Deltoidal icositetrahedron

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

• (±1, 0, 0), (0, ±1, 0), (0, 0, ±1)
• (0, ±1/2√2, ±1/2√2), (±1/2√2, 0, ±1/2√2), (±1/2√2, ±1/2√2, 0)
• (±(2√2+1)/7, ±(2√2+1)/7, ±(2√2+1)/7)

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

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

${\displaystyle {\begin{aligned}A&=6{\sqrt {29-2{\sqrt {2}}}}\,a^{2}\\V&={\sqrt {122+71{\sqrt {2}}}}\,a^{3}\end{aligned}}}$

The kites have three equal acute angles with value ${\displaystyle \arccos({\frac {1}{2}}-{\frac {1}{4}}{\sqrt {2}})\approx 81.578\,941\,881\,85^{\circ }}$ and one obtuse angle (between the short edges) with value ${\displaystyle \arccos(-{\frac {1}{4}}-{\frac {1}{8}}{\sqrt {2}})\approx 115.263\,174\,354\,45^{\circ }}$.

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 projections

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

Orthogonal projections

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

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

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

• Deltoidal hexecontahedron
• 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
• Pseudo-deltoidal icositetrahedron

• Williams, Robert (1979). The Geometrical Foundation of Natural Structure: A Source Book of Design. Dover Publications, Inc. ISBN 0-486-23729-X. (Section 3-9)
• 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-Strass, ISBN 978-1-56881-220-5 (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal icosikaitetrahedron)

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