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Small stellated dodecahedron

 
Small Stellated Dodecahedron

 
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Small stellated dodecahedron
 
 
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In geometry
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. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic 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 isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
. It also shares the same edge arrangement as the great icosahedron
Great icosahedron

In geometry, the great icosahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 20 intersecting triangular faces, with five triangles meeting at each vertex in a pentagrammic sequence....
.

It is considered the first of three stellations of the dodecahedron
List of Wenninger polyhedron models

This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger.The book was written as a guide book to building polyhedra as physical models....
.

If the pentagram
Pentagram

A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek language word pe?t???a???? , a noun form of pe?t???a???? or pe?t???a???? , a word meaning roughly "five-lined" or "five lines"....
mic faces are considered as 5 triangular faces, it shares the same surface topology as the pentakis dodecahedron
Pentakis dodecahedron

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....
, but with much taller isosceles triangle faces.


A transparent model of the small stellated dodecahedron (See also Animated)
an also be constructed as the first of four stellation
Stellation

Stellation is a process of constructing new polygons , new polyhedron 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
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
, and referenced as Wenninger model [W20
List of Wenninger polyhedron models

This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger.The book was written as a guide book to building polyhedra as physical models....
].

The stellation facets for construction are:

l 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.






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In geometry
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. It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic 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 isany polyhedron having 20 faces, but usually a regular icosahedron is implied, which has equilateral triangle s as faces....
. It also shares the same edge arrangement as the great icosahedron
Great icosahedron

In geometry, the great icosahedron is a Kepler-Poinsot polyhedra. It is one of four nonconvex regular polyhedra. It is composed of 20 intersecting triangular faces, with five triangles meeting at each vertex in a pentagrammic sequence....
.

It is considered the first of three stellations of the dodecahedron
List of Wenninger polyhedron models

This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger.The book was written as a guide book to building polyhedra as physical models....
.

If the pentagram
Pentagram

A pentagram is the shape of a five-pointed star drawn with five straight strokes. The word pentagram comes from the Greek language word pe?t???a???? , a noun form of pe?t???a???? or pe?t???a???? , a word meaning roughly "five-lined" or "five lines"....
mic faces are considered as 5 triangular faces, it shares the same surface topology as the pentakis dodecahedron
Pentakis dodecahedron

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....
, but with much taller isosceles triangle faces.

Smallstellateddodecahedron

A transparent model of the small stellated dodecahedron (See also Animated)

As a stellation

It can also be constructed as the first of four stellation
Stellation

Stellation is a process of constructing new polygons , new polyhedron 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
Dodecahedron

A dodecahedron is any polyhedron with twelve faces, but usually a regular dodecahedron is meant: a Platonic solid composed of twelve regular pentagonal faces, with three meeting at each vertex....
, and referenced as Wenninger model [W20
List of Wenninger polyhedron models

This table contains an indexed list of the Uniform and stellated polyhedra from the book Polyhedron Models, by Magnus Wenninger.The book was written as a guide book to building polyhedra as physical models....
].

The stellation facets for construction are:
First Stellation of Dodecahedron Facets

Paper construction

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.

A 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....
 for creating a small stellated dodecahedron might look something like this:

In art

  • A small stellated dodecahedron is seen featured in Gravitation
    Gravitation (M. C. Escher)

    Gravitation is a mixed media work by the Netherlands 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....
     by M. C. Escher
    M. C. Escher

    Maurits Cornelis Escher , usually referred to as M.C. Escher , was a Netherlands Graphic arts. He is known for his often mathematically-inspired woodcuts, lithography, and mezzotints....
    .
  • It can also be seen in a mosaic
    Mosaic

    Mosaic is the art of creating images with an assemblage of small pieces of colored glass, stone, or other material. It may be a technique of Decorative arts, an aspect of interior decoration or of cultural and spiritual significance as in a cathedral....
     by Paolo Uccello
    Paolo Uccello

    Paolo Uccello was an Italy painter 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 study trying to grasp the exact vanishing point....
     circa 1430.


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