Pentagonal hexecontahedron

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Pentagonal hexecontahedron
(Click here for rotating model)
Type Catalan solid
Coxeter diagram CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Conway notation gD
Face type V3.
DU29 facets.png

irregular pentagon
Faces 60
Edges 150
Vertices 92
Vertices by type 12 {5}
20+60 {3}
Symmetry group I, 1/2H3, [5,3]+, (532)
Rotation group I, [5,3]+, (532)
Dihedral angle 153°10′43″
Properties convex, face-transitive chiral
Snub dodecahedron ccw.png
Snub dodecahedron
(dual polyhedron)
Pentagonal hexecontahedron Net

In geometry, a pentagonal hexecontahedron is a Catalan solid, dual of the snub dodecahedron. It has two distinct forms, which are mirror images (or "enantiomorphs") of each other. It is also well-known to be the Catalan solid with the most vertices. Among the Catalan and Archimedean solids, it has the second largest number of vertices, after the truncated icosidodecahedron, which has 120 vertices.


The pentagonal hexecontahedron can be constructed from a snub dodecahedron without taking the dual. Pentagonal pyramids are added to the 12 pentagonal faces of the snub dodecahedron, and triangular pyramids are added to the 20 triangular faces that do not share an edge with a pentagon. The pyramid heights are adjusted to make them coplanar with the other 60 triangular faces of the snub dodecahedron. The result is the pentagonal hexecontahedron.[1]


The faces are irregular pentagons with two long edges and three short edges. The ratio of edge lengths is about 1:1.7489525667362. The faces have four obtuse angles of about 118.13662 degrees each, and one acute angle (between the two long edges) of about 67.45351 degrees. The dihedral angle is about 153.178732558 degrees between all faces.

Note that the face centers of the snub dodecahedron cannot serve directly as vertices of the pentagonal hexecontahedron: the four triangle centers lie in one plane but the pentagon center does not; it needs to be radially pushed out to make it coplanar with the triangle centers. Consequently, the vertices of the pentagonal hexecontahedron do not all lie on the same sphere and by definition it is not a zonohedron.


Isohedral variations can be constructed with pentagonal faces with 3 edge lengths.

This variation shown can be constructed by adding pyramids to 12 pentagonal faces and 20 triangular faces of a snub dodecahedron such that the new triangular faces are coparallel to other triangles and can be merged into the pentagon faces.

Pentagonal hexecontahedron variation0.png
Snub dodecahedron with augmented pyramids and merged faces
Pentagonal hexecontahedron variation.png
Example variation
Pentagonal hexecontahedron variation net.png

Orthogonal projections[edit]

The pentagonal hexecontahedron has three symmetry positions, two on vertices, and one mid-edge.

Orthogonal projections
[3] [5]+ [2]
Image Dual snub dodecahedron A2.png Dual snub dodecahedron H2.png Dual snub dodecahedron e1.png
Snub dodecahedron A2.png Snub dodecahedron H2.png Snub dodecahedron e1.png

Related polyhedra and tilings[edit]

Spherical pentagonal hexecontahedron

This polyhedron is topologically related as a part of sequence of polyhedra and tilings of pentagons with face configurations (V3.3.3.3.n). (The sequence progresses into tilings the hyperbolic plane to any n.) These face-transitive figures have (n32) rotational symmetry.

See also[edit]


External links[edit]