Octagonal tiling

In geometry, the octagonal tiling is a regular tiling of the hyperbolic plane. It is represented by Schläfli symbol of {8,3}, having three regular octagons around each vertex. It also has a construction as a truncated order-8 square tiling, t{4,8}.

Octagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic regular tiling
Vertex configuration83
Schläfli symbol{8,3}
t{4,8}
Wythoff symbol3 | 8 2
2 8 | 4
4 4 4 |
Coxeter diagram

Symmetry group[8,3], (*832)
[8,4], (*842)
[(4,4,4)], (*444)
DualOrder-8 triangular tiling
PropertiesVertex-transitive, edge-transitive, face-transitive

Uniform colorings

Like the hexagonal tiling of the Euclidean plane, there are 3 uniform colorings of this hyperbolic tiling. The dual tiling V8.8.8 represents the fundamental domains of [(4,4,4)] symmetry.

Regular Truncations

{8,3}

t{4,8}

t{4[3]}
= =
Dual tiling

{3,8}
=

=

= =

This tiling is topologically part of sequence of regular polyhedra and tilings with Schläfli symbol {n,3}.

And also is topologically part of sequence of regular tilings with Schläfli symbol {8,n}.

n82 symmetry mutations of regular tilings: 8n
Space Spherical Compact hyperbolic Paracompact
Tiling
Config. 8.8 83 84 85 86 87 88 ...8

From a Wythoff construction there are ten hyperbolic uniform tilings that can be based from the regular octagonal tiling.

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.

See also

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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