Small triambic icosahedron
Encyclopedia
| Small triambic icosahedron | |||||||
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| Type | Dual uniform polyhedron Uniform polyhedron A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive... |
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| Index | DU30, 2/59, W26 | ||||||
| Elements Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent... (As a star polyhedron) |
F = 20, E = 60 V = 32 (χ = −8) |
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| Symmetry group | icosahedral Icosahedral symmetry A regular icosahedron has 60 rotational symmetries, and a symmetry order of 120 including transformations that combine a reflection and a rotation... (Ih) |
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| Dual polyhedron Dual polyhedron In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another... |
small ditrigonal icosidodecahedron Small ditrigonal icosidodecahedron In geometry, the small ditrigonal icosidodecahedron is a nonconvex uniform polyhedron, indexed as U30.-Related polyhedra:Its convex hull is 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 triambic icosahedron is the dual
Dual polyhedron
In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another...
to the uniform
Uniform polyhedron
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...
small ditrigonal icosidodecahedron
Small ditrigonal icosidodecahedron
In geometry, the small ditrigonal icosidodecahedron is a nonconvex uniform polyhedron, indexed as U30.-Related polyhedra:Its convex hull is a regular dodecahedron...
. It is composed of 20 intersecting isogonal hexagon faces
Face (geometry)
In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
. It has 60 edges
Edge (geometry)
In geometry, an edge is a one-dimensional line segment joining two adjacent zero-dimensional vertices in a polygon. Thus applied, an edge is a connector for a one-dimensional line segment and two zero-dimensional objects....
and 32 vertices
Vertex (geometry)
In geometry, a vertex is a special kind of point that describes the corners or intersections of geometric shapes.-Of an angle:...
, and Euler characteristic
Euler characteristic
In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic is a topological invariant, a number that describes a topological space's shape or structure regardless of the way it is bent...
of −8.
If the intersected hexagonal faces are divided and new edges created, this figure becomes the triakis icosahedron
Triakis icosahedron
In geometry, the triakis icosahedron is an Archimedean dual solid, or a Catalan solid. Its dual is the truncated dodecahedron.It can be seen as an icosahedron with triangular pyramids augmented to each face; that is, it is the Kleetope of the icosahedron...
. The descriptive name triakis icosahedron represents a topological construction starting with an 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....
and adding short triangular pyramids
Pyramid (geometry)
In geometry, a pyramid is a polyhedron formed by connecting a polygonal base and a point, called the apex. Each base edge and apex form a triangle. It is a conic solid with polygonal base....
. With the proper pyramid height this figure becomes a Catalan solid
Catalan solid
In mathematics, a Catalan solid, or Archimedean dual, is a dual polyhedron to an Archimedean solid. The Catalan solids are named for the Belgian mathematician, Eugène Catalan, who first described them in 1865....
by the same name and the dual of the truncated dodecahedron
Truncated dodecahedron
In geometry, the truncated dodecahedron is an Archimedean solid. It has 12 regular decagonal faces, 20 regular triangular faces, 60 vertices and 90 edges.- Geometric relations :...
.
The nonconvex uniform polyhedra great stellated dodecahedron and great dodecahedron, as viewed as surface topologies, can also be constructed as icosahedron with pyramids, the first with much tall pyramids, and the second with inverted ones.
It is also a uniform dual
Uniform polyhedron
A uniform polyhedron is a polyhedron which has regular polygons as faces and is vertex-transitive...
, and is the dual of the small ditrigonal icosidodecahedron
Small ditrigonal icosidodecahedron
In geometry, the small ditrigonal icosidodecahedron is a nonconvex uniform polyhedron, indexed as U30.-Related polyhedra:Its convex hull is a regular dodecahedron...
. Other uniform duals which are also stellations of the icosahedron are the medial triambic icosahedron and the great triambic icosahedron
Great triambic icosahedron
In geometry, the great triambic icosahedron and medial triambic icosahedron are visually identical dual uniform polyhedra. The exterior surface also represents the De1f1 stellation of the icosahedron. The only way to differentiate these two polyhedra is to mark which intersections between edges are...
.
As a stellation
This figure is also the first stellationStellation
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...
of the 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....
, and given as Wenninger model index 26.
This stellation is a popular subject for construction in modular origami
Modular origami
Modular origami, or unit origami, is a paperfolding technique which uses multiple sheets of paper to create a larger and more complex structure than would be possible using single-piece origami techniques...
, often made of thirty Sonobe
Sonobe
The Sonobe module is a unit used to build modular origami, created by Mitsunobu Sonobe.Each individual unit is folded from a square sheet of paper, of which only one face is visible in the finished module; many ornamented variants of the plain Sonobe unit that expose both sides of the paper have...
units.