Triakis truncated tetrahedral honeycomb

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Triakis truncated tetrahedral honeycomb
Triakis truncated tetrahedral honeycomb.jpg
Cell type Triakis truncated tetrahedron
Face types hexagon
isosceles triangle
Coxeter group Ã3×2, [[3[4]]] (double)
Space group Fd3m (227)
PropertiesCell-transitive

The triakis truncated tetrahedral honeycomb is a space-filling tessellation (or honeycomb) in Euclidean 3-space made up of triakis truncated tetrahedra. It was discovered in 1914. [1] [2]

Contents

Voronoi tessellation

It is the Voronoi tessellation of the carbon atoms in diamond, [3] [4] which lie in the diamond cubic crystal structure.

Being composed entirely of triakis truncated tetrahedra, it is cell-transitive.

Relation to quarter cubic honeycomb

It can be seen as the uniform quarter cubic honeycomb where its tetrahedral cells are subdivided by the center point into 4 shorter tetrahedra, and each adjoined to the adjacent truncated tetrahedral cells.

HC A1-P1.png

See also

Related Research Articles

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The order-6 cubic honeycomb is a paracompact regular space-filling tessellation in hyperbolic 3-space. It is paracompact because it has vertex figures composed of an infinite number of facets, with all vertices as ideal points at infinity. With Schläfli symbol {4,3,6}, the honeycomb has six ideal cubes meeting along each edge. Its vertex figure is an infinite triangular tiling. Its dual is the order-4 hexagonal tiling honeycomb.

In geometry, the trigonal trapezohedral honeycomb is a uniform space-filling tessellation in Euclidean 3-space. Cells are identical trigonal trapezohedra or rhombohedra. Conway, Burgiel, and Goodman-Strauss call it an oblate cubille.

In geometry, a plesiohedron is a special kind of space-filling polyhedron, defined as the Voronoi cell of a symmetric Delone set. Three-dimensional Euclidean space can be completely filled by copies of any one of these shapes, with no overlaps. The resulting honeycomb will have symmetries that take any copy of the plesiohedron to any other copy.

References

  1. Föppl, L. (1914). "Der Fundamentalbereich des Diamantgitters". Phys. Z. 15: 191–193.
  2. Grünbaum, B.; Shephard, G. C. (1980). "Tilings with Congruent Tiles". Bull. Amer. Math. Soc. 3 (3): 951–973. doi: 10.1090/s0273-0979-1980-14827-2 .
  3. Conway, John. "Voronoi Polyhedron". geometry.puzzles. Retrieved 20 September 2012.
  4. Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). The Symmetries of Things. p. 332. ISBN   978-1568812205.