Deltoidal icositetrahedron

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.

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.

^[9]

Stellation

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