In
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 ...
, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron is a
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. Each polyhedron can contain either star polygon faces, star polygon vertex figures, ...
, indexed as . It has 92 faces (80
triangles
A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC.
In Euclidean geometry, any three points, when non-collinear ...
and 12
pentagrams), 150 edges, and 60 vertices.
It is given a
Schläfli symbol
Cartesian coordinates
Cartesian coordinates for the vertices of a great retrosnub icosidodecahedron are all the
even permutation
In mathematics, when ''X'' is a finite set with at least two elements, the permutations of ''X'' (i.e. the bijective functions from ''X'' to ''X'') fall into two classes of equal size: the even permutations and the odd permutations. If any total ...
s of
: (±2α, ±2, ±2β),
: (±(α−βτ−1/τ), ±(α/τ+β−τ), ±(−ατ−β/τ−1)),
: (±(ατ−β/τ+1), ±(−α−βτ+1/τ), ±(−α/τ+β+τ)),
: (±(ατ−β/τ−1), ±(α+βτ+1/τ), ±(−α/τ+β−τ)) and
: (±(α−βτ+1/τ), ±(−α/τ−β−τ), ±(−ατ−β/τ+1)),
with an even number of plus signs, where
: α = ξ−1/ξ
and
: β = −ξ/τ+1/τ
2−1/(ξτ),
where τ = (1+)/2 is the
golden mean and
ξ is the smaller positive real
root
In vascular plants, the roots are the organs of a plant that are modified to provide anchorage for the plant and take in water and nutrients into the plant body, which allows plants to grow taller and faster. They are most often below the su ...
of ξ
3−2ξ=−1/τ, namely
:
Taking the
odd permutations of the above coordinates with an odd number of plus signs gives another form, the
enantiomorph of the other one. Taking the odd permutations with an even number of plus signs or vice versa results in the same two figures rotated by 90 degrees.
The circumradius for unit edge length is
:
where
is the appropriate root of
. The four positive real roots of the
sextic in
:
are the circumradii of the
snub dodecahedron
In geometry, the snub dodecahedron, or snub icosidodecahedron, is an Archimedean solid, one of thirteen convex isogonal nonprismatic solids constructed by two or more types of regular polygon faces.
The snub dodecahedron has 92 faces (the most ...
(U
29),
great snub icosidodecahedron (U
57),
great inverted snub icosidodecahedron (U
69), and great retrosnub icosidodecahedron (U
74).
See also
*
List of uniform polyhedra
*
Great snub icosidodecahedron
*
Great inverted snub icosidodecahedron
References
External links
*
Uniform polyhedra
{{Polyhedron-stub