Triakis truncated tetrahedron

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Triakis truncated tetrahedron
Triakis truncated tetrahedron.png
Type Plesiohedron
Faces 4 hexagons
12 isosceles triangles
Edges 30
Vertices 16
Conway notation k3tT
Dual polyhedron 16|Order-3 truncated triakis tetrahedron
Properties convex

Triakis truncated tetrahedron.gif

In geometry, the triakis truncated tetrahedron is a convex polyhedron made from 4 hexagons and 12 isosceles triangles. It can be used to tessellate three-dimensional space, making the triakis truncated tetrahedral honeycomb. [1] [2]

Contents

The triakis truncated tetrahedron is the shape of the Voronoi cell of the carbon atoms in diamond, which lie on the diamond cubic crystal structure. [3] [4] As the Voronoi cell of a symmetric space pattern, it is a plesiohedron. [5]

Construction

For space-filling, the triakis truncated tetrahedron can be constructed as follows:

  1. Truncate a regular tetrahedron such that the big faces are regular hexagons.
  2. Add an extra vertex at the center of each of the four smaller tetrahedra that were removed.

See also

Related Research Articles

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Triakis truncated tetrahedral honeycomb

The triakis truncated tetrahedral honeycomb is a space-filling tessellation in Euclidean 3-space made up of triakis truncated tetrahedra. It was discovered in 1914.

Order-6 cubic honeycomb

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, 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.

Ideal polyhedron Type of polyhedron in hyperbolic geometry

In three-dimensional hyperbolic geometry, an ideal polyhedron is a convex polyhedron all of whose vertices are ideal points, points "at infinity" rather than interior to three-dimensional hyperbolic space. It can be defined as the convex hull of a finite set of ideal points. An ideal polyhedron has ideal polygons as its faces, meeting along lines of the hyperbolic space.

References

  1. Conway, John H.; Burgiel, Heidi; Goodman-Strauss, Chaim (2008). The Symmetries of Things. p. 332. ISBN   978-1568812205.
  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. Föppl, L. (1914). "Der Fundamentalbereich des Diamantgitters". Phys. Z. 15: 191–193.
  4. Conway, John. "Voronoi Polyhedron". geometry.puzzles. Retrieved 20 September 2012.
  5. Grünbaum, Branko; Shephard, G. C. (1980), "Tilings with congruent tiles", Bulletin of the American Mathematical Society , New Series, 3 (3): 951–973, doi: 10.1090/S0273-0979-1980-14827-2 , MR   0585178 .