Pentahexagonal pyritoheptacontatetrahedron

In geometry, a pentahexagonal pyritoheptacontatetrahedron is a near-miss Johnson solid with pyritohedral symmetry. This near-miss was discovered by Mason Green in 2006. It has 6 hexagonal faces, 12 pentagonal faces, and 56 triangles in 3 symmetry positions. Mason calls it a hexagonally expanded snubbed dodecahedron.^[1]

Pentahexagonal pyritoheptacontatetrahedron
Type	Near-miss Johnson solid Symmetrohedron
Faces	74: 6 hexagons 12 pentagons 8+24+24 non-equilateral triangles
Edges	132
Vertices	60
Vertex configuration	3.3.5.6 3.5.3.6 3.3.3.3.5
Symmetry group	Th, [3+,4], (3*2), order 24
Rotation group	T, [3,3]+, (332), order 12
Properties	convex
Net

Model built with polydron

With regular hexagons and pentagons it is a symmetrohedron.^[2] The triangles are not equilateral, with triangle-triangle edges compressed by 1.8%.

It has 3 vertex configurations, 3.3.5.6, 3.5.3.6, 3.3.3.3.5, with the last shared in the snub dodecahedron.

See also

• Tetrated dodecahedron has tetrahedral symmetry