Rectified truncated tetrahedron

In geometry, the rectified truncated tetrahedron is a polyhedron, constructed as a rectified, truncated tetrahedron. It has 20 faces: 4 equilateral triangles, 12 isosceles triangles, and 4 regular hexagons.

Rectified truncated tetrahedron
Faces20:
4 equilateral triangles
12 isosceles triangles
4 hexagons
Edges48
Vertices12+18
Schläfli symbolrt{3,3}
Conway notationatT
Symmetry groupTd, [3,3], (*332), order 24
Rotation groupT, [3,3]+, (332), order 12
Dual polyhedronJoined truncated tetrahedron
Propertiesconvex
Net

Topologically, the triangles corresponding to the tetrahedron's vertices are always equilateral, although the hexagons, while having equal edge lengths, do not have the same edge lengths with the equilateral triangles, having different but alternating angles, causing the other triangles to be isosceles instead.

The rectified truncated tetrahedron can be seen in sequence of rectification and truncation operations from the tetrahedron. Further truncation, and alternation operations creates two more polyhedra:

Name Truncated
tetrahedron
Rectified
truncated
tetrahedron
Truncated
rectified
truncated
tetrahedron
Snub
rectified
truncated
tetrahedron
Coxeter tT rtT trtT srtT
Conway atT btT stT
Image
Conway dtT = kT jtT mtT gtT
Dual

See also

References

    • Coxeter Regular Polytopes, Third edition, (1973), Dover edition, ISBN 0-486-61480-8 (pp. 145–154 Chapter 8: Truncation)
    • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5
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