Truncated tetrahedron

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}
Schläfli symbol	t{3,3}
Wythoff symbol	2 3 | 3
Coxeter–Dynkin
Symmetry	Td , [3,3], (*332)
Dihedral Angle
References	U02, C16, W6
Properties	Semiregular convex
Colored faces	3.6.6 (Vertex figure)
Triakis tetrahedron (dual polyhedron)	Net

In geometry, the truncated tetrahedron is an Archimedean solid. It has 4 regular hexagonal faces, 4 regular triangular faces, 12 vertices and 18 edges.

Area and volume

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

$A = 7\sqrt{3}a^2 \approx 12.12435565a^2$

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

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)

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 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.

Orthogonal projection

Orthogonal projection

Centered by	Edge normal	Face normal	Edge	Face/vertex
Image
Projective symmetry	[1]	[2]	[3]	[4]

Related polyhedra

Name	tetrahedron	rectified tetrahedron (octahedron)	truncated tetrahedron	cantellated tetrahedron (cuboctahedron)	omnitruncated tetrahedron (truncated octahedron)	Snub tetrahedron (icosahedron)
Schläfli	{3,3}	t1{3,3}	t0,1{3,3}	t0,2{3,3}	t0,1,2{3,3}	s{3,3}
Coxeter-Dynkin
Graph (A3)
Graph (A2)
Solid
Tiling

Use in architecture

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.

See also

• Truncated 5-cell – Similar uniform polytope in 4-dimensions

References

• 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)

External links

• Eric W. Weisstein, Truncated tetrahedron (Archimedean solid) at MathWorld.
• Richard Klitzing, 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