Truncated tetrahedron

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Truncated tetrahedron
Truncatedtetrahedron.jpg
(Click here for rotating model)
Type Archimedean solid
Uniform polyhedron
Elements F = 8, E = 18, V = 12 (χ = 2)
Faces by sides 4{3}+4{6}
Conway notation tT
Schläfli symbols t{3,3} = h2{4,3}
t0,1{3,3}
Wythoff symbol 2 3 | 3
Coxeter diagram CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
Symmetry group Td, A3, [3,3], (*332), order 24
Rotation group T, [3,3]+, (332), order 12
Dihedral Angle 3-6:109°28'16"
6-6:70°31'44"
References U02, C16, W6
Properties Semiregular convex
Truncated tetrahedron.png
Colored faces
Truncated tetrahedron vertfig.png
3.6.6
(Vertex figure)
Triakistetrahedron.jpg
Triakis tetrahedron
(dual polyhedron)
Truncated tetrahedron flat.svg
Net

In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 equilateral triangle faces, 12 vertices and 18 edges (of two types). It can be constructed by truncating all 4 vertices of a regular 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. The rectification of a tetrahedron produces an octahedron.[1]

A truncated tetrahedron is the Goldberg polyhedron GIII(1,1), containing triangular and hexagonal faces.

A truncated tetrahedron can be called a cantic cube, with Coxeter diagram, CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png, having half of the vertices of the cantellated cube (rhombicuboctahedron), CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png. There are two dual positions of this construction, and combining them creates the uniform compound of two truncated tetrahedra.

Area and volume[edit]

The area A and the volume V of a truncated tetrahedron of edge length a are:

A = 7\sqrt{3}a^2 \approx 12.12435565a^2
V = \frac{23}{12}\sqrt{2}a^3 \approx 2.710575995a^3.

Densest Packing[edit]

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.[2][3] 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.[2]

Cartesian coordinates[edit]

Cartesian coordinates for the 12 vertices of a truncated 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)
Truncated tetrahedron in unit cube.png Triangulated truncated tetrahedron.png UC54-2 truncated tetrahedra.png
Orthogonal projection showing Cartesian coordinates inside it bounding box: (±3,±3,±3). The hexagonal faces of the truncated tetrahedra can be divided into 6 coplanar equilateral triangles. The 4 new vertices have Cartesian coordinates:
(-1,-1,-1), (-1,+1,+1),
(+1,-1,+1), (+1,+1,-1).
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[edit]

Orthogonal projection
Centered by Edge normal Face normal Edge Face/vertex
Image Tetrahedron t01 ae.png Tetrahedron t01 af36.png 3-simplex t01.svg 3-simplex t01 A2.svg
Dual image Dual tetrahedron t01 ae.png Dual tetrahedron t01 af36.png Dual tetrahedron t01.png Dual tetrahedron t01 A2.png
Projective
symmetry
[1] [1] [3] [4]

Spherical tiling[edit]

The truncated tetrahedron can also be represented as a spherical tiling, and projected onto the plane via a stereographic projection. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.

Uniform tiling 332-t12.png Truncated tetrahedron stereographic projection triangle.png
triangle-centered
Truncated tetrahedron stereographic projection hexagon.png
hexagon-centered
Orthographic projection Stereographic projections

Friauf polyhedron[edit]

A lower symmetry version of the truncated tetrahedron (a truncated tetragonal disphenoid with order 8 D2d 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.[4] Its named after J. B. Friauf and his 1927 paper The crystal structure of the intermetallic compound MgCu2. [5]

Use in architecture[edit]

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.[6]


Truncated tetrahedral graph[edit]

Truncated tetrahedral graph
Tuncated tetrahedral graph.png
3-fold symmetry
Vertices 12[7]
Edges 18
Radius 3
Diameter 3[7]
Girth 3[7]
Automorphisms 24 (S4)[7]
Chromatic number 3[7]
Chromatic index 3[7]
Properties Hamiltonian, regular, 3-vertex-connected, planar graph

In the mathematical field of graph theory, a truncated tetrahedral graph is a Archimedean graph, the graph of vertices and edges of the truncated tetrahedron, one of the Archimedean solids. It has 12 vertices and 18 edges.[8] It is a connected cubic graph,[9] and connected cubic transitive graph.[10]

Circular Orthographic projections
Truncated tetrahedral graph.circo.svg 3-simplex t01.svg
4-fold symmetry
3-simplex t01 A2.svg
3-fold symmetry

Related polyhedra and tilings[edit]

Family of uniform tetrahedral polyhedra
Symmetry: [3,3], (*332) [3,3]+, (332)
Uniform polyhedron-33-t0.png Uniform polyhedron-33-t01.png Uniform polyhedron-33-t1.png Uniform polyhedron-33-t12.png Uniform polyhedron-33-t2.png Uniform polyhedron-33-t02.png Uniform polyhedron-33-t012.png Uniform polyhedron-33-s012.png
CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 3.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
{3,3} t{3,3} r{3,3} t{3,3} {3,3} rr{3,3} tr{3,3} sr{3,3}
Duals to uniform polyhedra
Tetrahedron.svg Triakistetrahedron.jpg Hexahedron.svg Triakistetrahedron.jpg Tetrahedron.svg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Dodecahedron.svg
V3.3.3 V3.6.6 V3.3.3.3 V3.6.6 V3.3.3 V3.4.3.4 V4.6.6 V3.3.3.3.3

It is also a part of a sequence of cantic polyhedra and tilings with vertex configuration 3.6.n.6. In this wythoff construction the edges between the hexagons represent degenerate digons.

*n33 orbifold symmetries of cantic tilings: 3.6.n.6
N33 fundamental domain t01.png Orbifold
*n32
Spherical Euclidean Hyperbolic Paracompact
*332 *333 *433 *533 *633... *∞33
Cantic figure Spherical cantic cube.png Uniform tiling 333-t12.png H2 tiling 334-6.png H2 tiling 335-6.png H2 tiling 336-6.png H2 tiling 33i-6.png
Vertex 3.6.2.6 3.6.3.6 3.6.4.6 3.6.5.6 3.6.6.6 3.6..6

Symmetry mutations[edit]

This polyhedron is topologically related as a part of sequence of uniform truncated polyhedra with vertex configurations (3.2n.2n), and [n,3] Coxeter group symmetry.

*n32 symmetry mutation of truncated spherical tilings: 3.2n.2n
Symmetry
*n32
[n,3]
Spherical Euclid. Compact hyperb. Paraco.
*232
[2,3]
*332
[3,3]
*432
[4,3]
*532
[5,3]
*632
[6,3]
*732
[7,3]
*832
[8,3]...
*∞32
[∞,3]
Truncated
figures
Spherical triangular prism.png Uniform tiling 332-t01-1-.png Uniform tiling 432-t01.png Uniform tiling 532-t01.png Uniform tiling 63-t01.png H2 tiling 237-3.png H2 tiling 238-3.png H2 tiling 23i-3.png
Config. 3.4.4 3.6.6 3.8.8 3.10.10 3.12.12 3.14.14 3.16.16 3.∞.∞
Triakis
figures
Spherical trigonal bipyramid.png Spherical triakis tetrahedron.png Spherical triakis octahedron.png Spherical triakis icosahedron.png Tiling Dual Semiregular V3-12-12 Triakis Triangular.svg Ord7 triakis triang til.png Ord8 triakis triang til.png Ord-infin triakis triang til.png
Config. V3.4.4 V3.6.6 V3.8.8 V3.10.10 V3.12.12 V3.14.14 V3.16.16 V3.∞.∞

See also[edit]

References[edit]

  1. ^ Chisholm, Matt; Avnet, Jeremy (1997). "Truncated Trickery: Truncatering". theory.org. Retrieved 2013-09-02. 
  2. ^ a b "Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces" (PDF). Dec 2011. 
  3. ^ "A Packing of Truncated Tetrahedra that Nearly Fills All of Space" (PDF). Sep 2011. 
  4. ^ http://met.iisc.ernet.in/~lord/webfiles/clusters/polyclusters.pdf
  5. ^ Friauf, J. B. The crystal structure of the intermetallic compound MgCu2 (1927) J. Am. Chem. Soc. 19, 3107-3114.
  6. ^ http://expo67.ncf.ca/man_the_producer_p1.html
  7. ^ a b c d e f An Atlas of Graphs, page=172, C105
  8. ^ An Atlas of Graphs, page=267, truncated tetrahedral graph
  9. ^ An Atlas of Graphs, page=130, connected cubic graphs, 12 vertices, C105
  10. ^ An Atlas of Graphs, page=161, connected cubic transitive graphs, 12 vertices, Ct11
  • Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press 

External links[edit]

References[edit]