Truncated order-4 heptagonal tiling

In geometry, the truncated order-4 heptagonal tiling is a uniform tiling of the hyperbolic plane. It has Schläfli symbol of t{7,4}.

Truncated heptagonal tiling

Poincaré disk model of the hyperbolic plane
TypeHyperbolic uniform tiling
Vertex configuration4.14.14
Schläfli symbolt{7,4}
Wythoff symbol7
2 7 7 |
Coxeter diagram
or
Symmetry group[7,4], (*742)
[7,7], (*772)
DualOrder-7 tetrakis square tiling
PropertiesVertex-transitive

Constructions

There are two uniform constructions of this tiling, first by the [7,4] kaleidoscope, and second by removing the last mirror, [7,4,1+], gives [7,7], (*772).

Two uniform constructions of 4.7.4.7
Name Tetraheptagonal Truncated heptahexagonal
Image
Symmetry [7,4]
(*742)
 [7,7] = [7,4,1+]
(*772)
=
Symbol t{7,4} tr{7,7}
Coxeter diagram

Symmetry

There is only one simple subgroup [7,7]+, index 2, removing all the mirrors. This symmetry can be doubled to 742 symmetry by adding a bisecting mirror.

Small index subgroups of [7,7]
Type Reflectional Rotational
Index 1 2
Diagram
Coxeter
(orbifold)		 [7,7] =
(*772)		 [7,7]+ =
(772)

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.

See also
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