The rhizomic tiles at Shooter’s Hill: an application of Truchet tiles

Abstract

We discuss patterns formed by terracotta tiles found on a housing terrasse in London, UK and note their relationship to Truchet tiles. We consider some of the mathematical questions that arise when combining the tiles into 2 × 2 arrays and their use in the creation of woven strips. In particular, we determine equivalence classes of patterns, and using graph theory and combinatorics we provide results regarding the number of woven strips that meet particular criteria.

GRAPHICAL ABSTRACT

KEYWORDS:

• Rhizomic tiles
• Truchet tiles
• equivalence classes
• weaving
• group theory
• combinatorics
• graph theory

Acknowledgements

We would like to thank the unknown architect and bricklayer of the terrassed housing in London, whose design, and perhaps absent-mindedness started this whole adventure. We also wish to thank the anonymous reviewers for their insightful comments.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 The relationship to the original Truchet tiles means that some of the findings described here can be applied to them.

2 The subscript i for a tetra-tile T_i represents the index of the tetra-tile as found in Appendix 1.

3 This formula is referred to as MacMahon’s formula in Graham et al. (1994, p. 140), although in Lucas (1961/1891, p. 503) it is credited to M. le colonel Moreau.

4 The ‘warp threads’ are those that are put in place on the loom before the weaving begins. They are usually in tension during the weaving process. The other threads are called weft threads since they are being ‘weft’ into the fabric.

5 See Appendix 2.

6 The graph theory terminology that we use can be found in West (2001).

7 It is a #P-complete problem (West, 2001).

8 Note that the simpler design has two tetra-tiles that are transitive: $T_2^{\mathrm{\prime }}$ and $T_4^{\mathrm{\prime }}$ in Figure 13.

9 Note: the elements of ${\mathcal{S}}_{43}$ are in black and white, with eliminated members of reflected pairs placed alongside and grayed out.

Additional information

Funding

This work was supported by NSERC [grant number: #2017-04401].