Gyrobifastigium Calculator

Name: Gyrobifastigium - Geometry Calculator
Author: Jürgen Kummer

Calculations at a gyrobifastigium, the Johnson solid J₂₆. This is a right wedge with regular polygons as faces, on which another such, twisted for 90 degrees, is attached base to base.
Enter one value and choose the number of decimal places. Then click Calculate.

Formulas:

h = \sqrt{3} \cdot a

A = (4 + \sqrt{3}) \cdot a^{2}

V = \frac{\sqrt{3}}{2} \cdot a^{3}

Edge length and height 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.

The gyrobifastigium is the only Johnson solid, and also the only non-uniform convex polyhedron with exclusively regular faces, capable of tiling three-dimensional space using only copies of itself. This arrangement forms the so-called gyrated triangular prismatic honeycomb. The other polyhedra with exclusively regular faces that can tile space are the cube, the truncated octahedron, the uniform triangular prism (one half of the gyrobifastigium), and the uniform hexagonal prism. It is also the only Johnson solid based on a double wedge. These two characteristics make it unique among the 92 Johnson solids. It has one of the simplest forms among those, also the formulas for calculating its quantities are also relatively simple. The gyrobifastigium is not uniform, as its vertices are not all alike.

The gyrobifastigium possesses neither point symmetry nor reflectional symmetry. It exhibits rotational symmetry about three distinct axes, at rotations of 180 degrees and multiples thereof. These axes pass through the midpoints of opposite edges of the central square or through the two wedge edges, and they are mutually perpendicular.

Similar forms are found in chemistry and crystallography. In these fields, the gyrobifastigium serves as a model for certain coordination polyhedra and molecular structures, as well as for describing space-filling crystal packings.

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