HEDRAWEB

The second of four short write-ups on curious pairings of symmetric tiles. We start with a regular hexagon and a non-tiler borrowed from Bob Nungester's computational heap  which I named the seesaw. In combination they can produce an attractive periodic tiling: Hexagons will never touch one another but their orientations may differ (e.g., those highlighted in green).  I have also coloured in a few randomly selected groups of 4, 5 and 6 seesaws  surrounding hexagons (in red)  to give some idea of complexity: Below is (I believe) a picture listing of all unique coronas surrounding a hexagon that may appear in a tiling. A couple of six-fold symmetry examples: Two and three-fold symmetry.  All these tiling examples can easily continue: These drawings were created using the latest stable version of Arnaud Chéritat's applet: https://www.math.univ-toulouse.fr/~cheritat/AppletsDivers/Monotile/generic/v4.2/

The first of four short articles where I combine two shapes with mirror symmetry to form tilings. In the first example, I chose one of Bram Cohen's polygons which I have named the squid.  This shape  can tile easily (and rather boringly) on its own unless you include hexagonal formations of six squids (in grey) as shown below:  With this in mind, the squid now behaves differently and relies on  another shape which I have named the glider (in white), in order to tile the plane.  The two tiles in combination can produce an attractive periodic pattern  which can continue without limits and exhibits a structure that is possibly unique. The six squids that make up the smaller hexagons can be positioned in a clockwise or anticlockwise direction but everything else remains static: These drawings were created using the latest stable version of Arnaud Chéritat's applet: https://www.math.univ-toulouse.fr/~cheritat/AppletsDivers/Monotile/generic/v4.1/