HEDRAWEB

Continuing on the theme of symmetric figures, Ortwin Schenker suggested combining a Mystic (two Spectres) with a rotor: both are non-tilers.  Arnaud Chéritat was the first to respond with a periodic tiling.  He used a motif of three Mystics and one rotor.  Gaps are filled with single rotors: A cut-down version of Arnaud's motif can be used to create more complex periodic tilings.  The square unit can be of any size and with different arrangements: There are eight different combinations of the 'chair' tromino: Any one of these can be assembled to produce a nonperiodic tiling using a substitution rule: A beautiful flower motif: This simple motif produces an attractive spiral: These drawings were created using the latest stable version of Arnaud Chéritat's applet: https://www.math.univ-toulouse.fr/~cheritat/AppletsDivers/Monotile/generic/v4.3/

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/

Lately, I have been creating new shapes from gaps or voids which are left when placing other tiles such as Bob Nungester's gingerbread man. It transpires that one of these new shapes (below right) can be made up of  two of Bram Cohen's computer generated 'ducks' (below left) which I have named 'swans': Interestingly, swans can easily tile the plane in a variety of ways, displaying symmetries every which way you look.  Mirroring of the polygon doesn't work so it's effectively a single tile. There are five ways that an 'indent' can be filled: The first 'enclosure' (above top left) produces at least 17 different coronas (worked out by hand) and all can be expanded.  It is reasonable to assume that the other four will produce a similar number: I added a further three coronas to one of the examples above: This shape is most impressive when creating two or three-fold radial patterns as they are likely to be different every time. There are three

Ortwin Schenker named this shape the gingerbread man.  It  can only tile one way but hides a secret or two. This is one from hundreds of computer generated polygons supplied by Bob Nungester and Bram Cohen.  The shapes are all based on the Spectre, i.e., side lengths of one or two units and with similar angles.  I pulled out a couple of other tiles (below, far right) that coincidentally make up the top and lower half of the gingerbread man.  Both display mirror symmetry and can tessellate in various ways.   I later named them 'yin' and 'yang' These two polygons tile in similar ways.  There are six combinations of central cores (rotations of two or more tiles pivoting around a central point) for each.  First we have yin: And next we have yang: Interesting periodic patterns can be created easily by combining zigzag motifs. Also, more chaotic patterns can evolve from periodic strips. Below is a patch of yin with a yang void. And another patch but this time of yang with a y