3-fold symmetry
 | namesake =
 | vertices = 12<ref name=ag>An Atlas of Graphs, page=172, C105</ref>
 | edges = 18
 | automorphisms = 24 ([[Symmetric group|S<sub>4</sub>]])<ref name=ag/>
 | radius = 3
 | diameter = 3<ref name=ag/>
 | girth = 3<ref name=ag/>
 | chromatic_number = 3<ref name=ag/>
 | chromatic_index = 3<ref name=ag/>
 | fractional_chromatic_index =
 | properties = Hamiltonian, [[Regular graph|regular]], [[K-vertex-connected graph|3-vertex-connected]], [[Planar graph|planar graph]]
}}
In the [[Mathematics|mathematical]] field of [[Graph theory|graph theory]], a '''truncated tetrahedral graph''' is an [[Archimedean graph]], the 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>

{| class=wikitable
!Circular
|- align=center valign=top
|[[File:Truncated tetrahedral graph.circo.svg|160px]]
|}
{| class=wikitable
!colspan=2|Orthographic projections
|- align=center valign=top
|[[File:3-simplex t01.svg|160px]]<br />4-fold symmetry
|[[File:3-simplex t01 A2.svg|160px]]<br />3-fold symmetry
|}
{{Clear}}

==Related polyhedra and tilings==

{{Tetrahedron family}}

It is also a part of a sequence of cantic polyhedra and tilings with [[Vertex configuration|vertex configuration]] 3.6.''n''.6. In this [[Wythoff construction|wythoff construction]] the edges between the hexagons represent degenerate [[Digon|digon]]s.

===Symmetry mutations===
This polyhedron is topologically related as a part of sequence of uniform [[Truncation (geometry)|truncated]] polyhedra with [[Vertex configuration|vertex configuration]]s (3.2''n''.2''n''), and [''n'',3] [[Coxeter group]] symmetry.


== 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 | [[Physics:Crystal model|crystal model]]
File:Tetraedro truncado (Matemateca IME-USP).jpg | photos from different perspectives (Matemateca)
File:D4 truncated tetrahedron.JPG | 4-sided die
File:Permutohedron in simplex of order 4, with truncated tetrahedron (0-based).png | 12 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

==References==
<references />
* {{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
* {{citation|last1=Read|first1=R. C.|last2=Wilson|first2=R. J.|title=An Atlas of Graphs|publisher=Oxford University Press|year= 1998}}

==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

{{Polyhedron navigator}}

[[Category:Archimedean solids]]
[[Category:Truncated tilings]]
[[Category:Individual graphs]]
[[Category:Planar graphs]]

{{Sourceattribution|Truncated tetrahedron}}