Deltoidal icositetrahedron
Deltoidal icositetrahedron  

(Click here for rotating 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.
Contents
DimensionsEdit
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
Occurrences in nature and cultureEdit
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 projectionsEdit
The deltoidal icositetrahedron has three symmetry positions, all centered on vertices:
Projective symmetry 
[2]  [4]  [6] 

Image  
Dual image 
Related polyhedraEdit
The great triakis octahedron is a stellation of the deltoidal icositetrahedron.
Dyakis dodecahedronEdit
The deltoidal icositetrahedron is topologically equivalent to a cube whose faces are divided in quadrants. It can also be projected onto a regular octahedron, with kite faces, or more general quadrilaterals with pyritohedral symmetry. In Conway polyhedron notation, they represent an ortho operation to a cube or octahedron.
In crystallography a rotational variation is called a dyakis dodecahedron^{[3]}^{[4]} or diploid.^{[5]}
Octahedral, O_{h}, order 24  Pyritohedral, T_{h}, order 12  

Related polyhedra and tilingsEdit
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.
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 
See alsoEdit
 Deltoidal hexecontahedron
 Tetrakis hexahedron, another 24face 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
 Pseudodeltoidal icositetrahedron
ReferencesEdit
 ^ Conway, Symmetries of Things, p.284–286
 ^ https://etc.usf.edu/clipart/keyword/forms
 ^ Isohedron 24k
 ^ The Isometric Crystal System
 ^ The 48 Special Crystal Forms
 Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 048623729X. (Section 39)
 Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 9780521543255, MR 0730208 (The thirteen semiregular convex polyhedra and their duals, Page 23, Deltoidal icositetrahedron)
 The Symmetries of Things 2008, John H. Conway, Heidi Burgiel, Chaim GoodmanStrass, ISBN 9781568812205 [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal icosikaitetrahedron)
External linksEdit
 Eric W. Weisstein, Deltoidal icositetrahedron (Catalan solid) at MathWorld.
 Deltoidal (Trapezoidal) Icositetrahedron – Interactive Polyhedron model