Elongated gyrobifastigium
Template:Short description Template:Infobox polyhedron File:Elongated gyrobifastigium.stl
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 elongated with 4 triangles expanded into pentagons. The name of the gyrobifastigium comes from the Latin fastigium, meaning a sloping roof.Template:R 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.Template:R 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.Template:R A gable is the triangular portion of a wall between the edges of intersecting roof pitches.
Geometry
The elongated gyrobifastigium is the dual polyhedron of a snub disphenoid,Template:Cn one of 92 Johnson solids, 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.
Related figures
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.
Variations
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.[1]
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 | 
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Honeycomb
Like the gyrobifastigium, it can self-tessellate space. Polyhedra are tessellated by translation in the plane, and are stacked with alternate orientations.Template:R 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 |
See also
- Bidiakis cube, a cubic regular graph with twelve vertices and eighteen edges.
- Elongated dodecahedron, another space-filling polyhedron
- Elongated square bipyramid, a space-filling Johnson solid
- Trapezo-rhombic dodecahedron, a space-filling dual of triangular orthobicupola
References
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