logo for Joan Taylor's Tilings

Joan Taylor's Tilings

Home
Eve tiling and variants
Rhombus seed tilings
Square tilings
Triangle and 3-fold
symmetry tilings
Cross  & 3 key tiles
Eve to llama tiling
2 Squares tiling
Cube tiling
llama
Inflation of the llama island.  The llama is the smallest island of one chiral type that apears in the single hexagon tiling.
PDF 2.8MB
I'm Joan Taylor, an amateur mathematician living in Tasmania who became hooked on aperiodic tiling in 1990 upon a single glance at Penrose's rhombus tiling.  The idea of an aperiodic monotile captivated me from the beginning.  After many explorations and diversions I discovered a marked hexagonal tile which has gone some way towards this goal.  This knowledge was shared and extended with Joshua Socolar in 2009 and subsequently published:
   Socolar J.E.S.; Taylor J.M. An aperiodic hexagonal tile. J. Comb. Theory A 2011, 118, 2207-2231; arXiv:1003.4279.
    Taylor J.M. Aperiodicity of a functional monotile. Preprint, 2010; available from http://www.math.uni-bielefeld.de/sfb701/preprints/view/420.

The aim of this site is to make available some of my explorations and diversions to a wider public who may be able to find their own inspiration in one or more of my images.  Anyone is welcome to use this material provided they acknowledge where they sourced it.  I hope others will enjoy looking for matching rules for the tilings that lack them and making computer programs to draw them. 

All my drawings are hand drawn and only a few are in a polished state, most being working drawings.  Almost all were made with an aperiodic monotile in mind but matching rules proved elusive.  To add some interest to the collection I have supplied details of substitution rules for producing many of these tilings.  Most tilings make use of a mirror image tile which is coloured differently to distinguish it more easily - it could have been marked with a chiral motif - and make the tiling structure more apparent.

There are a couple of tricks for elucidating the tiling structure which the reader may care to try.  The most useful one is to view the image through almost closed eyes.  In this way details are less distracting.  Sometimes more information appears if the page is viewed from a shallow angle while it is slowly rotated.

To see a bigger image left click on the small image.  To obtain a high definition image left click on "PDF" below the small image.

Updates
11th October, 2015:  New material has been added to this page and to Cross & 3 key tiles.