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 elongated gyrobifastigium or gabled rhombohedron is a space-filling

octahedron In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at e ...

with 4 rectangles and 4 right-angled

pentagon In geometry, a pentagon (from the Greek language, 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 polygon, simple pentagon is ...

al faces.

Name

The first name is from the regular-faced

gyrobifastigium 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 ...

but

elongated Elongation may refer to: * Elongation (astronomy) * Elongation (geometry) * Elongation (plasma physics) * Part of transcription of DNA into RNA of all types, including mRNA, tRNA, rRNA, etc. * Part of translation (biology) of mRNA into proteins ...

with 4 triangles expanded into pentagons. The name of the gyrobifastigium comes from the Latin ''fastigium'', meaning a sloping roof. In the standard naming convention of the

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 each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johns ...

s, ''bi-'' means two solids connected at their bases, and ''gyro-'' means the two halves are twisted with respect to each other. The gyrobifastigium is first in a series of

gyrobicupola In geometry, a bicupola is a solid formed by connecting two cupolae on their bases. There are two classes of bicupola because each cupola (bicupola half) is bordered by alternating triangles and squares. If similar faces are attached together ...

, so this solid can also be called an ''elongated digonal gyrobicupola''. Geometrically it can also be constructed as the dual of a digonal

gyrobianticupola In geometry, a Johnson solid is a strictly convex polyhedron each face of which is a regular polygon. There is no requirement that each face must be the same polygon, or that the same polygons join around each vertex. An example of a Johns ...

. This construction is space-filling. The second name, ''gabled

rhombohedron In geometry, a rhombohedron (also called a rhombic hexahedron or, inaccurately, a rhomboid) is a three-dimensional figure with six faces which are rhombi. It is a special case of a parallelepiped where all edges are the same length. It can be us ...

'', is from Michael Goldberg's paper on space-filling octahedra, model 8-VI, the 6th of at least 49 space-filling octahedra. A

gable A gable is the generally triangular portion of a wall between the edges of intersecting roof pitches. The shape of the gable and how it is detailed depends on the structural system used, which reflects climate, material availability, and aest ...

is the triangular portion of a wall between the edges of intersecting roof pitches.

Geometry

The highest symmetry forms are D_2d, order 8, while if the underlying

rectangular cuboid In geometry, a cuboid is a hexahedron, a six-faced solid. Its faces are quadrilaterals. Cuboid means "like a cube", in the sense that by adjusting the length of the edges or the angles between edges and faces a cuboid can be transformed into a cub ...

is distorted into a

, the symmetry is reduced to 2-fold rotational symmetry, C₂, order 2. It has all 3-valence vertices and its dual has all triangular faces, including the

snub disphenoid In geometry, the snub disphenoid, Siamese dodecahedron, triangular dodecahedron, trigonal dodecahedron, or dodecadeltahedron is a convex polyhedron with twelve equilateral triangles as its faces. It is not a regular polyhedron because some ve ...

as a

deltahedron In geometry, a deltahedron (plural ''deltahedra'') is a polyhedron whose faces are all equilateral triangles. The name is taken from the Greek upper case delta (Δ), which has the shape of an equilateral triangle. There are infinitely many del ...

with all equilateral triangles. However the dual of the

is not space-filling because the pentagons are not right-angled.

Related figures

The elongated gyrobifastigium is the cell of the

isochoric Isochoric may refer to: *cell-transitive, in geometry *isochoric process, a constant volume process in chemistry or thermodynamics * Isochoric model In genetics, an isochore is a large region of genomic DNA (greater than 300 kilobases) with a hig ...

tridecachoron, a polychoron constructed from the dual of the 13-5 step prism, which has a

vertex figure.

Variations

A topologically distinct '' elongated gyrobifastigium'' has square and equilateral triangle faces, seen as 2

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''. ...

s augmented to a central

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 on ...

. This is a failed

for not being strictly convex.Convex regular-faced polyhedra with conditional edges
P3,2 This is also a space-filling polyhedron, and matches the geometry of the

gyroelongated 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 p ...

if the ''elongated gyrobifastigium'' are

dissected Dissection (from Latin ' "to cut to pieces"; also called anatomization) is the dismembering of the body of a deceased animal or plant to study its anatomical structure. Autopsy is used in pathology and forensic medicine to determine the cause ...

back into cubes and triangular prisms. The ''elongated gyrobifastigium'' must be based on a

to fill-space, while the angle of the roof is free, including allowing concave forms. If the roof has zero angle, the geometry becomes a

. The pentagons can also be made regular and the rectangles become trapezoids, and it will no longer be space-filling.

Honeycomb

Like the gyrobifastigium, it can self-tessellate space. Polyhedra are tessellated by translation in the plane, and are stacked with alternate orientations. The cross section of the polyhedron must be square or rhombic, while the

roof A roof ( : roofs or rooves) is the top covering of a building, including all materials and constructions necessary to support it on the walls of the building or on uprights, providing protection against rain, snow, sunlight, extremes of temp ...

angle is free, and can be negative, making a concave polyhedron. Rhombic forms require chiral (mirror image) polyhedral pairs to be space-filling.

References

External links

* * {{mathworld2 , urlname2 = JohnsonSolid , title2 = Johnson solid, , urlname = Gyrobifastigium , title = Gyrobifastigium Space-filling polyhedra