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.

Construction

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.

Geometry

The faces are irregular pentagons with two long edges and three short edges. Let $\xi\approx 0.943\,151\,259\,24$ be the real zero of the polynomial $x^3+2x^2-\phi^2$, where $\phi=\frac$ is the golden ratio.
Then the ratio $l$ of the edge lengths is given by:
:$l=\frac\approx 1.749\,852\,566\,74$.
The faces have four equal obtuse angles and one acute angle (between the two long edges). The obtuse angles equal $\arccos(-\xi/2)\approx 118.136\,622\,758\,62^$, and the acute one equals $\arccos(-\phi^2\xi/2+\phi)\approx 67.453\,508\,965\,51^$. The dihedral angle equals $\arccos(-\xi/(2-\xi))\approx 153.178\,732\,558\,45^$.
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 ''t''$t=\frac\approx0.471\,575\,629\,622$ .

Then the surface area (A) is:

$A=\frac\approx162.698\,964\,198b^2$.

And the volume (V) is:

$V=\frac\approx189.789\,852\,067b^3$.

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

:$\Psi = \frac \approx 0.98163$

Variations

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.

Orthogonal projections

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

Related polyhedra and tilings

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

*Truncated pentagonal hexecontahedron
* Amazon Spheres

References

* (Section 3-9)
* (The thirteen semiregular convex polyhedra and their duals, Page 29, Pentagonal hexecontahedron)
*''The Symmetries of Things'' 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss,

(Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 287, pentagonal hexecontahedron )

External links

*
Pentagonal Hexecontrahedron
– Interactive Polyhedron Model

Catalan solids
Chiral polyhedra

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