geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the medial pentagonal hexecontahedron is a nonconvex

isohedral In geometry, a tessellation of dimension (a plane tiling) or higher, or a polytope of dimension (a polyhedron) or higher, is isohedral or face-transitive if all its faces are the same. More specifically, all faces must be not merely congruent ...

polyhedron In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

. It is the

dual Dual or Duals may refer to: Paired/two things * Dual (mathematics), a notion of paired concepts that mirror one another ** Dual (category theory), a formalization of mathematical duality *** see more cases in :Duality theories * Dual (grammatical ...

of the

snub dodecadodecahedron In geometry, the snub dodecadodecahedron is a nonconvex uniform polyhedron, indexed as . It has 84 faces (60 triangles, 12 pentagons, and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schläfli symbol as a snub great dodecahedron. ...

. It has 60 intersecting irregular pentagonal faces.

Proportions

Denote the

golden ratio In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0, where the Greek letter phi ( ...

\phi

, and let

\xi\approx -0.409\,037\,788\,014\,42

be the smallest (most negative) real zero of the polynomial

P=8x^4-12x^3+5x+1

. Then each face has three equal angles of

\arccos(\xi)\approx 114.144\,404\,470\,43^

, one of

\arccos(\phi^2\xi+\phi)\approx 56.827\,663\,280\,94^

and one of

\arccos(\phi^\xi-\phi^)\approx 140.739\,123\,307\,76^

. Each face has one medium length edge, two short and two long ones. If the medium length is

2

, then the short edges have length :

1+\sqrt\approx 1.550\,761\,427\,20

, and the long edges have length :

1+\sqrt\approx 3.854\,145\,870\,08

. The

dihedral angle A dihedral angle is the angle between two intersecting planes or half-planes. In chemistry, it is the clockwise angle between half-planes through two sets of three atoms, having two atoms in common. In solid geometry, it is defined as the uni ...

equals

\arccos(\xi/(\xi+1))\approx 133.800\,984\,233\,53^

. The other real zero of the polynomial

P

plays a similar role for the

medial inverted pentagonal hexecontahedron In geometry, the inverted snub dodecadodecahedron (or vertisnub dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U60. It is given a Schläfli symbol sr. Cartesian coordinates Cartesian coordinates for the vertices of an invert ...

References

External links

*
Uniform polyhedra and duals
Dual uniform polyhedra {{polyhedron-stub