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| {{Semireg polyhedra db|Semireg polyhedron stat table|tT}}
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| 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]] at one third of the original edge length.
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| 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]].<ref>{{cite web |url=https://theory.org/geotopo/tt/html/truncatering.html |title=Truncated Trickery: Truncatering |first=Matt |last=Chisholm |first2=Jeremy |last2=Avnet |work=theory.org |year=1997 |accessdate=2013-09-02}}</ref>
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| A ''truncated tetrahedron'' is the [[Goldberg polyhedron]] G<sub>III</sub>(1,1), containing triangular and hexagonal faces.
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| 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]]), {{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]].
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| ==Area and volume==
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| The area ''A'' and the [[volume]] ''V'' of a truncated tetrahedron of edge length ''a'' are:
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| :<math>A = 7\sqrt{3}a^2 \approx 12.12435565a^2</math>
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| :<math>V = \frac{23}{12}\sqrt{2}a^3 \approx 2.710575995a^3.</math>
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| ==Densest Packing==
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| The densest packing of the Archimedean truncated tetrahedron is believed to be Φ = 207/208, as reported by two independent groups using [[Monte Carlo methods]].<ref name="Damasceno">{{cite web | url=http://arxiv.org/pdf/1109.1323v2.pdf | title=Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces | date= Dec 2011}}</ref><ref name="Jiao">{{cite web | url=http://arxiv.org/pdf/1107.2300v4.pdf | title=A Packing of Truncated Tetrahedra that Nearly Fills All of Space | date= Sep 2011}}</ref> Although no mathematical proof exists that this is the best possible packing for those shapes, 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" />
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| ==Cartesian coordinates==
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| [[Cartesian coordinates]] for the 12 vertices of a [[Truncation (geometry)|truncated]] [[tetrahedron]] centered at the origin, with edge length √8, are all permutations of (±1,±1,±3) with an even number of minus signs:
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| *(+3,+1,+1), (+1,+3,+1), (+1,+1,+3)
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| *(−3,−1,+1), (−1,−3,+1), (−1,−1,+3)
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| *(−3,+1,−1), (−1,+3,−1), (−1,+1,−3)
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| *(+3,−1,−1), (+1,−3,−1), (+1,−1,−3)
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| {|class=wikitable width=600
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| |- valign=top
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| |[[File:Truncated_tetrahedron_in_unit_cube.png|200px]]
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| |[[File:Triangulated truncated tetrahedron.png|200px]]
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| |[[Image:UC54-2 truncated tetrahedra.png|200px]]
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| |- valign=top
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| |[[Orthogonal projection]] showing Cartesian coordinates inside it [[bounding box]]: (±3,±3,±3).
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| |The hexagonal faces of the truncated tetrahedra can be divided into 6 coplanar equilateral triangles. The 4 new vertices have Cartesian coordinates:<BR> (-1,-1,-1), (-1,+1,+1),<BR>(+1,-1,+1), (+1,+1,1).
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| |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 [[Compound polyhedron#Uniform compounds|uniform compound polyhedron]].
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| |}
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| Another simple construction exists in 4-space as cells of the [[truncated 16-cell]], with vertices as coordinate permutation of:
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| :(0,0,1,2)
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| == Orthogonal projection ==
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| {| class=wikitable
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| |+ [[Orthogonal projection]]
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| !Centered by
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| !Edge normal
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| !Face normal
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| !Edge
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| !Face/vertex
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| |- align=center
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| !Image
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| |[[File:tetrahedron t01 ae.png|100px]]
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| |[[File:tetrahedron t01 af36.png|100px]]
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| |[[File:3-simplex t01.svg|100px]]
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| |[[File:3-simplex t01 A2.svg|100px]]
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| |- align=center
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| !Dual image
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| |[[File:Dual tetrahedron t01 ae.png|100px]]
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| |[[File:Dual tetrahedron t01 af36.png|100px]]
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| |[[File:Dual tetrahedron t01.png|100px]]
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| |[[File:Dual tetrahedron t01 A2.png|100px]]
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| |- align=center
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| |- align=center
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| !Projective<BR>symmetry
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| ![1]
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| ![1]
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| ![3]
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| ![4]
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| |}
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| ==Related polyhedra and tilings==
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| {{Tetrahedron family}}
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| {{Octahedral truncations}}
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| {{Cantic table}}
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| This polyhedron is topologically related as a part of sequence of uniform [[Truncation (geometry)|truncated]] polyhedra with [[vertex configuration]]s (3.2n.2n), and [n,3] [[Coxeter group]] symmetry.
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| {{Truncated figure1 table}}
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| === Friauf polyhedron ===
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| 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 [[complex metallic alloys]]. This form fits 5 Friauf polyhedra around an axis, giving a 72 degree [[dihedral angle]] on a subset of 6-6 edges.<ref>http://met.iisc.ernet.in/~lord/webfiles/clusters/polyclusters.pdf</ref> Its named after [[J. B. Friauf]] and his 1927 paper ''The crystal structure of the intermetallic compound MgCu2''. <ref>Friauf, J. B. ''The crystal structure of the intermetallic compound MgCu2'' (1927) J. Am. Chem. Soc. 19, 3107-3114.</ref>
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| ==Use in architecture==
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| 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>http://expo67.ncf.ca/man_the_producer_p1.html</ref>
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| ==See also==
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| *[[Quarter cubic honeycomb]] - Fills space using truncated tetrahedra and smaller tetrahedra
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| *[[Truncated 5-cell]] – Similar uniform polytope in 4-dimensions
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| *[[Truncated triakis tetrahedron]]
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| *[[Triakis truncated tetrahedron]]
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| *[[Octahedron]] – a rectified tetrahedron
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| ==References==
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| *{{The Geometrical Foundation of Natural Structure (book)}} (Section 3-9)
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| <references/>
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| ==External links==
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| *{{mathworld2 |urlname=TruncatedTetrahedron |title=Truncated tetrahedron |urlname2=ArchimedeanSolid |title2=Archimedean solid}}
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| *{{KlitzingPolytopes|polyhedra.htm|3D convex uniform polyhedra|x3x3o - tut}}
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| *[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]
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| *[http://www.mathconsult.ch/showroom/unipoly/ The Uniform Polyhedra]
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| *[http://www.georgehart.com/virtual-polyhedra/vp.html Virtual Reality Polyhedra] The Encyclopedia of Polyhedra
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| {{Archimedean solids}}
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| {{Polyhedron navigator}}
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| [[Category:Archimedean solids]]
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