Truncated tetrahedron

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

${\displaystyle {\begin{aligned}A&=7{\sqrt {3}}a^{2}&&\approx 12.124\,355\,65a^{2}\\V&={\tfrac {23}{12}}{\sqrt {2}}a^{3}&&\approx 2.710\,575\,995a^{3}.\end{aligned}}}$

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 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 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:

(0,0,1,2)

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

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
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
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

Family of uniform tetrahedral polyhedra
Symmetry: [3,3], (*332)							[3,3]+, (332)
{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
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

	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.

*n32 symmetry mutation of truncated spherical tilings: t{n,3}
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
Symbol	t{2,3}	t{3,3}	t{4,3}	t{5,3}	t{6,3}	t{7,3}	t{8,3}	t{∞,3}
Triakis figures
Config.	V3.4.4	V3.6.6	V3.8.8	V3.10.10	V3.12.12	V3.14.14	V3.16.16	V3.∞.∞

• 
  drawing in De divina proportione (1509)
• 
  drawing in Perspectiva Corporum Regularium (1568)
• 
  crystal model
• 
  photos from different perspectives (Matemateca)
• 
  4-sided die
• 
  12 permutations of ${\displaystyle (4,2,0,0)}$ (brown)

• 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

• 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

Wikimedia Commons has media related to Truncated tetrahedron.

• Weisstein, Eric W., "Truncated tetrahedron" ("Archimedean solid") at MathWorld.
  • Weisstein, Eric W. "Truncated tetrahedral graph". MathWorld.
• Klitzing, Richard. "3D convex uniform polyhedra x3x3o - tut".
• Editable printable net of a truncated tetrahedron with interactive 3D view
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra