Hexakis Octahedron Calculator

Name: Hexakis Octahedron - Geometry Calculator
Author: Jürgen Kummer

Calculations at a hexakis octahedron or disdyakis dodecahedron, the dual body of the truncated cuboctahedron. A hexakis octahedron is a polyhedron with identical, but irregular triangle faces. It has twelve vertices with four edges, eight vertices with six edges and six vertices with eight edges.
Enter one value and choose the number of decimal places. Then click Calculate.

Formulas:

a = \frac{2}{7} \cdot d \cdot \sqrt{60 + 6 \sqrt{2}}

b = \frac{3}{7} \cdot d \cdot \sqrt{12 + 6 \sqrt{2}}

c = \frac{2}{7} \cdot d \cdot \sqrt{30 - 3 \sqrt{2}}

b = \frac{3}{14} \cdot a \cdot (1 + 2 \sqrt{2})

c = \frac{a}{14} \cdot (10 - \sqrt{2})

A = \frac{3}{7} \cdot a^{2} \cdot \sqrt{543 + 176 \sqrt{2}}

V = \frac{a^{3}}{28} \cdot \sqrt{6 \cdot (986 + 607 \sqrt{2})}

r_{m} = \frac{a}{4} \cdot (1 + 2 \sqrt{2})

r_{i} = \frac{a}{2} \cdot \sqrt{\frac{402 + 195 \sqrt{2}}{194}}

\frac{A}{V} = \frac{12 \cdot \sqrt{543 + 176 \sqrt{2}}}{a \cdot \sqrt{6 \cdot (986 + 607 \sqrt{2})}}

The hexakis octahedron is a Catalan solid. Edge lengths, diagonals and radiuses have the same unit (e.g. meter), the area has this unit squared (e.g. square meter), the volume has this unit to the power of three (e.g. cubic meter). A/V has this unit ^-1.

Hexakisoctahedron translates to sixfold octahedron, owing to its six times eight, that is 48-faces. Correspondingly, disdyakisdodecahedron means two times two times twelve faces.
The nature of the vertices of the hexakisicosahedron, and of the Catalan solids in general, corresponds to the faces of their respective dual Archimedean solids, in this case, the truncated cuboctahedron. Equal angles meet at each of its vertices. The triangular faces of the hexakisoctahedron possess irrational side lengths and three distinct acute angles. They appear almost like right triangles. However, the largest angle between the sides b and c measures only approximately 87 degrees. Like the deltoidal icositetrahedron, the hexakisoctahedron can be bisected in various directions without cutting through any of its original edges. The resulting cross-section takes also the form of a regular octagon with an edge length of a. If one of these halves is rotated by 45 degrees, a different shape with identical dimensions is formed, a shape that could be termed as pseudo-hexakisoctahedron.

In nature, the hexakisoctahedron occurs as a crystal form, for instance in diamonds and metal crystals such as copper. It is used, for example, as a 48-sided gaming die.

↑ up