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In
The
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 c ...
, the snub dodecadodecahedron 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 84 faces (60 triangle
A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), 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, an ...
s, 12 pentagon
In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°.
A pentagon may be simpl ...
s, and 12 pentagram
A pentagram (sometimes known as a pentalpha, pentangle, or star pentagon) is a regular five-pointed star polygon, formed from the diagonal line segments of a convex (or simple, or non-self-intersecting) regular pentagon. Drawing a circle aroun ...
s), 150 edges, and 60 vertices. It is given a Schläfli symbol
In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations.
The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...
as a 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" ...
great dodecahedron
In geometry, the great dodecahedron is a Kepler–Poinsot polyhedron, with Schläfli symbol and Coxeter–Dynkin diagram of . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagonal faces (six pairs of parallel pentagon ...
.
Cartesian coordinates
Cartesian coordinates
A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...
for the vertices of a snub dodecadodecahedron 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 or ...
s of
: (±2α, ±2, ±2β),
: (±(α+β/τ+τ), ±(-ατ+β+1/τ), ±(α/τ+βτ-1)),
: (±(-α/τ+βτ+1), ±(-α+β/τ-τ), ±(ατ+β-1/τ)),
: (±(-α/τ+βτ-1), ±(α-β/τ-τ), ±(ατ+β+1/τ)) and
: (±(α+β/τ-τ), ±(ατ-β+1/τ), ±(α/τ+βτ+1)),
with an even number of plus signs, where
: β = (α2/τ+τ)/(ατ−1/τ),
where τ = (1+)/2 is the golden mean and
α is the 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 sur ...
of τα4−α3+2α2−α−1/τ, or approximately 0.7964421.
Taking the odd 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 or ...
s 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.
Related polyhedra
Medial pentagonal hexecontahedron
medial pentagonal hexecontahedron
In geometry, the medial pentagonal hexecontahedron is a nonconvex isohedral polyhedron. It is the dual of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.
Proportions
Denote the golden ratio by \phi, and let \xi\a ...
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 th ...
. It is the dual
Dual or Duals may refer to:
Paired/two things
* Dual (mathematics), a notion of paired concepts that mirror one another
** Dual (category theory), a formalization of mathematical duality
*** see more cases in :Duality theories
* Dual (grammatical ...
of the snub dodecadodecahedron. It has 60 intersecting irregular pentagonal faces.
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 ...
* Inverted snub dodecadodecahedron
In geometry, the inverted snub dodecadodecahedron (or vertisnub dodecadodecahedron) is a nonconvex uniform polyhedron, indexed as U60. It is given a Schläfli symbol sr.
Cartesian coordinates
Cartesian coordinates for the vertices of an invert ...
References
*External links
* * Uniform polyhedra {{Polyhedron-stub