Deltoidal icositetrahedron

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Deltoidal icositetrahedron
Deltoidalicositetrahedron.jpg
(rotating and 3D model)
Type Catalan
Conway notation oC or deC
Coxeter diagramCDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.png
Face polygon DU10 facets.png
Kite with 3 equal acute angles & 1 obtuse angle
Faces24, congruent
Edges24 short + 24 long = 48
Vertices8 (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 Oh, BC3, [4,3], *432
Rotation group O, [4,3]+, (432)
Dihedral angle same value for short & long edges:

Dual polyhedron Rhombicuboctahedron
Propertiesconvex, face-transitive
Deltoidalicositetrahedron net.png
Net
Eeuwig water, Weiwerd, crop.jpg
D24 deltoidal icositetrahedron.JPG
D.i. as artwork and die
Perspectiva Corporum Regularium 36b.jpg
Perspectiva Corporum Regularium 34b.jpg
D.i. projected onto cube and octahedron in Perspectiva Corporum Regularium
Modell, Kristallform Disdodekaeders (Dyakisdodekaeder) -Krantz 374-.jpg
Deltoidal icositetrahedron octahedral gyro.png
Dyakis dodecahedron crystal model and projection onto octahedron

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 pseudo-deltoidal icositetrahedron, whose dual is the pseudorhombicuboctahedron; but the actual and pseudo-d.i. are not to be confused with each other.

Contents

Cartesian coordinates

Polyhedron small rhombi 6-8 dual max.png

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 regular-octagon equators, lying in three orthogonal planes.

Dimensions and angles

Dimensions

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:

Angles

For a deltoidal icositetrahedron, each kite face has:

Side Lengths

In a deltoidal icositetrahedron, each face is a kite-shaped quadrilateral. The side lengths of these kites can be expressed in the ratio 0.7731900694928638:1 Specifically, the side adjacent to the obtuse angle has a length of approximately 0.707106785, while the side adjacent to the acute angle has a length of approximately 0.914213565.

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.

In Guardians of The Galaxy Vol. 3, the device containing the files about the experiments carried on Rocket Raccoon has the shape of a deltoidal icositetrahedron.

Orthogonal projections

The deltoidal icositetrahedron has three symmetry positions, all centered on vertices:

Orthogonal projections
Projective
symmetry
[2][4][6]
Image Dual cube t02 f4b.png Dual cube t02 B2.png Dual cube t02.png
Dual
image
Cube t02 f4b.png 3-cube t02 B2.svg 3-cube t02.svg

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

Disdyakis 12 in deltoidal 24.png Disdyakis 12.png Disdyakis 12 untruncated to dyakis 12 horizontal with traces.png Tetartoid dark horizontal (with traces of dyakis 12).png
Deltoidal
icositetrahedron
Disdyakis
dodecahedron
Dyakis
dodecahedron
Tetartoid

Dyakis dodecahedron

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.

3D model of a dyakis dodecahedron Perfect diploid.stl
3D model of a dyakis dodecahedron

[8]

Stellation

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 4-fold 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{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}
CDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel node 1.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.pngCDel node h.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node.png
= CDel nodes 11.pngCDel split2.pngCDel node.png
CDel node h0.pngCDel 4.pngCDel node 1.pngCDel 3.pngCDel node 1.png
= CDel nodes 11.pngCDel split2.pngCDel node 1.png
CDel node h0.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png
= CDel nodes.pngCDel split2.pngCDel node 1.png
CDel node 1.pngCDel 4.pngCDel node h.pngCDel 3.pngCDel node h.pngCDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png =
CDel nodes 10ru.pngCDel split2.pngCDel node.png or CDel nodes 01rd.pngCDel split2.pngCDel node.png
CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node 1.png =
CDel nodes 10ru.pngCDel split2.pngCDel node 1.png or CDel nodes 01rd.pngCDel split2.pngCDel node 1.png
CDel node h.pngCDel 3.pngCDel node h.pngCDel 4.pngCDel node h0.png =
CDel node h.pngCDel split1.pngCDel nodes hh.png
Uniform polyhedron-43-t0.svg Uniform polyhedron-43-t01.svg Uniform polyhedron-43-t1.svg
Uniform polyhedron-33-t02.png
Uniform polyhedron-43-t12.svg
Uniform polyhedron-33-t012.png
Uniform polyhedron-43-t2.svg
Uniform polyhedron-33-t1.png
Uniform polyhedron-43-t02.png
Rhombicuboctahedron uniform edge coloring.png
Uniform polyhedron-43-t012.png Uniform polyhedron-43-s012.png Uniform polyhedron-33-t0.png Uniform polyhedron-33-t2.png Uniform polyhedron-33-t01.png Uniform polyhedron-33-t12.png Uniform polyhedron-43-h01.svg
Uniform polyhedron-33-s012.svg
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
CDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node fh.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel node fh.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel 4.pngCDel node.png
CDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel node f1.pngCDel 4.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel node.pngCDel 3.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel node fh.pngCDel 3.pngCDel node fh.pngCDel 3.pngCDel node fh.png
Octahedron.svg Triakisoctahedron.jpg Rhombicdodecahedron.jpg Tetrakishexahedron.jpg Hexahedron.svg Deltoidalicositetrahedron.jpg Disdyakisdodecahedron.jpg Pentagonalicositetrahedronccw.jpg Tetrahedron.svg Triakistetrahedron.jpg Dodecahedron.svg

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 face-transitive figures have (*n32) reflectional symmetry.

*n32 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.
Spherical trigonal bipyramid.svg
V3.4.2.4
Spherical rhombic dodecahedron.png
V3.4.3.4
Spherical deltoidal icositetrahedron.png
V3.4.4.4
Spherical deltoidal hexecontahedron.png
V3.4.5.4
Tiling Dual Semiregular V3-4-6-4 Deltoidal Trihexagonal.svg
V3.4.6.4
Deltoidal triheptagonal tiling.svg
V3.4.7.4
H2-8-3-deltoidal.svg
V3.4.8.4
Deltoidal triapeirogonal til.png
V3.4..4

See also

Related Research Articles

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References

  1. Conway, Symmetries of Things, p. 284–286.
  2. "Keyword: "forms" | ClipArt ETC".
  3. "Kite" . Retrieved 6 October 2019.
  4. 1 2 3 4 5 Weisstein, Eric W. "Deltoidal Icositetrahedron". mathworld.wolfram.com. Retrieved 2022-10-06.
    In this MathWorld entry, the small rhombicuboctahedron has edge length so this s.r.c.o.h. has circumradius and midradius so this s.r.c.o.h.'s dual with respect to their common midsphere is the deltoidal icositetrahedron with inradius ××
  5. Isohedron 24k
  6. The Isometric Crystal System
  7. The 48 Special Crystal Forms
  8. 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.