geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, the gyrobifastigium is a polyhedron that is constructed by attaching a triangular prism to square face of another one. It is an example of a

Johnson solid In geometry, a Johnson solid, sometimes also known as a Johnson–Zalgaller solid, is a convex polyhedron whose faces are regular polygons. They are sometimes defined to exclude the uniform polyhedrons. There are ninety-two Solid geometry, s ...

. 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 prism In geometry, a triangular prism or trigonal prism is a Prism (geometry), prism with 2 triangular bases. If the edges pair with each triangle's vertex and if they are perpendicular to the base, it is a ''right triangular prism''. A right triangul ...

s along corresponding square faces, giving a quarter-turn to one prism. These prisms cover the square faces so the resulting polyhedron has four

equilateral triangle An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the ...

s and four

square In geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...

s, making eight faces in total, an

octahedron In geometry, an octahedron (: octahedra or octahedrons) is any polyhedron with eight faces. One special case is the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. Many types of i ...

. Because its faces are all

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either ''convex ...

s and it is

convex Convex or convexity may refer to: Science and technology * Convex lens, in optics Mathematics * Convex set, containing the whole line segment that joins points ** Convex polygon, a polygon which encloses a convex set of points ** Convex polytop ...

, the gyrobifastigium is a

, indexed as

J_

. 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.

Cartesian coordinates In geometry, a Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of real numbers called ''coordinates'', which are the signed distances to the point from two fixed perpendicular o ...

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

as follows:

\left(\pm\frac,\pm\frac,0\right),\left(0,\pm\frac,\frac\right),\left(\pm\frac,0,-\frac\right).

Properties

To calculate the formula for the

surface area The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the d ...

and

volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...

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
 A &= \left(4+\sqrt\right)a^2 \approx 5.73205a^2, \\
 V &= \left(\frac\right)a^3 \approx 0.86603a^3.
\end

Related figures

The Schmitt–Conway–Danzer biprism (also called a SCD prototileForcing 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 In Euclidean geometry, a parallelogram is a simple polygon, simple (non-list of self-intersecting polygons, self-intersecting) quadrilateral with two pairs of Parallel (geometry), parallel sides. The opposite or facing sides of a parallelogram a ...

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 In plane discrete geometry, the einstein problem asks about the existence of a single prototile that by itself forms an aperiodic set of prototiles; that is, a shape that can tessellate space but only in a nonperiodic way. Such a shape is call ...

. 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 A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...

truncated octahedron In geometry, the truncated octahedron is the Archimedean solid that arises from a regular octahedron by removing six pyramids, one at each of the octahedron's vertices. The truncated octahedron has 14 faces (8 regular hexagon, hexagons and 6 Squa ...

, and

hexagonal prism In geometry, the hexagonal prism is a Prism (geometry), prism with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 face (geometry), faces, 18 Edge (geometry), edges, and 12 vertex (geometry), vertices.. As a semiregular polyhedro ...

) and it is the only Johnson solid capable of doing so. The

dual polyhedron In geometry, every polyhedron is associated with a second dual structure, where the vertices of one correspond to the faces of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other ...

of the gyrobifastigium is combinatorially equivalent to a

with two of its opposite faces subdivided into triangles by diagonals that are not parallel to each other. A combinatorially equivalent form of the

elongated gyrobifastigium In geometry, the elongated gyrobifastigium or gabled rhombohedron is a space-filling octahedron with 4 rectangles and 4 right-angled pentagonal faces. Name The first name is from the regular-faced gyrobifastigium but Elongation (geometry), elon ...

, a related space-filling polyhedron, can be obtained by instead subdividing two opposite faces into rectangles by midlines, again choosing the subdivision lines to be non-parallel.

References

External links

* {{Johnson solids navigator Johnson solids Space-filling polyhedra