Deltoidal icositetrahedron  

(rotating and 3D model)  
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  O_{h}, BC_{3}, [4,3], *432 
Rotation group  O, [4,3]^{+}, (432) 
Dihedral angle  138°07′05″ arccos(−7 + 4√2/17) 
Dual polyhedron  rhombicuboctahedron 
Properties  convex, facetransitive 
Net 
In geometry, a deltoidal icositetrahedron (also a trapezoidal icositetrahedron, tetragonal icosikaitetrahedron,^{[1]} tetragonal trisoctahedron^{[2]} and strombic icositetrahedron) is a Catalan solid. Its dual polyhedron is the rhombicuboctahedron.
Cartesian coordinates for a suitably sized deltoidal icositetrahedron centered at the origin are:
The long edges of this deltoidal icosahedron have length √(2√2) ≈ 0.765367.
The 24 faces are kites.^{[3]} 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
The kites have three equal acute angles with value and one obtuse angle (between the short edges) with value .
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.
The deltoidal icositetrahedron has three symmetry positions, all centered on vertices:
Projective symmetry 
[2]  [4]  [6] 

Image  
Dual image 
The solid's projection onto a cube divides its squares into quadrants. The projection onto an octahedron divides its triangles into kite faces. In Conway polyhedron notation this represents an ortho operation to a cube or octahedron.
The solid (dual of the small rhombicuboctahedron) is similar 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^{[4]}^{[5]} or diploid.^{[6]} It is common in crystallography.
It can be created by enlarging 24 of the 48 faces of the disdyakis dodecahedron. The tetartoid can be created by enlarging 12 of its 24 faces. ^{[7]}
The great triakis octahedron is a stellation of the deltoidal icositetrahedron.
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 rhombicuboctahedron for a dual, since for the rhombicuboctahedron the centers of its squares and its triangles are at different distances from the 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 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 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 