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, ½H3, [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 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.

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
Family of uniform icosahedral polyhedra
Symmetry: [5,3], (*532) [5,3]+, (532)
Uniform polyhedron-53-t0.png Uniform polyhedron-53-t01.png Uniform polyhedron-53-t1.png Uniform polyhedron-53-t12.png Uniform polyhedron-53-t2.png Uniform polyhedron-53-t02.png Uniform polyhedron-53-t012.png Uniform polyhedron-53-s012.png
CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node 1.png CDel node 1.pngCDel 5.pngCDel node 1.pngCDel 3.pngCDel node 1.png CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
{5,3} t{5,3} r{5,3} t{3,5} {3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
Icosahedron.svg Triakisicosahedron.jpg Rhombictriacontahedron.svg Pentakisdodecahedron.jpg Dodecahedron.svg Deltoidalhexecontahedron.jpg Disdyakistriacontahedron.jpg Pentagonalhexecontahedronccw.jpg
V5.5.5 V3.10.10 V3.5.3.5 V5.6.6 V3. V3.4.5.4 V4.6.10 V3.

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.

n32 symmetry mutations of snub tilings:
Spherical Euclidean Compact hyperbolic Paracomp.
232 332 432 532 632 732 832 ∞32
Spherical trigonal antiprism.png Spherical snub tetrahedron.png Spherical snub cube.png Spherical snub dodecahedron.png Uniform tiling 63-snub.png Uniform tiling 73-snub.png Uniform tiling 83-snub.png Uniform tiling i32-snub.png
Uniform tiling 432-t0.png Uniform tiling 532-t0.png Spherical pentagonal icositetrahedron.png Spherical pentagonal hexecontahedron.png Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg Ord7 3 floret penta til.png Order-3-infinite floret pentagonal tiling.png
Config. V3. V3. V3. V3. V3. V3. V3. V3.3.3.3.∞


External links[edit]