Pentagonal hexecontahedron

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Pentagonal hexecontahedron
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 V3.3.3.3.5
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
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.

Orthogonal projections[edit]

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

Orthogonal projections
Projective
symmetry
[3] [5]+ [2]
Image Dual snub dodecahedron A2.png Dual snub dodecahedron H2.png Dual snub dodecahedron e1.png
Dual
image
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)
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
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
{5,3} t{5,3} r{5,3} 2t{5,3}=t{3,5} 2r{5,3}={3,5} rr{5,3} tr{5,3} sr{5,3}
Duals to uniform polyhedra
CDel node f1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node.png CDel node.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node f1.png CDel node f1.pngCDel 5.pngCDel node f1.pngCDel 3.pngCDel node f1.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 3.pngCDel node fh.png
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 V3.3.3.3.3 V3.4.5.4 V4.6.10 V3.3.3.3.5

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.

Dimensional family of snub polyhedra and tilings: 3.3.3.3.n
Symmetry
n32
[n,3]+
Spherical Euclidean Compact hyperbolic Paracompact
232
[2,3]+
D3
332
[3,3]+
T
432
[4,3]+
O
532
[5,3]+
I
632
[6,3]+
P6
732
[7,3]+
832
[8,3]+...
∞32
[∞,3]+
Snub
figure
Spherical trigonal antiprism.png
3.3.3.3.2
Spherical snub tetrahedron.png
3.3.3.3.3
Spherical snub cube.png
3.3.3.3.4
Spherical snub dodecahedron.png
3.3.3.3.5
Uniform tiling 63-snub.png
3.3.3.3.6
Uniform tiling 73-snub.png
3.3.3.3.7
Uniform tiling 83-snub.png
3.3.3.3.8
Uniform tiling i32-snub.png
3.3.3.3.∞
Coxeter
Schläfli
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{2,3}
CDel node h.pngCDel 3.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{3,3}
CDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{4,3}
CDel node h.pngCDel 5.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{5,3}
CDel node h.pngCDel 6.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{6,3}
CDel node h.pngCDel 7.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{7,3}
CDel node h.pngCDel 8.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{8,3}
CDel node h.pngCDel infin.pngCDel node h.pngCDel 3.pngCDel node h.png
sr{∞,3}
Snub
dual
figure
Uniform tiling 432-t0.png
V3.3.3.3.2
Uniform tiling 532-t0.png
V3.3.3.3.3
Spherical pentagonal icositetrahedron.png
V3.3.3.3.4
Spherical pentagonal hexecontahedron.png
V3.3.3.3.5
Tiling Dual Semiregular V3-3-3-3-6 Floret Pentagonal.svg
V3.3.3.3.6
Ord7 3 floret penta til.png
V3.3.3.3.7
V3.3.3.3.8 Order-3-infinite floret pentagonal tiling.png
V3.3.3.3.∞
Coxeter CDel node fh.pngCDel 2x.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 5.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 6.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 7.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel 8.pngCDel node fh.pngCDel 3.pngCDel node fh.png CDel node fh.pngCDel infin.pngCDel node fh.pngCDel 3.pngCDel node fh.png

References[edit]

External links[edit]