Gyrobifastigium



In geometry, the gyrobifastigium is a poiyhedron that is constructed by attaching a triangular prism to square face of another one. It is an example of a Johnson solid. It is the only Johnson solid that can tile three-dimensional space.

Construction and its naming
The gyrobifastigium can be constructed by attaching two triangular prisms along corresponding square faces, giving a quarter-turn to one prism. These prisms cover the square faces so the resulting polyhedron has four equilateral triangles and four squares, making eight faces in total, an octahedron. Because its faces are all regular polygons and it is convex, the gyrobifastigium is classified as the Johnson solid that is enumerated as twenty-sixth Johnson solid $$ J_{26} $$.

The name of the gyrobifastigium comes from the Latin fastigium, meaning a sloping roof. In the standard naming convention of the Johnson solids, bi- means two solids connected at their bases, and gyro- means the two halves are twisted with respect to each other.

The gyrobifastigium's place in the list of Johnson solids, immediately before the bicupolas, is explained by viewing it as a digonal gyrobicupola. Just as the other regular cupolas have an alternating sequence of squares and triangles surrounding a single polygon at the top (triangle, square or pentagon), each half of the gyrobifastigium consists of just alternating squares and triangles, connected at the top only by a ridge.

Cartesian coordinates for the gyrobifastigium with regular faces and unit edge lengths may easily be derived from the formula of the height of unit edge length $ h = \frac{\sqrt{3}}{2} $ as follows: $$\left(\pm\frac{1}{2},\pm\frac{1}{2},0\right),\left(0,\pm\frac{1}{2},\frac{\sqrt{3}+1}{2}\right),\left(\pm\frac{1}{2},0,-\frac{\sqrt{3}+1}{2}\right).$$

Properties
To calculate the formula for the surface area and volume of a gyrobifastigium with regular faces and with edge length $$ a $$, one may adapt the corresponding formulae for the triangular prism. Its surface area $$ A $$ can be obtained by summing the area of four equilateral triangles and four squares, whereas its volume $$ V $$ by slicing it off into two triangular prisms and adding their volume. That is: $$ \begin{align} A &= \left(4+\sqrt{3}\right)a^2 \approx 5.73205a^2, \\ V &= \left(\frac{\sqrt{3}}{2}\right)a^3 \approx 0.86603a^3. \end{align} $$

Related figures
The Schmitt–Conway–Danzer biprism (also called a SCD prototile ) is a polyhedron topologically equivalent to the gyrobifastigium, but with parallelogram and irregular triangle faces instead of squares and equilateral triangles. Like the gyrobifastigium, it can fill space, but only aperiodically or with a screw symmetry, not with a full three-dimensional group of symmetries. Thus, it provides a partial solution to the three-dimensional einstein problem.

The gyrated triangular prismatic honeycomb can be constructed by packing together large numbers of identical gyrobifastigiums. The gyrobifastigium is one of five convex polyhedra with regular faces capable of space-filling (the others being the cube, truncated octahedron, triangular prism, and hexagonal prism) and it is the only Johnson solid capable of doing so.