geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

, the gyrobifastigium is the 26th

Johnson solid In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that isohedral, each face must be the same polygon, or that the same polygons join around each Vertex (geometry), ver ...

(). It can be constructed by joining two face-regular

triangular prism In geometry, a triangular prism is a three-sided prism; it is a polyhedron made of a triangular base, a translated copy, and 3 faces joining corresponding sides. A right triangular prism has rectangular sides, otherwise it is ''oblique''. A unif ...

s along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile three-dimensional space. It is also the vertex figure of the nonuniform

duoantiprism In geometry of 4 dimensions or higher, a double prism or duoprism is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an -polytope and an -polytope is an -polytope, whe ...

(if and are greater than 2). Despite the fact that would yield a geometrically identical equivalent to the Johnson solid, it lacks a

circumscribed sphere In geometry, a circumscribed sphere of a polyhedron is a sphere that contains the polyhedron and touches each of the polyhedron's vertices. The word circumsphere is sometimes used to mean the same thing, by analogy with the term ''circumcircle' ...

that touches all vertices, except for the case which represents a uniform

great duoantiprism In geometry, the great duoantiprism is the only uniform star-duoantiprism solution in 4-dimensional geometry. It has Schläfli symbol or Coxeter diagram , constructed from 10 pentagonal antiprisms, 10 pentagrammic crossed-antiprisms, and 50 ...

. Its dual, the

elongated tetragonal disphenoid In geometry, the gyrobifastigium is the 26th Johnson solid (). It can be constructed by joining two face-regular triangular prisms along corresponding square faces, giving a quarter-turn to one prism. It is the only Johnson solid that can tile ...

, can be found as cells of the duals of the

History and name

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 A triangle is a polygon with three Edge (geometry), edges and three Vertex (geometry), vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, an ...

square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90-degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length adj ...

pentagon In geometry, a pentagon (from the Greek πέντε ''pente'' meaning ''five'' and γωνία ''gonia'' meaning ''angle'') is any five-sided polygon or 5-gon. The sum of the internal angles in a simple pentagon is 540°. A pentagon may be simpl ...

), each half of the gyrobifastigium consists of just alternating squares and triangles, connected at the top only by a ridge.

Honeycomb

The

gyrated triangular prismatic honeycomb The triangular prismatic honeycomb or triangular prismatic cellulation is a space-filling tessellation (or honeycomb) in Euclidean 3-space. It is composed entirely of triangular prisms. It is constructed from a triangular tiling extruded into pri ...

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 In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only r ...

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 with hexagonal base. Prisms are polyhedrons; this polyhedron has 8 faces, 18 edges, and 12 vertices.. Since it has 8 faces, it is an octahedron. However, the term ''octahedron'' is primarily used to ...

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

Cartesian coordinates

Cartesian coordinates A Cartesian coordinate system (, ) in a plane is a coordinate system that specifies each point uniquely by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular oriented lines, measured in t ...

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).

To calculate

formula In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a ''chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betwee ...

e for the

surface area The surface area 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 definition of arc ...

and

volume Volume is a measure of occupied 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). The de ...

of a gyrobifastigium with regular faces and with edge length ''a'', one may simply adapt the corresponding formulae for the triangular prism: :

A=\left(4+\sqrt\right)a^2\approx5.73205a^2,

V=\left(\frac\right)a^3\approx0.86603a^3.

Topologically equivalent polyhedra

Schmitt–Conway–Danzer biprism

The Schmitt–Conway–Danzer biprism (also called a SCD prototile) is a polyhedron topologically equivalent to the gyrobifastigium, but with

parallelogram In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...

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 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 called an "einst ...

.Tiling Space with a Schmitt-Conway Biprism
wolfram demonstrations

Dual

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 has 8 faces: 4

isosceles triangle In geometry, an isosceles triangle () is a triangle that has two sides of equal length. Sometimes it is specified as having ''exactly'' two sides of equal length, and sometimes as having ''at least'' two sides of equal length, the latter versio ...

s, corresponding to the valence-3 vertices of the gyrobifastigium, and 4

s corresponding to the valence-4 equatorial vertices.

References

External links

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