In geometry

Geometry

Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....

, the great dodecicosahedron is a nonconvex uniform polyhedron

Nonconvex uniform polyhedron

In geometry, a uniform star polyhedron is a self-intersecting uniform polyhedron. They are also sometimes called nonconvex polyhedra to imply self-intersecting...

, indexed as U₆₃. Its vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

 is a crossed quadrilateral.

It has a composite Wythoff symbol

Wythoff symbol

In geometry, the Wythoff symbol was first used by Coxeter, Longeut-Higgens and Miller in their enumeration of the uniform polyhedra. It represents a construction by way of Wythoff's construction applied to Schwarz triangles....

, 3 5/3 (3/2 5/2) |, requiring two different Schwarz triangle

Schwarz triangle

In geometry, a Schwarz triangle, named after Hermann Schwarz is a spherical triangle that can be used to tile a sphere, possibly overlapping, through reflections in its edges. They were classified in ....

s to generate it: (3 5/3 3/2) and (3 5/3 5/2).

Its vertex figure

Vertex figure

In geometry a vertex figure is, broadly speaking, the figure exposed when a corner of a polyhedron or polytope is sliced off.-Definitions - theme and variations:...

 6.10/3.6/5.10/7 is also ambiguous, having two clockwise and two counterclockwise faces around each vertex.

Related polyhedra

It shares its vertex arrangement

Vertex arrangement

In geometry, a vertex arrangement is a set of points in space described by their relative positions. They can be described by their use in polytopes....

 with the truncated dodecahedron

Truncated dodecahedron

In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.- Geometric relations :...

. It additionally shares its edge arrangement with the great icosicosidodecahedron

Great icosicosidodecahedron

In geometry, the great icosicosidodecahedron is a nonconvex uniform polyhedron, indexed as U48. Its vertex figure is a crossed quadrilateral.- Related polyhedra :It shares its vertex arrangement with the truncated dodecahedron...

 (having the hexagonal faces in common) and the great ditrigonal dodecicosidodecahedron

Great ditrigonal dodecicosidodecahedron

In geometry, the great ditrigonal dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U42.- Related polyhedra :It shares its vertex arrangement with the truncated dodecahedron...

 (having the decagrammic faces in common).

Truncated dodecahedron Truncated dodecahedron In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.- Geometric relations :...

Great icosicosidodecahedron Great icosicosidodecahedron In geometry, the great icosicosidodecahedron is a nonconvex uniform polyhedron, indexed as U48. Its vertex figure is a crossed quadrilateral.- Related polyhedra :It shares its vertex arrangement with the truncated dodecahedron...

Great ditrigonal dodecicosidodecahedron Great ditrigonal dodecicosidodecahedron In geometry, the great ditrigonal dodecicosidodecahedron is a nonconvex uniform polyhedron, indexed as U42.- Related polyhedra :It shares its vertex arrangement with the truncated dodecahedron...

Great dodecicosahedron