Deltoidal icositetrahedron

Deltoidal icositetrahedron Type Catalan
Conway notation oC or deC
Coxeter diagram     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°07′05″
arccos(−7 + 42/17)
Dual polyhedron rhombicuboctahedron
Properties convex, face-transitive Net

In geometry, a deltoidal icositetrahedron (also a trapezoidal icositetrahedron, tetragonal icosikaitetrahedron,, tetragonal trisoctahedron and strombic icositetrahedron) is a Catalan solid. Its dual polyhedron is the rhombicuboctahedron.

Dimensions

The 24 faces are kites. The short and long edges of each kite are in the ratio 1:(2 − 1/2) ≈ 1:1.292893...

If its smallest edges have length a, its surface area and volume are

{\begin{aligned}A&=6{\sqrt {29-2{\sqrt {2}}}}\,a^{2}\\V&={\sqrt {122+71{\sqrt {2}}}}\,a^{3}\end{aligned}}

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.

Orthogonal projections

The deltoidal icositetrahedron has three symmetry positions, all centered on vertices:

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.

When projected onto a sphere (see right), it can be seen that the edges make up the edges of an octahedron and cube arranged in their dual positions. It can also be seen that the threefold corners and the fourfold corners can be made to have the same distance to the center. In that case the resulting icositetrahedron will no longer have a rombicubeoctahedron for a dual, since for the rombicubeoctahedron the centers of its squares and its triangles are at different distances from the center.

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.