Abstract simplicial complex
| Deltoidal icositetrahedron | |
|---|---|
| Deltoidal icositetrahedron Click on picture for large version. Click here for spinning version. | |
| Type | Catalan |
| Coxeter diagram | Template:CDD |
| Face polygon | kite |
| Faces | 24 |
| Edges | 48 |
| Vertices | 26 = 6 + 8 + 12 |
| Face configuration | V3.4.4.4 |
| Symmetry group | Oh, BC3, [4,3], *432 |
| Rotation group | O, [4,3]+, (432) |
| Dihedral angle | 138° 7' 5" |
| Dual polyhedron | rhombicuboctahedron |
| Properties | convex, face-transitive |
| Deltoidal icositetrahedron Net | |
In geometry, a deltoidal icositetrahedron (also a trapezoidal icositetrahedron and tetragonal icosikaitetrahedron) is a Catalan solid which looks a bit like an overinflated cube. Its dual polyhedron is the rhombicuboctahedron.
Dimensions
The 24 faces are deltoids or kites, 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 .
Occurrences in nature and culture
The deltoidal icositetrahedron is a crystal habit often formed by the mineral analcime and occasionally garnet. The shape is often called a trapezohedron in mineral contexts, although in solid geometry that name has another meaning.
Related polyhedra
The deltoidal icositetrahedron is topologically equivalent to a cube whose faces are divided in quadrants.
The great triakis octahedron is a stellation of the deltoidal icositetrahedron.
Related polyhedra and tilings
The deltoidal icositetrahedron is one of a family of duals to the uniform polyhedra related to the cube and regular octahedron.
Template:Octahedral truncations
This polyhedron is topologically related as a part of sequence of deltoidal polyhedra with face figure (V3.4.n.4), and continues as tilings of the hyperbolic plane. These face-transitive figures have (*n32) reflectional symmetry.
See also
- Deltoidal hexecontahedron
- "The Haunter of the Dark", a story by H.P. Lovecraft, whose plot involves this figure
References
- Template:The Geometrical Foundation of Natural Structure (book) (Section 3-9)
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External links
- Template:Mathworld2
- Deltoidal (Trapezoidal) Icositetrahedron – Interactive Polyhedron model