A geometric honeycomb is a space-filling of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions.
The order-6 tetrahedral honeycomb has a second construction as a uniform honeycomb, with Schläfli symbol {3,3[3]}. This construction contains alternating types, or colors, of tetrahedral cells. In Coxeter notation, this half symmetry is represented as [3,3,6,1+] ↔ [3,((3,3,3))], or [3,3[3]]: ↔ .
Related polytopes and honeycombs
The order-6 tetrahedral honeycomb is similar to the two-dimensional infinite-order triangular tiling, {3,∞}. Both tessellations are regular, and only contain triangles and ideal vertices.
The order-6 tetrahedral honeycomb is also a regular hyperbolic honeycomb in 3-space, and one of 11 which are paracompact.
N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
N.W. Johnson: Geometries and Transformations, (2018) Chapter 13: Hyperbolic Coxeter groups
March 09, 2023
order, tetrahedral, honeycomb, perspective, projection, viewwithin, poincaré, disk, modeltype, hyperbolic, regular, honeycombparacompact, uniform, honeycombschläfli, symbols, coxeter, diagrams, cells, faces, triangle, edge, figure, hexagon, vertex, figure, tri. Order 6 tetrahedral honeycombPerspective projection viewwithin Poincare disk modelType Hyperbolic regular honeycombParacompact uniform honeycombSchlafli symbols 3 3 6 3 3 3 Coxeter diagrams Cells 3 3 Faces triangle 3 Edge figure hexagon 6 Vertex figure triangular tilingDual Hexagonal tiling honeycombCoxeter groups V 3 displaystyle overline V 3 3 3 6 P 3 displaystyle overline P 3 3 3 3 Properties Regular quasiregularIn hyperbolic 3 space the order 6 tetrahedral honeycomb is a paracompact regular space filling tessellation or honeycomb It is paracompact because it has vertex figures composed of an infinite number of faces and has all vertices as ideal points at infinity With Schlafli symbol 3 3 6 the order 6 tetrahedral honeycomb has six ideal tetrahedra around each edge All vertices are ideal with infinitely many tetrahedra existing around each vertex in a triangular tiling vertex figure 1 A geometric honeycomb is a space filling of polyhedral or higher dimensional cells so that there are no gaps It is an example of the more general mathematical tiling or tessellation in any number of dimensions Honeycombs are usually constructed in ordinary Euclidean flat space like the convex uniform honeycombs They may also be constructed in non Euclidean spaces such as hyperbolic uniform honeycombs Any finite uniform polytope can be projected to its circumsphere to form a uniform honeycomb in spherical space Contents 1 Symmetry constructions 2 Related polytopes and honeycombs 2 1 Rectified order 6 tetrahedral honeycomb 2 2 Truncated order 6 tetrahedral honeycomb 2 3 Bitruncated order 6 tetrahedral honeycomb 2 4 Cantellated order 6 tetrahedral honeycomb 2 5 Cantitruncated order 6 tetrahedral honeycomb 2 6 Runcinated order 6 tetrahedral honeycomb 2 7 Runcitruncated order 6 tetrahedral honeycomb 2 8 Runcicantellated order 6 tetrahedral honeycomb 2 9 Omnitruncated order 6 tetrahedral honeycomb 3 See also 4 ReferencesSymmetry constructions Edit Subgroup relations The order 6 tetrahedral honeycomb has a second construction as a uniform honeycomb with Schlafli symbol 3 3 3 This construction contains alternating types or colors of tetrahedral cells In Coxeter notation this half symmetry is represented as 3 3 6 1 3 3 3 3 or 3 3 3 Related polytopes and honeycombs EditThe order 6 tetrahedral honeycomb is similar to the two dimensional infinite order triangular tiling 3 Both tessellations are regular and only contain triangles and ideal vertices The order 6 tetrahedral honeycomb is also a regular hyperbolic honeycomb in 3 space and one of 11 which are paracompact 11 paracompact regular honeycombs 6 3 3 6 3 4 6 3 5 6 3 6 4 4 3 4 4 4 3 3 6 4 3 6 5 3 6 3 6 3 3 4 4 This honeycomb is one of 15 uniform paracompact honeycombs in the 6 3 3 Coxeter group along with its dual the hexagonal tiling honeycomb 6 3 3 family honeycombs 6 3 3 r 6 3 3 t 6 3 3 rr 6 3 3 t0 3 6 3 3 tr 6 3 3 t0 1 3 6 3 3 t0 1 2 3 6 3 3 3 3 6 r 3 3 6 t 3 3 6 rr 3 3 6 2t 3 3 6 tr 3 3 6 t0 1 3 3 3 6 t0 1 2 3 3 3 6 The order 6 tetrahedral honeycomb is part of a sequence of regular polychora and honeycombs with tetrahedral cells 3 3 p polytopesSpace S3 H3Form Finite Paracompact NoncompactName 3 3 3 3 3 4 3 3 5 3 3 6 3 3 7 3 3 8 3 3 Image Vertexfigure 3 3 3 4 3 5 3 6 3 7 3 8 3 It is also part of a sequence of honeycombs with triangular tiling vertex figures Hyperbolic uniform honeycombs p 3 6 and p 3 3 vte Form Paracompact NoncompactName 3 3 6 3 3 3 4 3 6 4 3 3 5 3 6 5 3 3 6 3 6 6 3 3 7 3 6 7 3 3 8 3 6 8 3 3 3 6 3 3 Image Cells 3 3 4 3 5 3 6 3 7 3 8 3 3 Rectified order 6 tetrahedral honeycomb Edit Rectified order 6 tetrahedral honeycombType Paracompact uniform honeycombSemiregular honeycombSchlafli symbols r 3 3 6 or t1 3 3 6 Coxeter diagrams Cells r 3 3 3 6 Faces triangle 3 Vertex figure hexagonal prismCoxeter groups V 3 displaystyle overline V 3 3 3 6 P 3 displaystyle overline P 3 3 3 3 Properties Vertex transitive edge transitiveThe rectified order 6 tetrahedral honeycomb t1 3 3 6 has octahedral and triangular tiling cells arranged in a hexagonal prism vertex figure Perspective projection view within Poincare disk modelr p 3 6 vte Space H3Form Paracompact NoncompactName r 3 3 6 r 4 3 6 r 5 3 6 r 6 3 6 r 7 3 6 r 3 6 Image Cells 3 6 r 3 3 r 4 3 r 5 3 r 6 3 r 7 3 r 3 Truncated order 6 tetrahedral honeycomb Edit Truncated order 6 tetrahedral honeycombType Paracompact uniform honeycombSchlafli symbols t 3 3 6 or t0 1 3 3 6 Coxeter diagrams Cells t 3 3 3 6 Faces triangle 3 hexagon 6 Vertex figure hexagonal pyramidCoxeter groups V 3 displaystyle overline V 3 3 3 6 P 3 displaystyle overline P 3 3 3 3 Properties Vertex transitiveThe truncated order 6 tetrahedral honeycomb t0 1 3 3 6 has truncated tetrahedron and triangular tiling cells arranged in a hexagonal pyramid vertex figure Bitruncated order 6 tetrahedral honeycomb Edit The bitruncated order 6 tetrahedral honeycomb is equivalent to the bitruncated hexagonal tiling honeycomb Cantellated order 6 tetrahedral honeycomb Edit Cantellated order 6 tetrahedral honeycombType Paracompact uniform honeycombSchlafli symbols rr 3 3 6 or t0 2 3 3 6 Coxeter diagrams Cells r 3 3 r 3 6 x 6 Faces triangle 3 square 4 hexagon 6 Vertex figure isosceles triangular prismCoxeter groups V 3 displaystyle overline V 3 3 3 6 P 3 displaystyle overline P 3 3 3 3 Properties Vertex transitiveThe cantellated order 6 tetrahedral honeycomb t0 2 3 3 6 has cuboctahedron trihexagonal tiling and hexagonal prism cells arranged in an isosceles triangular prism vertex figure Cantitruncated order 6 tetrahedral honeycomb Edit Cantitruncated order 6 tetrahedral honeycombType Paracompact uniform honeycombSchlafli symbols tr 3 3 6 or t0 1 2 3 3 6 Coxeter diagrams Cells tr 3 3 t 3 6 x 6 Faces square 4 hexagon 6 Vertex figure mirrored sphenoidCoxeter groups V 3 displaystyle overline V 3 3 3 6 P 3 displaystyle overline P 3 3 3 3 Properties Vertex transitiveThe cantitruncated order 6 tetrahedral honeycomb t0 1 2 3 3 6 has truncated octahedron hexagonal tiling and hexagonal prism cells connected in a mirrored sphenoid vertex figure Runcinated order 6 tetrahedral honeycomb Edit The bitruncated order 6 tetrahedral honeycomb is equivalent to the bitruncated hexagonal tiling honeycomb Runcitruncated order 6 tetrahedral honeycomb Edit The runcitruncated order 6 tetrahedral honeycomb is equivalent to the runcicantellated hexagonal tiling honeycomb Runcicantellated order 6 tetrahedral honeycomb Edit The runcicantellated order 6 tetrahedral honeycomb is equivalent to the runcitruncated hexagonal tiling honeycomb Omnitruncated order 6 tetrahedral honeycomb Edit The omnitruncated order 6 tetrahedral honeycomb is equivalent to the omnitruncated hexagonal tiling honeycomb See also EditConvex uniform honeycombs in hyperbolic space Regular tessellations of hyperbolic 3 space Paracompact uniform honeycombsReferences Edit Coxeter The Beauty of Geometry 1999 Chapter 10 Table III 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 Chapter 16 17 Geometries on Three manifolds I II Norman Johnson Uniform Polytopes Manuscript N W Johnson The Theory of Uniform Polytopes and Honeycombs Ph D Dissertation University of Toronto 1966 N W Johnson Geometries and Transformations 2018 Chapter 13 Hyperbolic Coxeter groups Retrieved from https en wikipedia org w index php title Order 6 tetrahedral honeycomb amp oldid 1081823974, wikipedia, wiki, book, books, library,