Pentagonal hexecontahedron

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 has 92 vertices that span 60 pentagonal faces. It is 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.

Properties[edit]

3D model of a pentagonal hexecontahedron

The faces are irregular pentagons with two long edges and three short edges. Let 

ξ≈0.94315125924

 be the real zero of the polynomial 

x3+2x2−ϕ2

. Then the ratio 

l

 of the edge lengths is given by:

l=1+ξ2−ξ2≈1.74985256674.

The faces have four equal obtuse angles and one acute angle (between the two long edges). The obtuse angles equal 

arccos⁡(−ξ/2)≈118.13662275862∘

, and the acute one equals 

arccos⁡(−ϕ2ξ/2+ϕ)≈67.45350896551∘

. The dihedral angle equals 

arccos⁡(−ξ/(2−ξ))≈153.2∘

.

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.

To find the volume and surface area of a pentagonal hexecontahedron, denote the shorter side of one of the pentagonal faces as 

b

, and set a constant[1]

t=44+12ϕ(9+81ϕ−15)3+44+12ϕ(9−81ϕ−15)3−412≈0.472.

Then the surface area (

A

) is:

A=30b2⋅(2+3t)⋅1−t21−2t2≈162.698b2.

And the volume (

V

) is:

V=5b3(1+t)(2+3t)(1−2t2)⋅1−2t≈189.789b3.

Using these, one can calculate the measure of sphericity for this shape:

Ψ=π13(6V)23A≈0.982

Construction[edit]

Combining a unit circumradius icosahedron (12) centered at the origin with a chiral snub dodecahedron (60) combined with a dodecahedron of the same non-unity circumradius (20) to construct the pentagonal hexecontahedron

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.[2]

https://en.wikipedia.org/wiki/Pentagonal_hexecontahedron