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The title refers to tiles as devised by the British physicist Sir Roger Penrose (who has a keen interest in recreational mathematics), and as may be imagined, are somewhat more involved than with more 'ordinary' examples. More specifically, these are not one but two tiles that in conjunction tessellate in an non-periodic manner (hence their uniqueness), of which an excellent introduction, at a broadly accessible level is to be found in an Scientific American article of January 1977 'Extraordinary nonperiodic tiling that enriches the theory of tiles'. A feature of these tiles for life-like motif purposes is that due to their non-periodic nature, each line has to represent four elements, this being in contrast to the 'normal' two of those without this aspect. Consequently, this makes the finding of quality life-like motifs most difficult.
Examples of 'Penrose tilings' include:
Kites and Darts This set of tiles consists of two tiles derived from a rhombus, further details of which are in the above article. The tiles can be arranged in numerous distinct ways, forming specific patterns of a non-periodic nature which have their own titles, with examples such as 'Cartwheels', 'Infinite Sun' and 'Infinite Star'. These first came to the attention of recreational mathematicians in the above article, of which they are liberally discussed and illustrated. 'Thick and Thin' Rhombs These consist of two rhombs which tile in a non-periodic manner, also discussed in the 1977 article (page 120). 'Loaded Wheelbarrow' Tile A lesser known tiling is that of the 'loaded wheelbarrow', based on a single tile, so named by Martin Gardner (on account of its shape bearing a resemblance, albeit somewhat rudimentary) in an Scientific American article of August 1975 'More About Tiling the Plane: The Possibilities of Polyominoes, Polyiamonds and Polyhexes' in which a single tile is shown (this being set as a puzzle, with the answer and the tessellation shown in the next month's article). The examples following show my efforts at forming life-like creatures with these. As such, these are generally of an inherently lower quality than with my normal standard, this being due to the tiles unique 'arrangement', in which a single line is repeated in a manner that is not conducive to the addition of life-like forms. This being so, their 'limitations' in this matter thus have to be accepted, albeit this should not be seen as an excuse for inferior examples.
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