Deltoidal icositetrahedron

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	Oh, BC3, [4,3], *432
Rotation group	O, [4,3]^+, (432)
Dihedral angle	same value for short & long edges: ${\displaystyle \arccos \left(-{\frac {7+4{\sqrt {2}}}{17}}\right)}$ ${\displaystyle \approx 138^{\circ }07'05''}$
Dual polyhedron	Rhombicuboctahedron
Properties	convex, face-transitive
Net

D.i. as artwork and die

D.i. projected onto cube and octahedron in Perspectiva Corporum Regularium

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.

Description

A deltoidal icositetrahedron is a Catalan solid with 24-sided kites. All of its faces are congruent, each has three interior angles approximately 81.8 degrees and one angle 115.4 degrees. The dihedral angle between every two kites is 138.1 degrees. The deltoidal icositetrahedron has 44 edges, and 26 vertices – eight vertices surrounded by three kites and eighteen vertices by four kites. Its dual polyhedron is the rhombicuboctahedron, an Archimedean solid.^[3] Deltoidal icositetrahedron and deltoidal hexecontahedron are two Catalan solids with kite faces only.^[4]

Dimensions and angles

Dimensions

The deltoidal icositetrahedron with long body diagonal length D = 2 has:

• short body diagonal length:

${\displaystyle d={\frac {2{\sqrt {3}}\left(2{\sqrt {2}}+1\right)}{7}}\approx 1.894\,580;}$

• long edge length:^[5]

${\displaystyle S={\sqrt {2-{\sqrt {2}}}}\approx 0.765\,367;}$

• short edge length:^[5]

${\displaystyle s={\frac {\sqrt {20-2{\sqrt {2}}}}{7}}\approx 0.591\,980;}$

• inradius:^[5]

${\displaystyle r={\sqrt {\frac {7+4{\sqrt {2}}}{17}}}\approx 0.862\,856.}$

${\displaystyle r}$ 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:

${\displaystyle {\frac {S}{s}}=2-{\frac {1}{\sqrt {2}}}\approx 1.292\,893.}$

The deltoidal icositetrahedron with short edge length ${\displaystyle s}$ has:

• area:^[5]

${\displaystyle A=6{\sqrt {29-2{\sqrt {2}}}}\,s^{2};}$

• volume:^[5]

${\displaystyle V={\sqrt {122+71{\sqrt {2}}}}\,s^{3}.}$

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.

Related polyhedra

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).

Deltoidal icositetrahedron	Disdyakis dodecahedron	Dyakis dodecahedron	Tetartoid

Dyakis dodecahedron

A variant with pyritohedral symmetry is called a dyakis dodecahedron^[6][7] or diploid.^[8] 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

^[9]

Stellation

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

See also

• Tetrakis hexahedron, another 24-face 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.

References

1. Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). The Symmetries of Things. AK Peters. p. 286. ISBN 978-1-56881-220-5.
2. "Keyword: "forms" | ClipArt ETC".
3. Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. p. 81.
4. Williams (1979), p. 93.
5. Weisstein, Eric W. "Deltoidal Icositetrahedron". mathworld.wolfram.com. Retrieved 2022-10-06.
   In this MathWorld entry, the small rhombicuboctahedron has edge length ${\displaystyle a=1;}$ so this s.r.c.o.h. has circumradius ${\displaystyle R={\frac {\sqrt {5+2{\sqrt {2}}}}{2}}}$ and midradius ${\displaystyle \rho ={\sqrt {1+{\frac {\sqrt {2}}{2}}}};}$ so this s.r.c.o.h.'s dual with respect to their common midsphere is the deltoidal icositetrahedron with inradius ${\displaystyle r_{_{a=1}}={\frac {\rho ^{2}}{R}}={\sqrt {\frac {2\left(7+4{\sqrt {2}}\right)}{17}}}={\sqrt {2}}}$×${\displaystyle {\sqrt {\frac {7+4{\sqrt {2}}}{17}}}={\sqrt {2}}}$×${\displaystyle r_{_{D=2}}.}$
6. Isohedron 24k
7. The Isometric Crystal System
8. The 48 Special Crystal Forms
9. 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.

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:10.1017/CBO9780511569371, ISBN 978-0-521-54325-5, 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 Goodman-Strauss, ISBN 978-1-56881-220-5 [1] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal icosikaitetrahedron)

External links

• Weisstein, Eric W., "Deltoidal icositetrahedron" ("Catalan solid") at MathWorld.
• Deltoidal (Trapezoidal) Icositetrahedron – Interactive Polyhedron model