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 stellapentakis dodecahedron (or great astropentakis dodecahedron) 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 of the

truncated great icosahedron In geometry, the truncated great icosahedron (or great truncated icosahedron) is a nonconvex uniform polyhedron, indexed as U55. It has 32 faces (12 pentagrams and 20 hexagons), 90 edges, and 60 vertices. It is given a Schläfli symbol t or t0,1 ...

. It has 60 intersecting triangular faces.

Proportions

The triangles have one angle of

\arccos(-\frac-\frac\sqrt)\approx 138.891\,114\,686\,59^

and two of

\arccos(\frac+\frac\sqrt)\approx 20.554\,442\,656\,71^

. 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 123.320\,065\,258\,47^

. Part of each triangle lies within the solid, hence is invisible in solid models.

References

External links

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