Pentahexagonal pyritoheptacontatetrahedron

{{short description|Near-miss Johnson solid with 74 faces}}

{{Infobox polyhedron

| image = Pyritohedral_near-miss_johnson.png

| type = Near-miss Johnson solid
Symmetrohedron

| euler =

| faces = 74:
6 hexagons
12 pentagons
8+24+24 non-equilateral triangles

| edges = 132

| vertices = 60

| vertex_config = {{math|3.3.5.6
3.5.3.6
3.3.3.3.5}}

| schläfli =

| wythoff =

| conway =

| coxeter =

| symmetry = {{math|T_h, [3⁺,4], (3*2),}} order 24

| rotation_group = {{math|T, [3,3]⁺, (332),}} order 12

| surface_area =

| volume =

| angle =

| dual =

| properties = convex

| vertex_figure =

| net = Pyritohedral_near-miss_johnson-net.png

}}

File:Pyritohedral_near-miss_johnson-polydron.jpg]]

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.[http://www.orchidpalms.com/polyhedra/acrohedra/nearmiss/Tetrated%20Dodecahedra.html Near Misses based on dodecahedra]

With regular hexagons and pentagons it is a symmetrohedron.{{citation|contribution=Symmetrohedra: Polyhedra from Symmetric Placement of Regular Polygons|title=Bridges: Mathematical Connections in Art, Music and Science|year=2001|url=https://archive.bridgesmathart.org/2001/bridges2001-21.pdf|first1=Craig S.|last1=Kaplan|first2=George W.|last2=Hart|author2-link=George W. Hart}}. 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.