Deltoidal icositetrahedron

In geometry, the deltoidal icositetrahedron (or trapezoidal icositetrahedron, tetragonal icosikaitetrahedron, tetragonal trisoctahedron, strombic icositetrahedron) is a Catalan solid.

Description

A deltoidal icositetrahedron is a Catalan solid with 24 sides that are 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. Deltoidal icositetrahedron and deltoidal hexecontahedron are two Catalan solids with kite faces only.

Dimensions and angles

Dimensions

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

• short body diagonal length:

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

• long edge length:

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

• short edge length:

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

• inradius:

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

r {\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:

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

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

• area:

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

• volume:

V = 122 + 71 2 s 3 . {\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).

Dyakis dodecahedron

A variant with pyritohedral symmetry is called a dyakis dodecahedron or diploid. 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.

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.

• Wenninger, Magnus (1983), Dual Models, Cambridge University Press, doi:, ISBN 978-0-521-54325-5, MR (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 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal icosikaitetrahedron)

External links

• Weisstein, Eric W., "" ("") at MathWorld.
• – Interactive Polyhedron model