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
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 ...
.
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
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
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 ...
, 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:
See also
*
Compound of two tetrahedra
In geometry, a compound of two tetrahedra is constructed by two overlapping tetrahedra, usually implied as regular tetrahedra.
Stellated octahedron
There is only one uniform polyhedral compound, the stellated octahedron, which has octahedral s ...
*
Compound of cube and octahedron
The compound of cube and octahedron is a polyhedron which can be seen as either a polyhedral stellation or a compound. Construction
The 14 Cartesian coordinates of the vertices of the compound are.
: 6: (±2, 0, 0), ( 0, ±2, 0), ( 0, 0, ±2)
: ...
*
Compound of dodecahedron and icosahedron
In geometry, this polyhedron can be seen as either a polyhedral stellation or a compound.
As a compound
It can be seen as the compound of an icosahedron and dodecahedron. It is one of four compounds constructed from a Platonic solid or Keple ...
*
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
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