Gyrobifastigium

The gyrobifastigium, or gybef, is one of the 92 Johnson solids (J26). It consists of 4 triangles and 4 squares. It can be constructed by attaching two triangular prisms, seen as digonal cupolas, at one of their square faces so that their opposite edges are perpendicular. As such, it could be considered to be a digonal gyrobicupola.

Vertex coordinates
A gyrobifastigium of edge length 1 has vertices given by the following coordinates:


 * $$\left(±\frac12,\,0,\,\frac{\sqrt3}2\right),$$
 * $$\left(±\frac12,\,±\frac12,\,0\right),$$
 * $$\left(0,\,±\frac12,\,-\frac{\sqrt3}2\right).$$

In vertex figures
The gyrobifastigium appears as the vertex figure of the nonuniform triangular duoantiprism. This vertex figure has an edge length of 1, and has no corealmic realization, because the Johnson gyrobifastigium has no circumscribed sphere.

Variants made by changing the two edges perpendicular to the symmetry axis also appear as the vertex figure of the nonuniform duoantiprisms made out of two regular polygons. The symmetry of the gyrofastigium is BC2×A1/2, order 8 if the two polygons are identical, otherwise the symmetry is A1×A1×I, order 4.