| Elongated gyrobifastigium Gabled rhombohedron | |
|---|---|
 | |
| Type | Stereohedron |
| Faces | 4 rectangles 4 irregular pentagons |
| Edges | 18 |
| Vertices | 12 |
| Vertex configuration | (4) 4.4.5 (8) 4.5.5 |
| Symmetry group | D2d, [2+,4], (2*2), order 8 |
| Rotation group | D2, [2,2]+, (222), order 4 |
| Dual polyhedron | Snub disphenoid |
| Properties | convex, space-filling |
| Net | |
 | |
In geometry, the elongated gyrobifastigium or gabled rhombohedron is a space-filling octahedron with 4 rectangles and 4 right-angled pentagonal faces.
The first name is from the regular-faced gyrobifastigium but elongated with 4 triangles expanded into pentagons. The name of the gyrobifastigium comes from the Latin fastigium, meaning a sloping roof. [1] 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. [2] The gyrobifastigium is first in a series of gyrobicupola, so this solid can also be called an elongated digonal gyrobicupola. Geometrically, it can also be constructed as the dual of a digonal gyrobianticupola. This construction is space-filling.
The second name, gabled rhombohedron , is from Michael Goldberg's paper on space-filling octahedra, model 8-VI, the 6th of at least 49 space-filling octahedra. [3] A gable is the triangular portion of a wall between the edges of intersecting roof pitches.
The elongated gyrobifastigium is the dual polyhedron of a snub disphenoid, one of 92 Johnson solids, as well as a deltahedron for having twelve equilateral triangular faces,[ citation needed ] sharing the same three-dimensional dihedral symmetry as antiprismatic of order 8. If the underlying rectangular cuboid is distorted into a rhombohedron, the symmetry is reduced to two-fold rotational symmetry, C2, order 2.
The elongated gyrobifastigium is the cell of the isochoric tridecachoron, a polychoron constructed from the dual of the 13-5 step prism, which has a snub disphenoid vertex figure.
A topologically distinct elongated gyrobifastigium has square and equilateral triangle faces, seen as 2 triangular prisms augmented to a central cube. This is a failed Johnson solid for not being strictly convex. [4]
This is also a space-filling polyhedron, and matches the geometry of the gyroelongated triangular prismatic honeycomb if the elongated gyrobifastigium is dissected back into cubes and triangular prisms.
 Coplanar square and triangles |
The elongated gyrobifastigium must be based on a rectangular cuboid or rhombohedron 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 cube or rectangular cuboid.
The pentagons can also be made regular and the rectangles become trapezoids, and it will no longer be space-filling.
| Type | Space-filling | Not space-filling | ||||
|---|---|---|---|---|---|---|
| Image |  Equilateral pentagons |  Rhombic |  Coplanar |  Concave |  Dual of snub disphenoid |  Regular pentagons |
| Net |  |  |  |  |  |  |
Like the gyrobifastigium, it can self-tessellate space. Polyhedra are tessellated by translation in the plane, and are stacked with alternate orientations. [3] The cross-section of the polyhedron must be square or rhombic, while the roof angle is free, and can be negative, making a concave polyhedron. Rhombic forms require chiral (mirror image) polyhedral pairs to be space-filling.
 Equilateral variation |  Rhombic variation |  Convex variation |  Coplanar-faced variation |  Concave variation |
