Order-6-4 triangular honeycomb
In the geometry of hyperbolic 3-space , the order-6-4 triangular honeycomb is a regular space-filling tessellation (or honeycomb ) with Schläfli symbol {3,6,4}.
Geometry
It has four triangular tiling {3,6} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-4 hexagonal tiling vertex arrangement .
It has a second construction as a uniform honeycomb, Schläfli symbol {3,61,1 }, Coxeter diagram, Coxeter notation the half symmetry is [3,6,4,1+ ] = [3,61,1 ].
It a part of a sequence of regular polychora and honeycombs with triangular tiling cells : {3,6,p }
{3,6,p} polytopes
Space
H3
Form
Paracompact
Noncompact
Name
{3,6,3} {3,6,4} {3,6,5} {3,6,6} ... {3,6,∞}
Image
Vertex
{6,3} {6,4} {6,5} {6,6} {6,∞}
Order-6-5 triangular honeycomb
In the geometry of hyperbolic 3-space , the order-6-3 triangular honeycomb is a regular space-filling tessellation (or honeycomb ) with Schläfli symbol {3,6,5}. It has five triangular tiling , {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-5 hexagonal tiling vertex arrangement .
Order-6-6 triangular honeycomb
Order-6-6 triangular honeycomb
Type
Regular honeycomb
Schläfli symbols {3,6,6}
Coxeter diagrams
Cells
{3,6}
Faces
{3}
Edge figure
{6}
Vertex figure
{6,6}
Dual
{6,6,3}
Coxeter group [3,6,6]
Properties
Regular
In the geometry of hyperbolic 3-space , the order-6-6 triangular honeycomb is a regular space-filling tessellation (or honeycomb ) with Schläfli symbol {3,6,6}. It has infinitely many triangular tiling , {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an order-6 triangular tiling vertex arrangement .
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(6,3,6)}, Coxeter diagram, + ] = [3,((6,3,6))].
Order-6-infinite triangular honeycomb
Order-6-infinite triangular honeycomb
Type
Regular honeycomb
Schläfli symbols {3,6,∞}
Coxeter diagrams
Cells
{3,6}
Faces
{3}
Edge figure
{∞}
Vertex figure
{6,∞}
Dual
{∞,6,3}
Coxeter group [∞,6,3]
Properties
Regular
In the geometry of hyperbolic 3-space , the order-6-infinite triangular honeycomb is a regular space-filling tessellation (or honeycomb ) with Schläfli symbol {3,6,∞}. It has infinitely many triangular tiling , {3,6}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many triangular tilings existing around each vertex in an infinite-order triangular tiling vertex arrangement .
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(6,∞,6)}, Coxeter diagram, + ] = [3,((6,∞,6))].
See also
References
Coxeter , Regular Polytopes ISBN 0-486-61480-8 The Beauty of Geometry: Twelve Essays (1999), Dover Publications, LCCN 99-35678 , ISBN 0-486-40919-8 Regular Honeycombs in Hyperbolic Space Archived 2016-06-10 at the Wayback Machine ) Table IIIJeffrey R. Weeks The Shape of Space, 2nd edition ISBN 0-8247-0709-5 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)
External links
Spherical Video: {3,6,∞} honeycomb with parabolic Möbius transform YouTube , Roice NelsonJohn Baez , Visual insights : {7,3,3} Honeycomb (2014/08/01) {7,3,3} Honeycomb Meets Plane at Infinity (2014/08/14)Danny Calegari , Kleinian, a tool for visualizing Kleinian groups, Geometry and the Imagination 4 March 2014. [3]
