Compound of five truncated tetrahedra

Compound of five truncated tetrahedra
Type Uniform compound
Index UC55
Polyhedra 5 truncated tetrahedra
Faces 20 triangles, 20 hexagons
Edges 90
Vertices 60
Dual Compound of five triakis tetrahedra
Symmetry group chiral icosahedral (I)
Subgroup restricting to one constituent chiral tetrahedral (T)

The compound of five truncated tetrahedra is a uniform polyhedron compound. It's composed of 5 truncated tetrahedra rotated around a common axis. It may be formed by truncating each of the tetrahedra in the compound of five tetrahedra. A far-enough truncation creates the compound of five octahedra. Its convex hull is a nonuniform snub dodecahedron.

3D model of a compound of five truncated tetrahedra

Cartesian coordinates

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Cartesian coordinates for the vertices of this compound are all the cyclic permutations of

(±1, ±1, ±3)
(±τ−1, ±(−τ−2), ±2τ)
(±τ, ±(−2τ−1), ±τ2)
(±τ2, ±(−τ−2), ±2)
(±(2τ−1), ±1, ±(2τ − 1))

with an even number of minuses in the choices for '±', where τ = (1+5)/2 is the golden ratio (sometimes written φ).

References

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  • Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society, 79: 447–457, doi:10.1017/S0305004100052440, MR 0397554.