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 great deltoidal hexecontahedron (or great sagittal 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 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

nonconvex great rhombicosidodecahedron In geometry, the nonconvex great rhombicosidodecahedron is a nonconvex uniform polyhedron, indexed as U67. It has 62 faces (20 triangles, 30 squares and 12 pentagrams), 120 edges, and 60 vertices. It is also called the quasirhombicosidodecahedron ...

. It is visually identical to the great rhombidodecacron. It has 60 intersecting cross quadrilateral faces, 120 edges, and 62 vertices. Its faces are darts. Part of each dart lies inside the solid, hence is invisible in solid models. It is also called a ''great strombic hexecontahedron.''

Proportions

The darts have two angles of

\arccos(\frac+\frac\sqrt)\approx 18.699\,407\,085\,15^

, one of

\arccos(-\frac+\frac\sqrt)\approx 91.512\,394\,720\,74^

and one of

360^-\arccos(-\frac-\frac\sqrt)\approx 231.088\,791\,108\,96^

. 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(\frac)\approx 91.553\,403\,672\,16^

. The ratio between the lengths of the long and short edges is

\frac\approx 1.259\,463\,815\,11

References

External links

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