Truncated tetrahedron

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.

Truncated tetrahedron

(Click here for rotating model)
TypeArchimedean solid
Uniform polyhedron
ElementsF = 8, E = 18, V = 12 (χ = 2)
Faces by sides4{3}+4{6}
Conway notationtT
Schläfli symbolst{3,3} = h2{4,3}
t0,1{3,3}
Wythoff symbol3
Coxeter diagram =
Symmetry groupTd, A3, [3,3], (*332), order 24
Rotation groupT, [3,3]+, (332), order 12
Dihedral angle3-6: 109°28′16′
6-6: 70°31′44″
ReferencesU02, C16, W6
PropertiesSemiregular convex

Colored faces
3.6.6
(Vertex figure)

Triakis tetrahedron
(dual polyhedron)
Net
3D model of a truncated tetrahedron

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, , having half of the vertices of the cantellated cube (rhombicuboctahedron), . There are two dual positions of this construction, and combining them creates the uniform compound of two truncated tetrahedra.

Area and volume

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

Densest Packing

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 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.[2]

Cartesian coordinates

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)
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). As solid this can represent a 3D dissection making 4 red octahedra and 6 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
Orthogonal projection
Centered by Edge normal Face normal Edge Face
Wireframe
Wireframe
Dual
Projective
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. This projection is conformal, preserving angles but not areas or lengths. Straight lines on the sphere are projected as circular arcs on the plane.


triangle-centered
hexagon-centered
Orthographic projection Stereographic projections

Friauf polyhedron

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] It is named after J. B. Friauf and his 1927 paper "The crystal structure of the intermetallic compound MgCu2".[5]

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.[6]

The Tetraminx puzzle has a truncated tetrahedral shape. This puzzle shows a dissection of a truncated tetrahedron into 4 octahedra and 6 tetrahedra. It contains 4 central planes of rotations.

Truncated tetrahedral graph
Truncated tetrahedral graph
3-fold symmetry
Vertices12[7]
Edges18
Radius3
Diameter3[7]
Girth3[7]
Automorphisms24 (S4)[7]
Chromatic number3[7]
Chromatic index3[7]
PropertiesHamiltonian, regular, 3-vertex-connected, planar graph
Table of graphs and parameters

In the mathematical field of graph theory, a truncated tetrahedral graph is an 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

4-fold symmetry
3-fold symmetry

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
Orbifold
*n32		 Spherical Euclidean Hyperbolic Paracompact
*332 *333 *433 *533 *633... *∞33
Cantic figure
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

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.

Examples
  • Truncated tetrahedron in rotation
  • Truncated tetrahedron (Matemateca IME-USP)
  • Truncated 4-sided die

See also

References
  1. Chisholm, Matt; Avnet, Jeremy (1997). "Truncated Trickery: Truncatering". theory.org. Retrieved 2013-09-02.
  2. Damasceno, Pablo F.; Engel, Michael; Glotzer, Sharon C. (Dec 2011). "Crystalline Assemblies and Densest Packings of a Family of Truncated Tetrahedra and the Role of Directional Entropic Forces". ACS Nano. 6 (2012): 609–614. arXiv:1109.1323. doi:10.1021/nn204012y. PMID 22098586.
  3. Jiao, Yang; Torquato, Sal (Sep 2011). "A Packing of Truncated Tetrahedra that Nearly Fills All of Space". arXiv:1107.2300 [cond-mat.soft].
  4. http://met.iisc.ernet.in/~lord/webfiles/clusters/polyclusters.pdf
  5. Friauf, J. B. (1927). "The crystal structure of the intermetallic compound MgCu2". J. Am. Chem. Soc. 49: 3107–3114. doi:10.1021/ja01411a017.
  6. http://expo67.ncf.ca/man_the_producer_p1.html
  7. 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
  • Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
  • Read, R. C.; Wilson, R. J. (1998), An Atlas of Graphs, Oxford University Press
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