It is a part of a sequence of regular honeycombs with order-8 triangular tiling cells: {3,8,p}.
It is a part of a sequence of regular honeycombs with octagonal tiling vertex figures: {p,8,3}.
It is a part of a sequence of self-dual regular honeycombs: {p,8,p}.
Order-8-4 triangular honeycomb
In the geometry of hyperbolic 3-space, the order-8-4 triangular honeycomb (or 3,8,4 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,4}.
It has four order-8 triangular tilings, {3,8}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-8 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 {3,81,1}, Coxeter diagram, 
, with alternating types or colors of order-8 triangular tiling cells. In Coxeter notation the half symmetry is [3,8,4,1+] = [3,81,1].
Order-8-6 triangular honeycomb
| Order-8-6 triangular honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbols | {3,8,6}
{3,(8,3,8)} |
| Coxeter diagrams |  
 =  |
| Cells | {3,8}  |
| Faces | {3} |
| Edge figure | {6} |
| Vertex figure | {8,6} 
{(8,3,8)}  |
| Dual | {6,8,3} |
| Coxeter group | [3,8,6] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-8-6 triangular honeycomb (or 3,8,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,6}. It has infinitely many order-8 triangular tiling, {3,8}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-8 triangular tilings existing around each vertex in an order-6 octagonal tiling, {8,6}, vertex figure.
Order-8-infinite triangular honeycomb
| Order-8-infinite triangular honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbols | {3,8,∞}
{3,(8,∞,8)} |
| Coxeter diagrams | 
=  |
| Cells | {3,8} |
| Faces | {3} |
| Edge figure | {∞} |
| Vertex figure | {8,∞} 
{(8,∞,8)}  |
| Dual | {∞,8,3} |
| Coxeter group | [∞,8,3]
[3,((8,∞,8))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-8-infinite triangular honeycomb (or 3,8,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {3,8,∞}. It has infinitely many order-8 triangular tiling, {3,8}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-8 triangular tilings existing around each vertex in an infinite-order octagonal tiling, {8,∞}, vertex figure.
It has a second construction as a uniform honeycomb, Schläfli symbol {3,(8,∞,8)}, Coxeter diagram, = , with alternating types or colors of order-8 triangular tiling cells. In Coxeter notation the half symmetry is [3,8,∞,1+] = [3,((8,∞,8))].
Order-8-3 apeirogonal honeycomb
In the geometry of hyperbolic 3-space, the order-8-3 apeirogonal honeycomb (or ∞,8,3 honeycomb) a regular space-filling tessellation (or honeycomb). Each infinite cell consists of an order-8 apeirogonal tiling whose vertices lie on a 2-hypercycle, each of which has a limiting circle on the ideal sphere.
The Schläfli symbol of the apeirogonal tiling honeycomb is {∞,8,3}, with three order-8 apeirogonal tilings meeting at each edge. The vertex figure of this honeycomb is an octagonal tiling, {8,3}.
The "ideal surface" projection below is a plane-at-infinity, in the Poincaré half-space model of H3. It shows an Apollonian gasket pattern of circles inside a largest circle.
Order-8-6 hexagonal honeycomb
| Order-8-6 hexagonal honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbols | {6,8,6}
{6,(8,3,8)} |
| Coxeter diagrams |
= |
| Cells | {6,8}  |
| Faces | {6} |
| Edge figure | {6} |
| Vertex figure | {8,6} 
{(5,3,5)}  |
| Dual | self-dual |
| Coxeter group | [6,8,6]
[6,((8,3,8))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-8-6 hexagonal honeycomb (or 6,8,6 honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {6,8,6}. It has six order-8 hexagonal tilings, {6,8}, around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many hexagonal tilings existing around each vertex in an order-6 octagonal tiling vertex arrangement.
It has a second construction as a uniform honeycomb, Schläfli symbol {6,(8,3,8)}, Coxeter diagram, , with alternating types or colors of cells. In Coxeter notation the half symmetry is [6,8,6,1+] = [6,((8,3,8))].
Order-8-infinite apeirogonal honeycomb
| Order-8-infinite apeirogonal honeycomb |
|---|
| Type | Regular honeycomb |
| Schläfli symbols | {∞,8,∞}
{∞,(8,∞,8)} |
| Coxeter diagrams |
↔ |
| Cells | {∞,8}  |
| Faces | {∞} |
| Edge figure | {∞} |
| Vertex figure | {8,∞}
{(8,∞,8)} |
| Dual | self-dual |
| Coxeter group | [∞,8,∞]
[∞,((8,∞,8))] |
| Properties | Regular |
In the geometry of hyperbolic 3-space, the order-8-infinite apeirogonal honeycomb (or ∞,8,∞ honeycomb) is a regular space-filling tessellation (or honeycomb) with Schläfli symbol {∞,8,∞}. It has infinitely many order-8 apeirogonal tiling {∞,8} around each edge. All vertices are ultra-ideal (existing beyond the ideal boundary) with infinitely many order-8 apeirogonal tilings existing around each vertex in an infinite-order octagonal tiling vertex figure.
It has a second construction as a uniform honeycomb, Schläfli symbol {∞,(8,∞,8)}, Coxeter diagram, , with alternating types or colors of cells.
