[math-fun] a Christmas carpet
Glad tilings of great joy: a Christmas carpet. Herewith the only greeting card I am ever likely to get around to sending, folks. Make the most of it --- but don't spy this at phone! In connection with Penrose tilings, `empire' denotes the subset of tile placements forced by some given subset of compatible placements; `country' denotes any connected region in such an empire. Modulo geometric congruence and de/inflation, there are just 7 distinct classes of country, identifiable by their simple polygonal boundary shapes. The graphic illustrates the empire of a `free wheel', or cartwheel for which a location is specified, but not an orientation. It somehow appeals to this supercilious misanthropist as vaguely festive, even to the extent of omitting any coffin for Tiny Tim: https://www.dropbox.com/s/ixfdnjax12kn7id/glad_tiling.png Amongst others, the recent paper by Fang Fang, Dugan Hammock, Klee Irwin, "Methods for Calculating Empires in Quasicrystals" (2017) https://www.mdpi.com/2073-4352/7/10/304/htm provides more concrete illustrations of 3 of the 7 home-country empires, populated with "Penrose rhomb" slings-and-arrows tiles, rather than visually preferable kites-and-darts. Figs. 9a, 9b, 9f illustrate instances of single worms (sandwiches) with increasing sizes; 9c, 8b, 9e cartwheels (coffins); 9d, 8a pentagonal stars. By the way, I am in the process of providing more material on this topic, available to anybody sufficiently reckless to enquire further. A Dickens of a lot more ... Bah! Humbug! WFL
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Fred Lunnon