Elongated gyrobifastigium
| 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.
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.[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.
Geometry
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, 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.[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 | ||||||
Honeycomb
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 |
See also
References
- ^ Rich, Anthony (1875), "Fastigium", in Smith, William (ed.), A Dictionary of Greek and Roman Antiquities, London: John Murray, pp. 523–524.
- ^ Berman, Martin (1971), "Regular-faced convex polyhedra", Journal of the Franklin Institute, 291 (5): 329–352, doi:10.1016/0016-0032(71)90071-8, MR 0290245.
- ^ a b Goldberg, Michael (January 1981), "On the space-filling octahedra", Geometriae Dedicata, 10 (1): 323–335, doi:10.1007/BF01447431, archived from the original on 22 December 2017.
- ^ Convex regular-faced polyhedra with conditional edges P3,2