Gyrobifastigium#Schmitt–Conway–Danzer biprism

The gyrobifastigium can be constructed by attaching two triangular prisms along corresponding square faces, giving a quarter-turn to one prism.{{r|darling}} These prisms cover the square faces so the resulting polyhedron has four equilateral triangles and four squares, making eight faces in total, an octahedron.{{r|berman}} Because its faces are all regular polygons and it is convex, the gyrobifastigium is a Johnson solid, indexed as $J\_\{26\}$.{{r|francis}}

The name of the gyrobifastigium comes from the Latin fastigium, meaning a sloping roof.{{citation|contribution-url=https://penelope.uchicago.edu/Thayer/E/Roman/Texts/secondary/SMIGRA*/Fastigium.html|editor-first=William|editor-last=Smith|editor-link=William Smith (lexicographer)|title=A Dictionary of Greek and Roman Antiquities|publisher=John Murray|location=London|year=1875|contribution=Fastigium|pages=523–524|first=Anthony|last=Rich}}. 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.{{r|berman}}

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 $h\; =\; \backslash frac\{\backslash sqrt\{3\}\}\{2\}$ as follows:

$$\backslash left(\backslash pm\backslash frac\{1\}\{2\},\backslash pm\backslash frac\{1\}\{2\},0\backslash right),\backslash left(0,\backslash pm\backslash frac\{1\}\{2\},\backslash frac\{\backslash sqrt\{3\}+1\}\{2\}\backslash right),\backslash left(\backslash pm\backslash frac\{1\}\{2\},0,-\backslash frac\{\backslash sqrt\{3\}+1\}\{2\}\backslash right).$$

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:{{r|berman}}

$$\backslash begin\{align\}$$

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}

{{multiple image

| image1 = SCD tile.svg

| caption1 = The Schmitt–Conway–Danzer biprism

| image2 = Gyrobifastigium honeycomb.png

| caption2 = The gyrobifastigium honeycomb

| total_width = 500

}}

{{anchor|Schmitt–Conway–Danzer biprism}}The Schmitt–Conway–Danzer biprism (also called a SCD prototile[https://arxiv.org/abs/1009.1419 Forcing Nonperiodicity With a Single Tile] Joshua E. S. Socolar and Joan M. Taylor, 2011) 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.{{r|senechal}}

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.{{r|ah06|kepler}}

• Elongated gyrobifastigium, a related space-filling polyhedron

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• {{mathworld2 | urlname2 = JohnsonSolid | title2 = Johnson solid| | urlname = Gyrobifastigium | title = Gyrobifastigium}}

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