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 snub 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 U
57. 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 can be represented by a
Schläfli symbol sr, and
Coxeter-Dynkin diagram .
This polyhedron is the
snub
A snub, cut or slight is a refusal to recognise an acquaintance by ignoring them, avoiding them or pretending not to know them. For example, a failure to greet someone may be considered a snub.
In Awards and Lists
For awards, the term "snub" ...
member of a family that includes the
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 ...
, the
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 ...
and 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 ...
.
In the book ''
Polyhedron Models'' by
Magnus Wenninger
Father Magnus J. Wenninger OSB (October 31, 1919Banchoff (2002)– February 17, 2017) was an American mathematician who worked on constructing polyhedron models, and wrote the first book on their construction.
Early life and education
Born to Ge ...
, the polyhedron is misnamed ''
great inverted snub icosidodecahedron
In geometry, the great inverted snub icosidodecahedron (or great vertisnub icosidodecahedron) is a uniform star polyhedron, indexed as U69. It is given a Schläfli symbol sr, and Coxeter-Dynkin diagram . In the book '' Polyhedron Models'' by M ...
'', and vice versa.
Cartesian coordinates
Cartesian coordinates for the vertices of a great snub 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 negative 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, or approximately −1.5488772.
Taking the
odd permutations of the above coordinates with an odd number of plus signs gives another form, the
enantiomorph
In geometry, a figure is chiral (and said to have chirality) if it is not identical to its mirror image, or, more precisely, if it cannot be mapped to its mirror image by rotations and translations alone. An object that is not chiral is said to ...
of the other one.
The circumradius for unit edge length is
:
where
is the appropriate root of
. The four positive real roots of the
sextic
In algebra, a sextic (or hexic) polynomial is a polynomial of degree six.
A sextic equation is a polynomial equation of degree six—that is, an equation whose left hand side is a sextic polynomial and whose right hand side is zero. More precis ...
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
In geometry, the great inverted snub icosidodecahedron (or great vertisnub icosidodecahedron) is a uniform star polyhedron, indexed as U69. It is given a Schläfli symbol sr, and Coxeter-Dynkin diagram . In the book '' Polyhedron Models'' by M ...
(U
69), and
great retrosnub icosidodecahedron
In geometry, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron is a nonconvex uniform polyhedron, indexed as . It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schlä ...
(U
74).
Related polyhedra
Great pentagonal hexecontahedron
The great pentagonal hexecontahedron (or great petaloid ditriacontahedron) 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 ...
and
dual to the uniform ''great snub icosidodecahedron''. It has 60 intersecting irregular pentagonal faces, 120 edges, and 92 vertices.
Proportions
Denote the
golden ratio
In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities. Expressed algebraically, for quantities a and b with a > b > 0,
where the Greek letter phi ( ...
by
. Let
be the negative zero of the polynomial
. Then each pentagonal face has four equal angles of
and one angle of
. Each face has three long and two short edges. The ratio
between the lengths of the long and the short edges is given by
:
.
The
dihedral angle
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 un ...
equals
. Part of each face lies inside the solid, hence is invisible in solid models. The other two zeroes of the polynomial
play a similar role in the description of the
great inverted pentagonal hexecontahedron and the
great pentagrammic hexecontahedron
In geometry, the great pentagrammic hexecontahedron (or great dentoid ditriacontahedron) is a nonconvex isohedral polyhedron. It is the dual of the great retrosnub icosidodecahedron. Its 60 faces are irregular pentagrams.
Proportions
Denote th ...
.
See also
*
List of uniform polyhedra
In geometry, a uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive ( transitive on its vertices, isogonal, i.e. there is an isometry mapping any vertex onto any other). It follows that all vertices are c ...
*
Great inverted snub icosidodecahedron
In geometry, the great inverted snub icosidodecahedron (or great vertisnub icosidodecahedron) is a uniform star polyhedron, indexed as U69. It is given a Schläfli symbol sr, and Coxeter-Dynkin diagram . In the book '' Polyhedron Models'' by M ...
*
Great retrosnub icosidodecahedron
In geometry, the great retrosnub icosidodecahedron or great inverted retrosnub icosidodecahedron is a nonconvex uniform polyhedron, indexed as . It has 92 faces (80 triangles and 12 pentagrams), 150 edges, and 60 vertices. It is given a Schlä ...
References
*
External links
*
*
{{Nonconvex polyhedron navigator
Uniform polyhedra