Elongated gyrobifastigium

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. 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 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. A gable is the triangular portion of a wall between the edges of intersecting roof pitches. The highest symmetry forms are D2d, order 8, while if the underlying rectangular cuboid is distorted into a rhombohedron, 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 as a deltahedron with all equilateral triangles. However the dual of the snub disphenoid is not space-filling because the pentagons are not right-angled. 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. This is also a space-filling polyhedron, and matches the geometry of the gyroelongated triangular prismatic honeycomb if the elongated gyrobifastigium are dissected back into cubes and triangular prisms.