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
TypeNear-miss Johnson solid
Symmetrohedron
Faces74:
6 hexagons
12 pentagons
8+24+24 non-equilateral triangles
Edges132
Vertices60
Vertex configuration3.3.5.6
3.5.3.6
3.3.3.3.5
Symmetry groupTh, [3+,4], (3*2), order 24
Rotation groupT, [3,3]+, (332), order 12
Propertiesconvex
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

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References

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  1. ^ Near Misses based on dodecahedra
  2. ^ Kaplan, Craig S.; Hart, George W. (2001), "Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons", Bridges: Mathematical Connections in Art, Music and Science (PDF).
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