“Trigonal trapezohedral honeycomb”的意思、由来-开放百科全书

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Trigonal trapezohedral honeycomb

释义

Related honeycombs and tilings
Dual tiling
See also
References

Trigonal trapezohedral honeycomb
(No image)
Type	Dual uniform honeycomb
Coxeter-Dynkin diagrams	labelh\|node_fh\|4\|node\|3\|node\|4\|node_fh\|labelh}}
Cell	Trigonal trapezohedron (1/4 of rhombic dodecahedron)
Faces	Rhombus
Space group	3}}m (227)
Coxeter group	×2, [4]">3^[4] (double)
vertex figures	{{CDD>node_fh\|4\|node\|3\|node}} \| {{CDD\|node_f1\|3\|node\|4\|node_fh}}
Dual	Quarter cubic honeycomb
Properties	Cell-transitive, Face-transitive

The trigonal trapezohedral honeycomb is a uniform space-filling tessellation (or honeycomb) in Euclidean 3-space. Cells are identical trigonal trapezohedron or rhombohedra. John Horton Conway calls it an oblate cubille.

Related honeycombs and tilings

This honeycomb can be seen as a rhombic dodecahedral honeycomb, with the rhombic dodecahedra dissected with its center into 4 trigonal trapezohedra or rhombohedra.

rhombic dodecahedral honeycomb	Rhombic dodecahedra dissection	Rhombic net

It is analogous to the regular hexagonal being dissectable into 3 rhombi and tiling the plane as a rhombille. The rhombille tiling is actually an orthogonal projection of the trigonal trapezohedral honeycomb. A different orthogonal projection produces the quadrille where the rhombi are distorted into squares.

Dual tiling

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

References

John H. Conway, Heidi Burgiel, Chaim Goodman-Strauss, (2008) The Symmetries of Things, {{isbn|978-1-56881-220-5}} (Chapter 21, Naming the Archimedean and Catalan polyhedra and tilings, Architectonic and Catoptric tessellations, p 292-298, includes all the nonprismatic forms)
Branko Grünbaum, Uniform tilings of 3-space. Geombinatorics 4(1994), 49 - 56.

1 : Honeycombs (geometry)

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Related honeycombs and tilings

Dual tiling

See also

References