Compound of ten truncated tetrahedra
Type	Uniform compound
Index	UC56
Polyhedra	10 truncated tetrahedra
Faces	40 triangles, 40 hexagons
Edges	180
Vertices	120
Symmetry group	icosahedral (Ih)
Subgroup restricting to one constituent	chiral tetrahedral (T)

This uniform polyhedron compound is a composition of 10 truncated tetrahedra, formed by truncating each of the tetrahedra in the compound of 10 tetrahedra. It also results from composing the two enantiomers of the compound of 5 truncated tetrahedra. It could also be called a truncated icosicosahedron.

Cartesian coordinates

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

(±1, ±1, ±3)

(±τ⁻¹, ±(−τ⁻²), ±2τ)

(±τ, ±(−2τ⁻¹), ±τ²)

(±τ², ±(−τ⁻²), ±2)

(±(2τ−1), ±1, ±(2τ − 1))

where τ = (1+√5)/2 is the golden ratio (sometimes written φ).

References

• Skilling, John (1976), "Uniform Compounds of Uniform Polyhedra", Mathematical Proceedings of the Cambridge Philosophical Society , 79 (3): 447–457, Bibcode:1976MPCPS..79..447S, doi:10.1017/S0305004100052440, MR   0397554, S2CID   123279687 .
• McCooey, Robert. "Uniform Polyhedron Compounds". Hedron Dude. Retrieved 24 June 2025.

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