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 great pentagrammic hexecontahedron (or great 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

great retrosnub icosidodecahedron In geometry, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron is a nonconvex uniform polyhedron, indexed as . It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schlä ...

. Its 60 faces are irregular pentagrams. Great pentagrammic hexecontahedron

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

. Let

\xi\approx 0.946\,730\,033\,56

be the largest positive zero of the polynomial

P = 8x^3-8x^2+\phi^

. Then each pentagrammic face has four equal angles of

\arccos(\xi)\approx 18.785\,633\,958\,24^

and one angle of

\arccos(-\phi^+\phi^\xi)\approx 104.857\,464\,167\,03^

. Each face has three long and two short edges. The ratio

l

between the lengths of the long and the short edges is given by :

l = \frac\approx 1.774\,215\,864\,94

. 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 60.901\,133\,713\,21^

. Part of each face lies inside the solid, hence is invisible in solid models. The other two zeroes of the polynomial

P

play a similar role in the description of the great pentagonal hexecontahedron and the great inverted pentagonal hexecontahedron.

References

External links

* Dual uniform polyhedra {{polyhedron-stub