Deltoidal icositetrahedron
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Type	Catalan 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....
Face polygon	kite Kite (geometry) In Euclidean geometry a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are next to each other. In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite each other rather than next to each other...
Faces	24
Edges	48
Vertices	26 = 6 + 8 + 12
Face configuration Face configuration In geometry, a face configuration is notational description of a face-transitive polyhedron. It represents a sequential count of the number of faces that exist at each vertex around a face....	V3.4.4.4
Symmetry group	Oh, [4,3], *432
Dihedral angle Dihedral angle In geometry, a dihedral or torsion angle is the angle between two planes.The dihedral angle of two planes can be seen by looking at the planes "edge on", i.e., along their line of intersection...	138° 6' 34"
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...	rhombicuboctahedron Rhombicuboctahedron In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each. Note that six of the squares only share vertices with the triangles...
Properties	convex, face-transitive
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...

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 ....

, a deltoidal icositetrahedron (also a trapezoidal icositetrahedron and tetragonal icosikaitetrahedron) is 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....

 which looks a bit like an overinflated cube. Its 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...

 is the rhombicuboctahedron

Rhombicuboctahedron

In geometry, the rhombicuboctahedron, or small rhombicuboctahedron, is an Archimedean solid with eight triangular and eighteen square faces. There are 24 identical vertices, with one triangle and three squares meeting at each. Note that six of the squares only share vertices with the triangles...

.

The 24 faces are deltoids or kites

Kite (geometry)

In Euclidean geometry a kite is a quadrilateral whose four sides can be grouped into two pairs of equal-length sides that are next to each other. In contrast, a parallelogram also has two pairs of equal-length sides, but they are opposite each other rather than next to each other...

, also called trapezia in the US and trapezoids in Britain. The short and long edges of each kite are in the ratio 1:1.292893...

If its smallest edges have length 1, its surface area is and its volume is .

Related polyhedra

The deltoidal icositetrahedron is topologically equivalent to a cube

Cube

In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...

 whose faces are divided in quadrants.

The great triakis octahedron

Great triakis octahedron

In geometry, the great triakis octahedron is the dual of the stellated truncated hexahedron . It has 24 intersecting isosceles triangle faces....

 is a stellation of the deltoidal icositetrahedron.

Occurrences in nature and culture

The deltoidal icositetrahedron is a crystal habit

Crystal habit

Crystal habit is an overall description of the visible external shape of a mineral. This description can apply to an individual crystal or an assembly of crystals or aggregates....

 often formed by the mineral analcime and occasionally garnet

Garnet

The garnet group includes a group of minerals that have been used since the Bronze Age as gemstones and abrasives. The name "garnet" may come from either the Middle English word gernet meaning 'dark red', or the Latin granatus , possibly a reference to the Punica granatum , a plant with red seeds...

. The shape is often called a trapezohedron in mineral contexts, although in solid geometry

Solid geometry

In mathematics, solid geometry was the traditional name for the geometry of three-dimensional Euclidean space — for practical purposes the kind of space we live in. It was developed following the development of plane geometry...

 that name has another meaning

Trapezohedron

The n-gonal trapezohedron, antidipyramid or deltohedron is the dual polyhedron of an n-gonal antiprism. Its 2n faces are congruent kites . The faces are symmetrically staggered.The n-gon part of the name does not reference the faces here but arrangement of vertices around an axis of symmetry...

.

The Shining Trapezohedron of the fictional Lovecraft Mythos was probably intended to refer to a crystal

Crystal

A crystal or crystalline solid is a solid material whose constituent atoms, molecules, or ions are arranged in an orderly repeating pattern extending in all three spatial dimensions. The scientific study of crystals and crystal formation is known as crystallography...

 of this shape.

See also

• Deltoidal hexecontahedron
  Deltoidal hexecontahedron
  In geometry, a deltoidal hexecontahedron is a catalan solid which looks a bit like an overinflated dodecahedron. It is sometimes also called the trapezoidal hexecontahedron or strombic hexecontahedron...
  
  
• "The Haunter of the Dark
  The Haunter of the Dark
  "The Haunter of the Dark" is a horror story in the Cthulhu Mythos genre. It was written by H. P. Lovecraft in November 1935, and published in the December 1936 edition of Weird Tales...
  
  ", a story by H.P. Lovecraft, whose plot involves this figure

External links

• Deltoidal (Trapezoidal) Icositetrahedron -- Interactive Polyhedron model