There are two different compounds of great icosahedron and great stellated dodecahedron: one is a

dual compound In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram. The outer vertices of a compound can be connected ...

and a

stellation In geometry, stellation is the process of extending a polygon in two dimensions, polyhedron in three dimensions, or, in general, a polytope in ''n'' dimensions to form a new figure. Starting with an original figure, the process extends specific el ...

of the

great icosidodecahedron In geometry, the great icosidodecahedron is a nonconvex uniform polyhedron, indexed as U54. It has 32 faces (20 triangles and 12 pentagrams), 60 edges, and 30 vertices. It is given a Schläfli symbol r. It is the rectification of the great stell ...

, the other is a stellation of the

icosidodecahedron In geometry, an icosidodecahedron is a polyhedron with twenty (''icosi'') triangular faces and twelve (''dodeca'') pentagonal faces. An icosidodecahedron has 30 identical vertices, with two triangles and two pentagons meeting at each, and 60 id ...

Dual compound

It can be seen as a

polyhedron compound In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram. The outer vertices of a compound can be connected ...

of a

great icosahedron In geometry, the great icosahedron is one of four Kepler–Poinsot polyhedra (nonconvex regular polyhedra), with Schläfli symbol and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeti ...

and

great stellated dodecahedron In geometry, the great stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 intersecting pentagrammic faces, with three pentagrams meeting at each ve ...

. It is one of five compounds constructed from a

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...

or Kepler-Poinsot solid, and its dual. It is a

of the

. It has

icosahedral symmetry In mathematics, and especially in geometry, an object has icosahedral symmetry if it has the same symmetries as a regular icosahedron. Examples of other polyhedra with icosahedral symmetry include the regular dodecahedron (the dual of the ...

(I_''h'') and it has the same

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. For example, a ''square vertex arrangement'' is understood to mean four points in a plane, equ ...

as a

great rhombic triacontahedron In geometry, the great rhombic triacontahedron is a nonconvex isohedral, isotoxal polyhedron. It is the dual of the great icosidodecahedron (U54). Like the convex rhombic triacontahedron it has 30 rhombic faces, 60 edges and 32 vertices (also ...

. This can be seen as one of the two three-dimensional equivalents of the compound of two pentagrams ( " decagram"); this series continues into the fourth dimension as compounds of star 4-polytopes.

Stellation of the icosidodecahedron

This

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 ...

is a

of the

, and given as Wenninger model index 61. It has the same vertex arrangement as a

rhombic triacontahedron In geometry, the rhombic triacontahedron, sometimes simply called the triacontahedron as it is the most common thirty-faced polyhedron, is a convex polyhedron with 30 rhombic faces. It has 60 edges and 32 vertices of two types. It is a Cata ...

, its convex hull. The stellation facets for construction are:

References

* , p. 90. * , pp. 51-53. *

Martyn Cundy Henry Martyn Cundy (23 December 1913 – 25 February 2005) was a mathematics teacher and professor in United Kingdom, Britain and Malawi as well as a singer, musician and poet. He was one of the founders of the School Mathematics Project to ref ...

and A. Rollett. "Great Icosahedron Plus Great Stellated Dodecahedron". §3.10.4 in ''

Mathematical Models A mathematical model is a description of a system using mathematical concepts and language. The process of developing a mathematical model is termed mathematical modeling. Mathematical models are used in the natural sciences (such as physics, b ...

'', 3rd ed. Stradbroke, England: Tarquin Pub., pp. 132-133, 1989.

External links

*
Great Stellated Dodecahedron plus Great Icosahedron - metallic paper model
*

VRML VRML (Virtual Reality Modeling Language, pronounced ''vermal'' or by its initials, originally—before 1995—known as the Virtual Reality Markup Language) is a standard file format for representing 3-dimensional (3D) interactive vector graphi ...

br>George Hart: VRML transparent model
Polyhedral stellation Polyhedral compounds {{polyhedron-stub

Dual compound

Stellation of the icosidodecahedron

See also

References

External links