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 ...

, the medial hexagonal hexecontahedron (or midly dentoid ditriacontahedron) 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 ...

. It is the dual of the

uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, ...

snub icosidodecadodecahedron In geometry, the snub icosidodecadodecahedron is a nonconvex uniform polyhedron, indexed as U46. It has 104 faces (80 triangles, 12 pentagons, and 12 pentagrams), 180 edges, and 60 vertices. As the name indicates, it belongs to the family of sn ...

Proportions

The faces of the medial hexagonal hexecontahedron are irregular nonconvex hexagons. 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 -0.377\,438\,833\,12

be the real zero of the polynomial

8x^3-4x^2+1

. The number

\xi

can be written as

\xi=-1/(2\rho)

, where

\rho

is the

plastic number In mathematics, the plastic number (also known as the plastic constant, the plastic ratio, the minimal Pisot number, the platin number, Siegel's number or, in French, ) is a mathematical constant which is the unique real solution of the cubic ...

. Then each face has four equal angles of

\arccos(\xi)\approx 112.175\,128\,045\,27^

, one of

\arccos(\phi^2\xi+\phi)\approx 50.958\,265\,917\,31^

and one of

360^-\arccos(\phi^\xi-\phi^)\approx 220.341\,221\,901\,59^

. Each face has two long edges, two of medium length and two short ones. If the medium edges have length

2

, the long ones have length

1+\sqrt\approx 4.121\,448\,816\,41

and the short ones

1-\sqrt\approx 0.453\,587\,559\,98

. 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 un ...

equals

\arccos(\xi/(\xi+1))\approx 127.320\,132\,197\,62^

References

External links

* Dual uniform polyhedra {{polyhedron-stub