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 small ditrigonal dodecacronic hexecontahedron (or fat star) 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

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, se ...

small ditrigonal dodecicosidodecahedron. It is visually identical to the small dodecicosacron. Its faces are darts. A part of each dart lies inside the solid, hence is invisible in solid models.

Proportions

Faces have two angles of

\arccos(\frac+\frac\sqrt)\approx 12.661\,078\,804\,43^

, one of

\arccos(-\frac-\frac\sqrt)\approx 116.996\,396\,851\,70^

and one of

360^-\arccos(-\frac-\frac\sqrt)\approx 217.681\,445\,539\,45^

. Its

dihedral angles 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 ...

equal

\arccos()\approx 146.230\,659\,755\,53^

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

\frac\approx 1.110\,008\,944\,41

References

External links

* Dual uniform polyhedra {{polyhedron-stub