Tetragonal disphenoid honeycomb

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Tetragonal disphenoid tetrahedral honeycomb
Type convex uniform honeycomb dual
Coxeter-Dynkin diagram CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node.png
Cell type Disphenoid tetrahedron.png
Tetragonal disphenoid
Face types isosceles triangle {3}
Vertex figure Tetrakishexahedron.jpg
tetrakis hexahedron
CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.png
Space group Im3m (229)
Symmetry [[4,3,4]]
Coxeter group {\tilde{C}}_3, [4,3,4]
Dual Bitruncated cubic honeycomb
Properties cell-transitive, face-transitive, vertex-transitive

The tetragonal disphenoid tetrahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of identical tetragonal disphenoidal cells. Cells are face-transitive with 4 identical isosceles triangle faces.

John Horton Conway calls this honeycomb a oblate tetrahedrille.

The disphenoid tetrahedral honeycomb is the dual of the uniform bitruncated cubic honeycomb.

Geometry[edit]

This honeycomb's vertex figure is a tetrakis cube: 24 disphenoids meet at each vertex. The union of these 24 disphenoids forms a rhombic dodecahedron. Each edge of the tessellation is surrounded by either four or six disphenoids, according to whether it forms the base or one of the sides of its adjacent isosceles triangle faces respectively. When an edge forms the base of its adjacent isosceles triangles, and is surrounded by four disphenoids, they form an irregular octahedron. When an edge forms one of the two equal sides of its adjacent isosceles triangle faces, the six disphenoids surrounding the edge form a special type of parallelepiped called a trigonal trapezohedron.

Disphenoid tetrah hc.png


Hexakis cubic honeycomb[edit]

Hexakis cubic honeycomb
Hexakis cubic honeycomb.png
Type Dual uniform honeycomb
Coxeter–Dynkin diagrams CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Cell Isosceles square pyramid Square pyramid.png
Faces Triangle
square
Space group
Fibrifold notation
Pm3m (221)
4:2
Coxeter group {\tilde{C}}_3, [4,3,4]
vertex figures Hexahedron.pngRhombic dodecahedron.jpg
CDel node f1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png, CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png
Dual Truncated cubic honeycomb
Properties Cell-transitive

The hexakis cubic honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. It can be seen as a cubic honeycomb with each cube subdivided by a center point into 6 elongated square pyramid cells.

John Horton Conway calls this honeycomb a pyramidille.

There are two types of planes of faces: one as a square tiling, and flattened triangular tiling with half of the triangles removed as holes.

Tiling
plane
Square tiling uniform coloring 1.png Hexakis cubic honeycomb triangular plane.png
Symmetry p4m, [4,4] (*442) pmm, [∞,2,∞] (*2222)

Related honeycombs[edit]

It is dual to the truncated cubic honeycomb with octahedral and truncated cubic cells:

Truncated cubic honeycomb.png

If the square pyramids of the pyramidille are joined on their bases, another honeycomb is created with identical vertices and edges, called a square bipyramidal honeycomb or oblate octahedrille, or the dual of the rectified cubic honeycomb.

It is analogous to the 2-dimensional tetrakis square tiling:

Tiling Dual Semiregular V4-8-8 Tetrakis Square.svg

Square bipyramidal honeycomb[edit]

Square bipyramidal honeycomb
Hexakis cubic honeycomb.png
Type Dual uniform honeycomb
Coxeter–Dynkin diagrams CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png
Cell Square bipyramids Square bipyramid.png
Faces Triangles
Space group
Fibrifold notation
Pm3m (221)
4:2
Coxeter group {\tilde{C}}_3, [4,3,4]
vertex figures Hexahedron.pngRhombic dodecahedron.jpg
CDel node f1.pngCDel 3.pngCDel node.pngCDel 4.pngCDel node.png, CDel node.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node.png
Dual Rectified cubic honeycomb
Properties Cell-transitive, Face-transitive

The square bipyramidal honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. John Horton Conway calls it an oblate octahedrille.

It can be seen as a cubic honeycomb with each cube subdivided by a center point into 6 elongated square pyramid cells. The original cubic honeycomb walls are removed, joining pairs of square pyramids into elongated square bipyramids (octahedron). Its vertex and edge framework is identical to the hexakis cubic honeycomb.

There is one type of plane with faces: a flattended triangular tiling with half of the triangles as holes. These cut face-diagonally through the original cubes. There are also square tiling plane that exist as nonface holes passing through the centers of the octahedral cells.

Tiling
plane
Koushi.JPG
Square tiling "holes"
Square bipyramidal honeycomb triangular plane.png
flattened triangular tiling
Symmetry p4m, [4,4] (*442) pmm, [∞,2,∞] (*2222)

Related honeycombs[edit]

It is dual to the rectified cubic honeycomb with octahedral and cuboctahedral cells:

Rectified cubic honeycomb.png

Phyllic disphenoidal honeycomb[edit]

Phyllic disphenoidal honeycomb
(No image)
Type Dual uniform honeycomb
Coxeter-Dynkin diagrams CDel node f1.pngCDel 4.pngCDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.png
Cell Half-turn tetrahedron diagram.png
Phyllic disphenoid
Faces Rhombus
Triangle
Space group
Fibrifold notation
Coxeter notation
Im3m (229)
8o:2
[[4,3,4]]
Coxeter group [4,3,4], {\tilde{C}}_3
vertex figures Disdyakis dodecahedron.pngOctagonal bipyramid.png
CDel node f1.pngCDel 3.pngCDel node f1.pngCDel 4.pngCDel node f1.png, CDel node f1.pngCDel 2.pngCDel node f1.pngCDel 4.pngCDel node f1.png
Dual Omnitruncated cubic honeycomb
Properties Cell-transitive, face-transitive

The phyllic disphenoidal honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. John Horton Conway calls it an eighth pyramidille.

Related honeycombs[edit]

It is dual to the omnitruncated cubic honeycomb:

Omnitruncated cubic honeycomb1.png

See also[edit]

References[edit]

  • Gibb, William (1990), "Paper patterns: solid shapes from metric paper", Mathematics in School 19 (3): 2–4 , reprinted in Pritchard, Chris, ed. (2003), The Changing Shape of Geometry: Celebrating a Century of Geometry and Geometry Teaching, Cambridge University Press, pp. 363–366, ISBN 0-521-53162-4 .
  • Senechal, Marjorie (1981), "Which tetrahedra fill space?", Mathematics Magazine (Mathematical Association of America) 54 (5): 227–243, doi:10.2307/2689983, JSTOR 2689983 .