Pentagonal hexecontahedron
The Pentagonhexakontaeder is a convex polyhedron composed of 60 pentagons and one of the the Catalan bodies. It is dual to the chamfered dodecahedron and has 92 vertices and 150 edges.
The following pictures show two mirror -image Pentagonhexakontaeder.
Mirror variant 2
Formation
By connecting the midpoints of five edges that collide in each corner of the room of the chamfered dodecahedron produces a Sehnenfünfeck whose radius of inscribed circle Tangentenfünfecks, the boundary surface of the Pentagonhexakontaeders, is at the same time. In this particular type, all dihedral angles equal ( ≈ 153 ° ), and there is a uniform sphere radius edges.
Hereinafter the term designates the cosine of the smaller central angle in the aforementioned Sehnenfünfeck; is the golden number.
Is the only real solution of the cubic equation.
Is the edge length of the chamfered dodecahedron, the resulting lengths of the sides are represented by Tangentenfünfecks
Thus, the two side lengths are in the following ratio to each other:
Related polyhedra
Inscribed dodecahedron
Inscribed icosahedron
Formulas
For the polyhedron
For the boundary surfaces
Application
In the U.S., a process has been patented in which 92 of the total of 332 wells ( " dimples " ) of a golf ball lying on the grid points of Pentagonhexakontaeders.