Truncated tetrahedron

Truncated tetrahedron
(Click here for rotating model)
Type	Archimedean solid
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 or (*332)
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.

If the truncated tetrahedron has unit edge lengths, its dual triakis tetrahedron has edge lengths and 3.

Area and volume

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

Cartesian coordinates

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.

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)

Use in architecture

Giant truncated tetrahedrons 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 tetrahedrons 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

• octahedron
• tetrahedron

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.
• The Uniform Polyhedra
• Virtual Reality Polyhedra The Encyclopedia of Polyhedra