Jump to content

Order-7-3 triangular honeycomb

Add links
From Wikipedia, the free encyclopedia
This is an old revision of this page, as edited by Tomruen (talk | contribs) at 07:23, 26 September 2017 (→Geometry). The present address (URL) is a permanent link to this revision, which may differ significantly from the current revision.
Order-7-3 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,7,3}
Coxeter diagrams
Cells {3,7}
Faces {3}
Edge figure {3}
Vertex figure {7,3}
Dual Self-dual
Coxeter group [3,7,3]
Properties Regular

In the geometry of hyperbolic 3-space, the order-7-3 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,7,3}.

Geometry

It has three order-7 triangular tiling {3,7} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an heptagonal tiling vertex figure.


Poincaré disk model
Ideal surface 		240px
Upper half space model

It a part of a sequence of self-dual regular honeycombs: {p,7,p}.

{3,p,3} polytopes
Space S3 H3
Form Finite Compact Paracompact Noncompact
{3,p,3} {3,3,3} {3,4,3} {3,5,3} {3,6,3} {3,7,3} {3,8,3} ... {3,∞,3}
Image
Cells
{3,3}
{3,4}
{3,5}
{3,6}
{3,7}
{3,8}
{3,∞}
Vertex
figure
{3,3}
{4,3}
{5,3}
{6,3}
{7,3}
{8,3}
{∞,3}

It is also a part of a sequence of regular honeycombs with order-7 triangular tiling cells: {3,7,p}.

Order-7-4 triangular honeycomb

Order-7-4 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,7,4}
Coxeter diagrams
=
Cells {3,7}
Faces {3}
Edge figure {4}
Vertex figure {7,4}
r{7,7}
Dual {4,7,3}
Coxeter group [3,7,4]
Properties Regular

In the geometry of hyperbolic 3-space, the order-7-4 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,7,4}.

It has four order-7 triangular tilings, {3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an order-4 hexagonal tiling vertex arrangement.


Poincaré disk model
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {3,71,1}, Coxeter diagram, , with alternating types or colors of order-7 triangular tiling cells. In Coxeter notation the half symmetry is [3,7,4,1+] = [3,71,1].

Order-7-5 triangular honeycomb

Order-7-5 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,7,5}
Coxeter diagrams
Cells {3,7}
Faces {3}
Edge figure {5}
Vertex figure {7,5}
Dual {5,7,3}
Coxeter group [3,7,5]
Properties Regular

In the geometry of hyperbolic 3-space, the order-7-3 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,7,5}. It has five order-7 triangular tiling, {3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an order-5 heptagonal tiling vertex figure.


Poincaré disk model
Ideal surface

Order-7-6 triangular honeycomb

Order-7-6 triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,7,6}
{3,(7,3,7)}
Coxeter diagrams
=
Cells {3,7}
Faces {3}
Edge figure {6}
Vertex figure {7,6}
{(7,3,7)}
Dual {7,7,3}
Coxeter group [3,7,6]
Properties Regular

In the geometry of hyperbolic 3-space, the order-7-6 triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,7,6}. It has infinitely many order-7 triangular tiling, {3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an order-6 heptagonal tiling, {7,6}, vertex figure.


Poincaré disk model
Ideal surface

Order-7-infinite triangular honeycomb

Order-7-infinite triangular honeycomb
Type Regular honeycomb
Schläfli symbols {3,7,∞}
{3,(7,∞,7)}
Coxeter diagrams
=
Cells {3,7}
Faces {3}
Edge figure {∞}
Vertex figure {7,∞}
{(7,∞,7)}
Dual {∞,7,3}
Coxeter group [∞,7,3]
[3,((7,∞,7))]
Properties Regular

In the geometry of hyperbolic 3-space, the order-7-infinite triangular honeycomb is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,7,∞}. It has infinitely many order-7 triangular tiling, {3,7}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-7 triangular tilings existing around each vertex in an infinite-order heptagonal tiling, {7,∞}, vertex figure.


Poincaré disk model
Ideal surface

It has a second construction as a uniform honeycomb, Schläfli symbol {3,(7,∞,7)}, Coxeter diagram, = , with alternating types or colors of order-7 triangular tiling cells. In Coxeter notation the half symmetry is [3,7,∞,1+] = [3,((7,∞,7))].

See also

References

  • Coxeter, Regular Polytopes, 3rd. ed., Dover Publications, 1973. ISBN 0-486-61480-8. (Tables I and II: Regular polytopes and honeycombs, pp. 294–296)
  • The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678, ISBN 0-486-40919-8 (Chapter 10, Regular Honeycombs in Hyperbolic Space) Table III
  • Jeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 (Chapters 16–17: Geometries on Three-manifolds I,II)
  • George Maxwell, Sphere Packings and Hyperbolic Reflection Groups, JOURNAL OF ALGEBRA 79,78-97 (1982) [1]
  • Hao Chen, Jean-Philippe Labbé, Lorentzian Coxeter groups and Boyd-Maxwell ball packings, (2013)[2]
  • Visualizing Hyperbolic Honeycombs arXiv:1511.02851 Roice Nelson, Henry Segerman (2015)
Retrieved from "https://en.wikipedia.org/w/index.php?title=Order-7-3_triangular_honeycomb&oldid=802456539"