Pentahexagonal pyritoheptacontatetrahedron

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

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]

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.

Dual Polyhedron

The dual of the pentahexagonal pyritoheptacontatetrahedron is made of kites, quadrilaterals, and pentagonal faces. It has 60 faces.

Image Animation

See also

References

  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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