Small stellated dodecahedron
Encyclopedia
In geometry
, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol {5/2,5}. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagram
mic faces, with five pentagrams meeting at each vertex.
It shares the same vertex arrangement
as the convex regular icosahedron
. It also shares the same edge arrangement as the great icosahedron.
It is considered the first of three stellations of the dodecahedron.
If the pentagram
mic faces are considered as 5 triangular faces, it shares the same surface topology as the pentakis dodecahedron
, but with much taller isosceles triangle faces.
. It also shares its edges with the great icosahedron.
This polyhedron is the truncation
of the great dodecahedron:
The truncated
small stellated dodecahedron looks like a dodecahedron on the surface, but it has 24 faces: 12 pentagons from the truncated vertices and 12 overlapping as (truncated pentagrams).
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, with Schläfli symbol {5/2,5}. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagram
Pentagram
A pentagram is the shape of a five-pointed star drawn with five straight strokes...
mic faces, with five pentagrams meeting at each vertex.
It shares the same vertex arrangement
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....
as the convex regular icosahedron
Icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
. It also shares the same edge arrangement as the great icosahedron.
It is considered the first of three stellations of the dodecahedron.
If the pentagram
Pentagram
A pentagram is the shape of a five-pointed star drawn with five straight strokes...
mic faces are considered as 5 triangular faces, it shares the same surface topology as the pentakis dodecahedron
Pentakis dodecahedron
In geometry, a pentakis dodecahedron is a Catalan solid. Its dual is the truncated icosahedron, an Archimedean solid.It can be seen as a dodecahedron with a pentagonal pyramid covering each face; that is, it is the Kleetope of the dodecahedron...
, but with much taller isosceles triangle faces.
Images
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| Spherical tiling | Stellation Stellation Stellation is a process of constructing new polygons , new polyhedra in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again... |
net Net (polyhedron) In geometry the net of a polyhedron is an arrangement of edge-joined polygons in the plane which can be folded to become the faces of the polyhedron... |
 This polyhedron also represents a spherical tiling with a density of 3. (One spherical pentagram face, outlined in blue, filled in yellow) |
 It can also be constructed as the first of three stellation Stellation Stellation is a process of constructing new polygons , new polyhedra in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again... s of the dodecahedron, and referenced as Wenninger model [W20]. |
 Small stellated dodecahedra can be constructed out of paper or cardstock by connecting together 12 five-sided isosceles pyramids in the same manner as the pentagons in a regular dodecahedron. With an opaque material, this visually represents the exterior portion of each pentagrammic face. |
In art
- It can also be seen in a mosaicMosaicMosaic is the art of creating images with an assemblage of small pieces of colored glass, stone, or other materials. It may be a technique of decorative art, an aspect of interior decoration, or of cultural and spiritual significance as in a cathedral...
by Paolo UccelloPaolo UccelloPaolo Uccello , born Paolo di Dono, was an Italian painter and a mathematician who was notable for his pioneering work on visual perspective in art. Giorgio Vasari in his book Lives of the Artists wrote that Uccello was obsessed by his interest in perspective and would stay up all night in his...
circa 1430. - It is central to two lithographs by M. C. EscherM. C. EscherMaurits Cornelis Escher , usually referred to as M. C. Escher , was a Dutch graphic artist. He is known for his often mathematically inspired woodcuts, lithographs, and mezzotints...
: Contrast (Order and Chaos) (1950) and GravitationGravitation (M. C. Escher)Gravitation is a mixed media work by the Dutch artist M. C. Escher which was completed in June, 1952. It was first printed as a black-and-white lithograph and then coloured by hand in watercolour....
(1952).
Floor mosaic Mosaic Mosaic is the art of creating images with an assemblage of small pieces of colored glass, stone, or other materials. It may be a technique of decorative art, an aspect of interior decoration, or of cultural and spiritual significance as in a cathedral... in St Mark's Basilica St Mark's Basilica The Patriarchal Cathedral Basilica of Saint Mark is the cathedral church of the Roman Catholic Archdiocese of Venice, northern Italy. It is the most famous of the city's churches and one of the best known examples of Byzantine architecture... , Venice Venice Venice is a city in northern Italy which is renowned for the beauty of its setting, its architecture and its artworks. It is the capital of the Veneto region... by Paolo Uccello Paolo Uccello Paolo Uccello , born Paolo di Dono, was an Italian painter and a mathematician who was notable for his pioneering work on visual perspective in art. Giorgio Vasari in his book Lives of the Artists wrote that Uccello was obsessed by his interest in perspective and would stay up all night in his... |
Related polyhedra
Its convex hull is the regular convex icosahedronIcosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
. It also shares its edges with the great icosahedron.
This polyhedron is the truncation
Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.- Uniform truncation :...
of the great dodecahedron:
The truncated
Truncation (geometry)
In geometry, a truncation is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex.- Uniform truncation :...
small stellated dodecahedron looks like a dodecahedron on the surface, but it has 24 faces: 12 pentagons from the truncated vertices and 12 overlapping as (truncated pentagrams).
| Name | Small stellated dodecahedron | Truncated small stellated dodecahedron | Dodecadodecahedron | Truncated great dodecahedron |
Great dodecahedron |
|---|---|---|---|---|---|
| Coxeter-Dynkin diagram Coxeter-Dynkin diagram In geometry, a Coxeter–Dynkin diagram is a graph with numerically labeled edges representing the spatial relations between a collection of mirrors... |
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