Elongated gyrobifastigium

From Wikipedia, the free encyclopedia
Elongated gyrobifastigium
Gabled rhombohedron
Dual digonal gyrobianticupola.png
TypeStereohedron
Faces4 rectangles
4 pentagons
Edges18
Vertices12
Vertex configuration(4) 4.4.5
(8) 4.5.5
Symmetry groupD2d, [2+,4], (2*2), order 8
Rotation groupD2, [2,2]+, (222), order 4
Dual polyhedronSnub disphenoid
Propertiesconvex, space-filling
Net
Elongated gyrobifastigium net.png

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. 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.[2] A gable is the triangular portion of a wall between the edges of intersecting roof pitches.

Geometry[]

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.[3] However the dual of the snub disphenoid is not space-filling because the pentagons are not right-angled.

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 are dissected back into cubes and triangular prisms.

Elongated digonal gyrobicupola.png
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 Elongated digonal gyrobicupola2.png
Equilateral pentagons 		Skew75 gabled rhombohedron.png
Rhombic 		Elongated gyrobifastigium conplanar.png
Coplanar 		Elongated gyrobifastigium concave.png
Concave 		Dual of snub disphenoid.png
Dual of snub disphenoid 		Elongated digonal gyrobicupola3.png
Regular pentagons
Net Elongated digonal gyrobicupola net2.png Skew75 gabled rhombohedron net.png Elongated gyrobifastigium coplanar net.png Elongated digonal gyrobicupola concave net.png Dual of snub disphenoid net.png Elongated digonal gyrobicupola net3.png

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

Elongated gyrobifastigium equilateral honeycomb.png
Equilateral variation 		Skew75 gabled rhombohedron honeycomb.png
Rhombic variation 		Honeycomb by dual of digonal gyrobianticupola.png
Convex variation 		Elongated gyrobifastigium coplanar honeycomb.png
Coplanar-faced variation 		Elongated gyrobifastigium concave honeycomb.png
Concave variation

See also[]

References[]

  1. ^ Rich, Anthony (1875), "Fastigium", in Smith, William (ed.), A Dictionary of Greek and Roman Antiquities, London: John Murray, pp. 523–524.
  2. ^ Goldberg, Michael, On the space-filling octahedra, Geometriae Dedicata, January 1981, Volume 10, Issue 1, pp 323–335 [1] PDF Archived 2017-12-22 at the Wayback Machine
  3. ^ Dual of Snub Disphenoid (J84)
  4. ^ Convex regular-faced polyhedra with conditional edges P3,2

External links[]

Retrieved from ""