Order-8 triangular tiling

In geometry, the order-8 triangular tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {3,8}, having eight regular triangles around each vertex.

Order-8 triangular tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration38
Schläfli symbol{3,8}
(3,4,3)
Wythoff symbol8 | 3 2
4 | 3 3
Coxeter diagram
Symmetry group[8,3], (*832)
[(4,3,3)], (*433)
[(4,4,4)], (*444)
DualOctagonal tiling
PropertiesVertex-transitive, edge-transitive, face-transitive

Uniform colorings

The half symmetry [1+,8,3] = [(4,3,3)] can be shown with alternating two colors of triangles:

Symmetry
Octagonal tiling with *444 mirror lines, .

From [(4,4,4)] symmetry, there are 15 small index subgroups (7 unique) by mirror removal and alternation operators. Mirrors can be removed if its branch orders are all even, and cuts neighboring branch orders in half. Removing two mirrors leaves a half-order gyration point where the removed mirrors met. In these images fundamental domains are alternately colored black and white, and mirrors exist on the boundaries between colors. Adding 3 bisecting mirrors across each fundamental domains creates 832 symmetry. The subgroup index-8 group, [(1+,4,1+,4,1+,4)] (222222) is the commutator subgroup of [(4,4,4)].

A larger subgroup is constructed [(4,4,4*)], index 8, as (2*2222) with gyration points removed, becomes (*22222222).

The symmetry can be doubled to 842 symmetry by adding a bisecting mirror across the fundamental domains. The symmetry can be extended by 6, as 832 symmetry, by 3 bisecting mirrors per domain.

Small index subgroups of [(4,4,4)] (*444)
Index 1 2 4
Diagram
Coxeter [(4,4,4)]
 [(1+,4,4,4)]
= 		[(4,1+,4,4)]
= 		[(4,4,1+,4)]
= 		[(1+,4,1+,4,4)]
 [(4+,4+,4)]
Orbifold *444 *4242 2*222 222×
Diagram
Coxeter [(4,4+,4)]
 [(4,4,4+)]
 [(4+,4,4)]
 [(4,1+,4,1+,4)]
 [(1+,4,4,1+,4)]
=
Orbifold 4*22 2*222
Direct subgroups
Index 2 4 8
Diagram
Coxeter [(4,4,4)]+
 [(4,4+,4)]+
= 		[(4,4,4+)]+
= 		[(4+,4,4)]+
= 		[(4,1+,4,1+,4)]+
=
Orbifold 444 4242 222222
Radical subgroups
Index 8 16
Diagram
Coxeter [(4,4*,4)] [(4,4,4*)] [(4*,4,4)] [(4,4*,4)]+ [(4,4,4*)]+ [(4*,4,4)]+
Orbifold *22222222 22222222
The {3,3,8} honeycomb has {3,8} vertex figures.

From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal and order-8 triangular tilings.

Drawing the tiles colored as red on the original faces, yellow at the original vertices, and blue along the original edges, there are 10 forms.

It can also be generated from the (4 3 3) hyperbolic tilings:

gollark: Okay, sure, it's probably™ fine, it can always be changed later probably.
gollark: Also, my *server* has a HDD, although I guess the DB is tiny and will likely just be cached in memory.
gollark: I am not worried about latency as much as throughput.
gollark: Okay, just run the function with some kind of fake message events, at, say, 100 message/s volume (that's the likely maximum...) and see if it explodes.
gollark: Wait, no, this can actually be benchmarked if you can... send fake messages somehow, hmmm.

See also
  • Order-8 tetrahedral honeycomb
  • Tilings of regular polygons
  • List of uniform planar tilings
  • List of regular polytopes

References
    • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 19, The Hyperbolic Archimedean Tessellations)
    • "Chapter 10: Regular honeycombs in hyperbolic space". The Beauty of Geometry: Twelve Essays. Dover Publications. 1999. ISBN 0-486-40919-8. LCCN 99035678.
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