In geometry, the deltoidal icositetrahedron (or trapezoidal icositetrahedron, tetragonal icosikaitetrahedron, tetragonal trisoctahedron, strombic icositetrahedron) is a Catalan solid. Its 24 faces are congruent kites. The deltoidal icositetrahedron, whose dual is the (uniform) rhombicuboctahedron, is tightly related to the pseudo-deltoidal icositetrahedron, whose dual is the pseudorhombicuboctahedron; but the actual and pseudo-d.i. are not to be confused with each other. Cartesian coordinates In the image above, the long body diagonals are those between opposite red vertices and between opposite blue vertices, and the short body diagonals are those between opposite yellow vertices.Cartesian coordinates for the vertices of the deltoidal icositetrahedron centered at the origin and with long body diagonal length 2 are: red vertices (lying in -fold symmetry axes): blue vertices (lying in -fold symmetry axes): yellow vertices (lying in -fold symmetry axes): For example, the point with coordinates is the intersection of the plane with equation and of the line with system of equations A deltoidal icositetrahedron has three regular-octagon equators, lying in three orthogonal planes. Dimensions and angles Dimensions The deltoidal icositetrahedron with long body diagonal length D = 2 has: short body diagonal length: long edge length: 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: Angles For a deltoidal icositetrahedron, each kite face has: three equal acute angles, with value: one obtuse angle (between the short edges), with value: 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 the name trapezohedron has another meaning. Orthogonal projections The deltoidal icositetrahedron has three symmetry positions, all centered on vertices: 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