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Order-7-3 triangular honeycomb

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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

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)
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