Deltoidal icositetrahedron

Deltoidal icositetrahedron
(Click here for rotating model)
Type	Catalan
Conway notation	oC or deC
Coxeter diagram
Face polygon	kite
Faces	24
Edges	48
Vertices	26 = 6 + 8 + 12
Face configuration	V3.4.4.4
Symmetry group	O_h, BC₃, [4,3], *432
Rotation group	O, [4,3]⁺, (432)
Dihedral angle	138°07′05″ arccos(−7 + 4√2/17)
Dual polyhedron	rhombicuboctahedron
Properties	convex, face-transitive
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.

Cartesian coordinates

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.

Dimensions

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

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

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 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

Related polyhedra

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

Dyakis dodecahedron

The deltoidal icositetrahedron is topologically equivalent to a cube whose faces are divided in quadrants. It can also be projected onto a regular octahedron, with kite faces, or more general quadrilaterals with pyritohedral symmetry. In Conway polyhedron notation, they represent an ortho operation to a cube or octahedron.

In crystallography a rotational variation is called a dyakis dodecahedron^[4]^[5] or diploid.^[6]

Octahedral, O_h, order 24		Pyritohedral, T_h, order 12

Related polyhedra and tilings

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{3^1,1}	t{3,4} t{3^1,1}	{3,4} {3^1,1}	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{3^1,1}

		=	=	=				= 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.

*n32 symmetry mutation of dual expanded tilings: V3.4.n.4
Symmetry *n32 [n,3]	Spherical				Euclid.	Compact hyperb.		Paraco.
Symmetry *n32 [n,3]	*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

References

^ Conway, Symmetries of Things, p.284–286
^ https://etc.usf.edu/clipart/keyword/forms
^ "Kite". Retrieved 6 October 2019.
^ Isohedron 24k
^ The Isometric Crystal System
^ The 48 Special Crystal Forms

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 [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

[1] Conway, Symmetries of Things, p.284–286

[2] ttps://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

[1]

[2]

[3]

[4]

[5]

[6]