The hexagonal tiling honeycomb , also known as the order-3 hexagonal tiling honeycomb , is a paracompact regular tiling of 3D hyperbolic space. Each cell is a hexagonal tiling whose vertices lie on a horosphere, a surface in hyperbolic space that approaches a single ideal point at infinity. 3 hexagonal tilings meet at each edge, and 4 meet at each vertex.
It can be seen as a truncated triangular tiling honeycomb or bitruncated order-6 hexagonal tiling honeycomb .
This honeycomb can be alternated into an alternated hexagonal tiling honeycomb , which is uniform.
The hexagonal tiling honeycomb has the following Coxeter diagrams :
x6o3o3o ( ) (full symmetry)
x3x6o3o ( ) (as truncated triangular tiling honeycomb )
o6x3x6o ( ) (as bitruncated order-6 hexagonal tiling honeycomb )
o6x3x3x3*b ( ) (half symmetry of bitruncated order-6 hexagonal tiling honeycomb)
x3x3x3x3*a3*c *b3*d ( ) (quarter symmetry of bitruncated order-6 hexagonal tiling honeycomb)
The hexagonal faces of the hexagonal tiling honeycomb form the regular skew polyhedron {6,4∣6} .
truncationsName OBSA Schläfli symbol CD diagram Image Hexagonal tiling honeycomb hexah {6,3,3} Truncated hexagonal tiling honeycomb thexah t{6,3,3} Rectified hexagonal tiling honeycomb rihexah r{6,3,3} Tetrahedral-hexagonal tiling honeycomb tehexah 2t{6,3,3} Rectified tetrahedral honeycomb rath r{3,3,6} Truncated tetrahedral honeycomb tath t{3,3,6} Tetrahedral honeycomb thon {3,3,6} Small rhombated hexagonal tiling honeycomb srihexah rr{6,3,3} Great rhombated hexagonal tiling honeycomb grihexah tr{6,3,3} Small rhombated tetrahedral honeycomb srath rr{3,3,6} Great rhombated tetrahedral honeycomb grath tr{3,3,6} Small prismated hexagonal tiling honeycomb sidpithexah t0,3 {6,3,3} Prismatorhombated tetrahedral honeycomb prath t0,1,3 {6,3,3} Prismatorhombated hexagonal tiling honeycomb prihexah t0,1,3 {3,3,6} Great prismated hexagonal tiling honeycomb gidpithexah t0,1,2,3 {6,3,3}
truncationsName OBSA Schläfli symbol CD diagram Image Triangular tiling honeycomb trah {3,6,3} Truncated triangular tiling honeycomb = Hexagonal tiling honeycomb hexah t{3,6,3} Rectified triangular tiling honeycomb ritrah r{3,6,3} Distriangular tiling honeycomb ditrah 2t{3,6,3} Rectified triangular tiling honeycomb ritrah r{3,6,3} Truncated triangular tiling honeycomb = Hexagonal tiling honeycomb hexah t{3,6,3} Triangular tiling honeycomb trah {3,6,3} Small rhombated triangular tiling honeycomb sritrah rr{3,6,3} Great rhombated triangular tiling honeycomb gritrah tr{3,6,3} Small rhombated triangular tiling honeycomb sritrah rr{3,6,3} Great rhombated triangular tiling honeycomb gritrah tr{3,6,3} Small prismated triangular tiling honeycomb spidditrah t0,3 {3,6,3} Prismatorhombated triangular tiling honeycomb pritrah t0,1,3 {3,6,3} Prismatorhombated triangular tiling honeycomb pritrah t0,1,3 {3,6,3} Great prismated triangular tiling honeycomb spidditrah t0,1,2,3 {3,6,3}
truncationsName OBSA Schläfli symbol CD diagram Image Order-6 hexagonal tiling honeycomb hihexah {6,3,6} Rectified order-6 hexagonal tiling honeycomb rihihexah r{6,3,6} Rectified order-6 hexagonal tiling honeycomb rihihexah r{6,3,6} Order-6 hexagonal tiling honeycomb hihexah {6,3,6} Truncated order-6 hexagonal tiling honeycomb thihexah t{6,3,6} Small rhombated order-6 hexagonal tiling honeycomb srihihexah rr{6,3,6} Small prismated order-6 hexagonal tiling honeycomb spiddihexah t0,3 {6,3,6} Dishexagonal tiling honeycomb = Hexagonal tiling honeycomb hexah 2t{6,3,6} Small rhombated order-6 hexagonal tiling honeycomb srihihexah rr{6,3,6} Truncated order-6 hexagonal tiling honeycomb thihexah t{6,3,6} Great rhombated order-6 hexagonal tiling honeycomb grihihexah tr{6,3,6} Prismatorhombated order-6 hexagonal tiling honeycomb prihihexah t0,1,3 {6,3,6} Prismatorhombated order-6 hexagonal tiling honeycomb prihihexah t0,1,3 {6,3,6} Great rhombated order-6 hexagonal tiling honeycomb grihihexah tr{6,3,6} Great prismated order-6 hexagonal tiling honeycomb gipiddihexah t0,1,2,3 {6,3,6}
Convex polychoraSchläfli-Hess polychora Honeycombs Compounds Hyperbolic honeycombs
Skew polychora
Full rank
Finite
Honeycombs Hyperbolic honeycombs
Higher dimensional
Finite Apeirochora
(0,2,2,3) (0,2,3,2) (0,3,2,3) (2,3,2,3) (2,3,3,3) (3,1,3,3) (3,2,3,2) (3,2,3,3) (3,3,2,3)
Abstract polychoraComplex polychora