{| class=wikitable align="right" width="250"

!bgcolor=#e7dcc3 colspan=2|Deltoidal icositetrahedron

|-

|align=center colspan=2|290px|Deltoidal icositetrahedron<br>(rotating and 3D model)

|-

|bgcolor=#e7dcc3|Type||Catalan

|-

|bgcolor=#e7dcc3|Conway notation||oC or deC

|-

|bgcolor=#e7dcc3|Coxeter diagram||

|-

|bgcolor=#e7dcc3|Face polygon||60px<BR>Kite with 3 equal acute angles & 1 obtuse angle

|-

|bgcolor=#e7dcc3|Faces||24, congruent

|-

|bgcolor=#e7dcc3|Edges||24 short + 24 long = 48

|-

|bgcolor=#e7dcc3|Vertices||8 (connecting 3 short edges)<br>+ 6 (connecting 4 long edges)<br>+ 12 (connecting 4 alternate short & long edges)<br>= 26

|-

|bgcolor=#e7dcc3|Face configuration||V3.4.4.4

|-

|bgcolor=#e7dcc3|Symmetry group||O<sub>h</sub>, BC<sub>3</sub>, [4,3], *432

|-

|bgcolor=#e7dcc3|Rotation group||O, [4,3]<sup>+</sup>, (432)

|-

|bgcolor=#e7dcc3|Dihedral angle||same value for short & long edges:<br><math>\arccos \left( - \frac{7 + 4 \sqrt{2{17} \right)</math><br><math>\approx 138^{\circ} 07 ' 05 </math>

|-

|bgcolor=#e7dcc3|Dual polyhedron||Rhombicuboctahedron

|-

|bgcolor=#e7dcc3|Properties||convex, face-transitive

|-

|align=center colspan=2|150px|Deltoidal icositetrahedron<br>Net

|}

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

Description

thumb|3D model of a deltoidal icositetrahedron

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 81.6 degrees and one angle 115.3 degrees. The dihedral angle between every two kites is 138.1 degrees. The deltoidal icositetrahedron has 44 edges, and 26 vertices &ndash; 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:

::<math>d = \frac{2 \sqrt{3} \left( 2 \sqrt{2} + 1 \right) }{7} \approx 1.894\,580 ;</math>

  • long edge length:

::<math>S = \sqrt{2 - \sqrt{2 \approx 0.765\,367 ;</math>

  • short edge length: or diploid. It is common in crystallography.<br>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.

thumb|3D model 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.

References

  • (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, [https://web.archive.org/web/20100919143320/https://akpeters.com/product.asp?ProdCode=2205] (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, page 286, tetragonal icosikaitetrahedron)
  • Deltoidal (Trapezoidal) Icositetrahedron – Interactive Polyhedron model