Deltoidal icositetrahedron(rotating and 3D model) TypeCatalanConway notationoC or deCCoxeter diagramFace polygonKite with 3 equal acute angles & 1 obtuse angleFaces24, congruentEdges24 short + 24 long = 48Vertices8 (connecting 3 short edges)+ 6 (connecting 4 long edges)+ 12 (connecting 4 alternate short & long edges)= 26Face configurationV3.4.4.4Symmetry groupOh, BC3, [4,3], *432Rotation groupO, [4,3]+, (432)Dihedral anglesame value for short & long edges:Dual polyhedronRhombicuboctahedronPropertiesconvex, face-transitiveNet
In geometry, the deltoidal icositetrahedron (or trapezoidal icositetrahedron, tetragonal icosikaitetrahedron, tetragonal trisoctahedron, strombic icositetrahedron) is a Catalan solid.
Description
A deltoidal icositetrahedron is a Catalan solid with 24 sides that are kites. All of its faces are congruent, each has three interior angles approximately 81.6 degrees and one angle 115.3 degrees. The dihedral angle between every two kites is 138.1 degrees. The deltoidal icositetrahedron has 48 edges, and 26 vertices – eight vertices surrounded by three kites and eighteen vertices by four kites. Its dual polyhedron is the rhombicuboctahedron, an Archimedean solid. Deltoidal icositetrahedron and deltoidal hexecontahedron are two Catalan solids with kite faces only.
Dimensions and angles Dimensions The deltoidal icositetrahedron with long body diagonal length D = 2 has: short body diagonal length:
long edge length:In this MathWorld entry, the small rhombicuboctahedron has edge length so this s.r.c.o.h. has circumradius and midradius so this s.r.c.o.h.'s dual with respect to their common midsphere is the deltoidal icositetrahedron with inradius ××
short edge length:
inradius:
is the distance from the center to any face plane; it may be calculated by normalizing the equation of plane above, replacing (x, y, z) with (0, 0, 0), and taking the absolute value of the result.
A deltoidal icositetrahedron has its long and short edges in the ratio:
The deltoidal icositetrahedron with short edge length has: area:
volume:
Side Lengths In a deltoidal icositetrahedron, each face is a kite-shaped quadrilateral. The side lengths of these kites can be expressed in the ratio 0.7731900694928638:1. Specifically, the side adjacent to the obtuse angle has a length of approximately 0.707106785, while the side adjacent to the acute angle has a length of approximately 0.914213565.
Occurrences in nature 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 the name trapezohedron has another meaning.
Related polyhedra The deltoidal icositetrahedron's projection onto a cube divides its squares into quadrants. The projection onto a regular octahedron divides its equilateral triangles into kite faces. In Conway polyhedron notation this represents an ortho operation to a cube or octahedron.
The deltoidal icositetrahedron (dual of the small rhombicuboctahedron) is tightly related to the disdyakis dodecahedron (dual of the great rhombicuboctahedron). The main difference is that the latter also has edges between the vertices on 3- and 4-fold symmetry axes (between yellow and red vertices in the images below).
Deltoidalicositetrahedron Disdyakisdodecahedron Dyakisdodecahedron Tetartoid
Dyakis dodecahedron
A variant with pyritohedral symmetry is called a dyakis dodecahedronIsohedron 24kThe Isometric Crystal System or diploid.The 48 Special Crystal Forms It is common in crystallography.A dyakis dodecahedron can be created by enlarging 24 of the 48 faces of a disdyakis dodecahedron. A tetartoid can be created by enlarging 12 of the 24 faces of a dyakis dodecahedron. Both is indicated in the two crystal models in the top right corner of this photo. A visual demonstration can be seen here and here.
Stellation The great triakis octahedron is a stellation of the deltoidal icositetrahedron.
See also Tetrakis hexahedron, another 24-face Catalan solid which looks a bit like an overinflated cube. "The Haunter of the Dark", a story by H.P. Lovecraft, whose plot involves this figure.
References
(The thirteen semiregular convex polyhedra and their duals, Page 23, Deltoidal icositetrahedron) The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal icosikaitetrahedron)
External links
Deltoidal (Trapezoidal) Icositetrahedron – Interactive Polyhedron model