Truncated tetrahedron

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Truncated tetrahedron
(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}
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
Polyhedron truncated 4a max.png
Colored faces
Polyhedron truncated 4a vertfig.svg
(Vertex figure)
Polyhedron truncated 4a dual max.png
Triakis tetrahedron
(dual polyhedron)
Polyhedron truncated 4a net.svg
3D model of a 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.

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:

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 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[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 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:
(−1,−1,−1), (−1,+1,+1),
(+1,−1,+1), (+1,+1,−1). As a solid, this can represent a 3D 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:


Orthogonal projection[edit]

Orthogonal projection
Centered by Edge normal Face normal Edge Face
Wireframe Polyhedron truncated 4a from redyellow max.png Polyhedron truncated 4a from blue max.png Polyhedron truncated 4a from red max.png Polyhedron truncated 4a from yellow max.png
Wireframe Tetrahedron t01 ae.png Tetrahedron t01 af36.png 3-simplex t01.svg 3-simplex t01 A2.svg
Dual Dual tetrahedron t01 ae.png Dual tetrahedron t01 af36.png Dual tetrahedron t01.png Dual tetrahedron t01 A2.png
[1] [1] [4] [3]

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


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[edit]

Truncated tetrahedral graph
Tuncated tetrahedral graph.png
3-fold symmetry
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]

Truncated tetrahedral graph.circo.svg
Orthographic projections
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.svg
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.

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
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..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: t{n,3}
Spherical Euclid. Compact hyperb. Paraco.
Spherical triangular prism.png Uniform tiling 332-t01-1-.png Uniform tiling 432-t01.png Uniform tiling 532-t01.png Uniform tiling 63-t01.svg Truncated heptagonal tiling.svg H2-8-3-trunc-dual.svg H2 tiling 23i-3.png
Symbol t{2,3} t{3,3} t{4,3} t{5,3} t{6,3} t{7,3} t{8,3} t{∞,3}
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 Order-7 triakis triangular tiling.svg H2-8-3-kis-primal.svg 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]


  1. ^ Chisholm, Matt; Avnet, Jeremy (1997). "Truncated Trickery: Truncatering". Retrieved 2013-09-02.
  2. ^ a b Damasceno, Pablo F.; Engel, Michael; Glotzer, Sharon C. (2012). "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. S2CID 12785227.
  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. ^[bare URL 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. ^ "Expo 67 - Man the Producer - page 1".
  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
  • 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

External links[edit]