Deltoidal icositetrahedron  

(rotating and 3D model)  
Type  Catalan 
Conway notation  oC or deC 
Coxeter diagram  
Face polygon  Kite with 3 equal acute angles & 1 obtuse angle 
Faces  24, congruent 
Edges  24 short + 24 long = 48 
Vertices  8 (connecting 3 short edges) + 6 (connecting 4 long edges) + 12 (connecting 4 alternate short & long edges) = 26 
Face configuration  V3.4.4.4 
Symmetry group  O_{h}, BC_{3}, [4,3], *432 
Rotation group  O, [4,3]^{+}, (432) 
Dihedral angle  same value for short & long edges: 
Dual polyhedron  Rhombicuboctahedron 
Properties  convex, facetransitive 
Net 
In geometry, the deltoidal icositetrahedron (or trapezoidal icositetrahedron, tetragonal icosikaitetrahedron,^{[1]} tetragonal trisoctahedron,^{[2]} strombic icositetrahedron) is a Catalan solid. Its 24 faces are congruent kites.^{[3]} The deltoidal icositetrahedron, whose dual is the (uniform) rhombicuboctahedron, is tightly related to the pseudodeltoidal icositetrahedron, whose dual is the pseudorhombicuboctahedron; but the actual and pseudod.i. are not to be confused with each other.
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:
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 regularoctagon equators, lying in three orthogonal planes.
The deltoidal icositetrahedron with long body diagonal length D = 2 has:
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:
For a deltoidal icositetrahedron, each kite face has:
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.
In Guardians of The Galaxy Vol. 3, the device containing the files about the experiments carried on Rocket Raccoon has the shape of a deitoidal icositetrahedron.
The deltoidal icositetrahedron has three symmetry positions, all centered on vertices:
Projective symmetry 
[2]  [4]  [6] 

Image  
Dual image 
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 4fold symmetry axes (between yellow and red vertices in the images below).
Deltoidal icositetrahedron 
Disdyakis dodecahedron 
Dyakis dodecahedron 
Tetartoid 
A variant with pyritohedral symmetry is called a dyakis dodecahedron^{[5]}^{[6]} or diploid.^{[7]} 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.
^{[8]}
The great triakis octahedron is a stellation of the deltoidal icositetrahedron.
The deltoidal icositetrahedron is a member 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 a cube and regular octahedron arranged in their dual positions. It can also be seen that the 3 and 4fold corners can be made to have the same distance to the center. In that case the resulting icositetrahedron will no longer have a rhombicuboctahedron for a dual, since the centers of the square and triangle faces of a rhombicuboctahedron are at different distances from its center.
Uniform octahedral polyhedra  

Symmetry: [4,3], (*432)  [4,3]^{+} (432) 
[1^{+},4,3] = [3,3] (*332) 
[3^{+},4] (3*2)  
{4,3}  t{4,3}  r{4,3} r{3^{1,1}} 
t{3,4} t{3^{1,1}} 
{3,4} {3^{1,1}} 
rr{4,3} s_{2}{3,4} 
tr{4,3}  sr{4,3}  h{4,3} {3,3} 
h_{2}{4,3} t{3,3} 
s{3,4} s{3^{1,1}} 
= 
= 
= 
= or 
= or 
=  




 
Duals to uniform polyhedra  
V4^{3}  V3.8^{2}  V(3.4)^{2}  V4.6^{2}  V3^{4}  V3.4^{3}  V4.6.8  V3^{4}.4  V3^{3}  V3.6^{2}  V3^{5} 
This polyhedron is a term of a sequence of topologically related deltoidal polyhedra with face configuration V3.4.n.4; this sequence continues with tilings of the Euclidean and hyperbolic planes. These facetransitive figures have (*n32) reflectional symmetry.
Symmetry *n32 [n,3] 
Spherical  Euclid.  Compact hyperb.  Paraco.  

*232 [2,3] 
*332 [3,3] 
*432 [4,3] 
*532 [5,3] 
*632 [6,3] 
*732 [7,3] 
*832 [8,3]... 
*∞32 [∞,3]  
Figure Config. 
V3.4.2.4 
V3.4.3.4 
V3.4.4.4 V3.4.5.4 
V3.4.6.4 
V3.4.7.4 
V3.4.8.4 
V3.4.∞.4 