Deltoidal icositetrahedron

Deltoidal icositetrahedron
Deltoidal icositetrahedron
(rotating and 3D model)
Type Catalan
Conway notation oC or deC
Coxeter diagram CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png
Face polygon DU10 facets.png
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
Deltoidal icositetrahedron
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.

D. i. as artwork and die
D. i. projected onto cube and octahedron in Perspectiva Corporum Regularium
Dyakis dodecahedron crystal model and projection onto an octahedron

Cartesian coordinatesEdit

Cartesian coordinates for a suitably sized deltoidal icositetrahedron centered at the origin are:

  • (±1, 0, 0), (0, ±1, 0), (0, 0, ±1)
  • (0, ±1/22, ±1/22), (±1/22, 0, ±1/22), (±1/22, ±1/22, 0)
  • (±(22+1)/7, ±(22+1)/7, ±(22+1)/7)

The long edges of this deltoidal icosahedron have length (2-2) ≈ 0.765367.

DimensionsEdit

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  .

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:

Orthogonal projections
Projective
symmetry
[2] [4] [6]
Image      
Dual
image
     

Related polyhedraEdit

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 4-fold symmetry axes (between yellow and red vertices in the images below).

       
Deltoidal
icositetrahedron
Disdyakis
dodecahedron
Dyakis
dodecahedron
Tetartoid

Dyakis dodecahedronEdit

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]

StellationEdit

The great triakis octahedron is a stellation of the deltoidal icositetrahedron.

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 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{31,1}
t{3,4}
t{31,1}
{3,4}
{31,1}
rr{4,3}
s2{3,4}
tr{4,3} sr{4,3} h{4,3}
{3,3}
h2{4,3}
t{3,3}
s{3,4}
s{31,1}
                                                     
     
=    
     
=    
     
=    
            =
    or    
      =
    or    
      =
   
     
 
 
 
 
 
 
 
           
 
Duals to uniform polyhedra
V43 V3.82 V(3.4)2 V4.62 V34 V3.43 V4.6.8 V34.4 V33 V3.62 V35
                                                                 
                                         
                     

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.

*n42 symmetry mutation of dual expanded tilings: V3.4.n.4
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

ReferencesEdit

  1. ^ Conway, Symmetries of Things, p.284–286
  2. ^ https://etc.usf.edu/clipart/keyword/forms
  3. ^ "Kite". Retrieved 6 October 2019.
  4. ^ Isohedron 24k
  5. ^ The Isometric Crystal System
  6. ^ The 48 Special Crystal Forms
  7. ^ Both is indicated in the two crystal models in the top right corner of this photo. A visual demonstration can be seen here and here.

External linksEdit