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The gingerbread man

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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
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The golem

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This shape which I have named the golem, was hand picked from hundreds of Spectre-like polygons (Spectroids) that were sent my way via Ortwin Schenker, Bob Nungester and Bram Cohen.   This golem has the same number of sides as the Spectre and also shares internal angles of 90, 120 and 270 degrees. Below left is the simplest of the periodic tilings.  Positioning of new tiles is forced in most cases.  That said, the pattern can deviate at any time by placing other rotations where there are choices (as shown in the example bottom right). The golem is a delight to tessellate (you can't go wrong!).  Below are a few more examples of periodic tilings.  These can be sewn together to make larger chaotic tilings with a bit of tweaking. Next up are a couple of radial patterns.  The example on the right is (I think) quite unusual in that three golem segments are pointing towards the centre whilst the other three are pointing outwards. The golem has mirror symmetry and fits to an underlying reg
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Determining chess 960 starting positions with a standard die (no rethrows)

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Chess 960 is a chess game variant invented by Bobby Fischer in 1996.   In Chess 960, the pawns are lined up on the second rank as in standard chess.   The first rank pieces are positioned randomly whilst conforming to certain caveats - the king must be placed somewhere between the rooks to allow castling; the bishops must stand on opposite coloured squares. I've worked out a system that allows me to select a starting arrangement for the first rank pieces using a standard six sided die without the need for rethrows.  My dice procedure is different from that of  I ngo Althöfer's (devised in 1998), where rethrows can occur.   Some information on how his system works can be seen here https://www.chessvariants.org/diffsetup.dir/fischer-random-setup.html The die I use to generate the random piece arrangement is a standard cube/hexahedron (one of the platonic solids).  O nly five or six throws of the die are needed to formulate one of the 960 starting positions. Rethrows add to the t
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Pseudo Islamic art-like patterns

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Three shapes, the b owtie, regular pentagon and rhombus (with markings)  are all that are needed to produce Islamic art-like patterns.   The large regular pentagon (on left) contains  three  smaller pentagons,  two  rhombi and one  bowtie. Ten pointed star (on right) is made up of  three  bowties,  two  regular pentagons and  one  rhombus. Ten rotations of each... Here's the sort of thing you can do... The batman below comprises  two regular pentagons,  one  rhombus and one bowtie. Below - one example (plain/decorative) of a periodic batman pattern and another including rhombi.
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