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 ditrigonal dodecacronic hexecontahedron (or great lanceal trisicosahedron) 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 ...

great ditrigonal dodecicosidodecahedron. Its faces are

kites A kite is a tethered heavier than air flight, heavier-than-air or lighter-than-air craft with wing surfaces that react against the air to create Lift (force), lift and Drag (physics), drag forces. A kite consists of wings, tethers and anchors. ...

. Part of each kite lies inside the solid, hence is invisible in solid models.

Proportions

Kite faces have two angles of

\arccos(\frac-\frac\sqrt)\approx 98.183\,872\,491\,81^

, one of

\arccos(-\frac+\frac\sqrt)\approx 112.296\,452\,073\,54^

and one of

\arccos(-\frac+\frac\sqrt)\approx 51.335\,802\,942\,83^

. 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 127.686\,523\,427\,48^

. The ratio between the lengths of the long edges and the short ones equals

\frac\approx 1.917\,288\,176\,70

References

* p. 62

External links

* Dual uniform polyhedra {{polyhedron-stub