Three Fractal Tiles =====Warren D. Smith====18 June 2015======== ********Fractal Tile #1:********* Let R obey R^3=R+1, R=-0.662358978622373+0.562279512062301*i arg(R)=139.671923191730 degrees |R|=0.86883696183271 |R|^2=0.75487766625 |R|^(-2)=1.32471795724474602596091 which is the real root of the original cubic, and wikipedia calls this the "plastic number." Because of the theorem that every non-integer algebraic integer is irrational, we know all roots of this cubic are irrational. Hence |R| and |R|^2 are irrational. Hence also any integer linear combination of the roots of the cubic, is irrational or zero. Then the "linear forms in logarithms" theorem by Alan Baker shows arg(R) is irrational measured in degrees, also. Then let tile T be defined by T = SUM(j>=0) R^j b_j where the bits b_j each are either 0 or 1, and the allowed bit sequences b0, b1, b2, ... are defined recursively by appending either 10 or 100 to the left of an allowed bit sequence. The number Q(N) of N-bit long allowed sequences then obeys the recurrence Q(N)=Q(N-2)+Q(N-3) and lim Q(N)^(1/N)=1.3247... Obviously T is a bounded set. Then T is tiled by T*R^2 and T*R^3, two rescaled & rotated copies of itself. By recursing forever, it follows that the three tiles T, T*R, T*R^2 together suffice to tile the plane. T is fractal in shape. The irrationality of arg(R) shows this tiling is necessarily aperiodic. ********Fractal Tile #2:********* Let R be a complex non-real root of x^8+3*x^6+4*x^4+2*x^2+1 = 0 with |R|^2=0.61803...=1/GoldenRatio and arg(R) is a nontrivial integer multiple of 72 degrees. (I have constructed this polynomial so those two things happen.) Define tile T by T = SUM(j>=0) R^j b_j where the bits b_j each are either 0 or 1, and the allowed bit sequences b0, b1, b2, ... are any binary sequence not containing two consecutive 1s. The number of N-bit long allowed sequences ending with a 1 at the most-significant end then is 1,1,2,3,5,8,13,21,... for N=1,2,3,... (Fibonacci numbers). Incidentally, it seems what I'd been calling the "Fibonaccci number system" is called "Zeckendorf's theorem" by wikipedia and "Zeckendorf Representation" by Weisstein, which is because it was invented way before Zeckendorf 1972, e.g. published by C.G. Lekkerkerker in Simon Stevin 29 (1952) 190-195. The allowed N-long sequences can be got by appending 1 to the left of an (N-1)-long one after replacing the 1 at its left end by 0; and appending 10 to the left of an (N-2)-long one. Then T is tiled by T*R and T*R^2, two rescaled & rotated copies of itself. By recursing forever, it follows that the two tiles T and S=T*R together suffice to tile the plane. T is fractal in shape. Is this tiling necessarily aperiodic? It does not involve any irrational (in degrees) rotations, all rotation angles are multiples of 72 degrees. But the two basic tiles T and S arise in the recursive process in numbers #T #S 1 0 to tile T 1 1 to tile T/R 2 1 to tile T/R^2 3 2 to tile T/R^3 5 3 to tile T/R^4 8 5 to tile T/R^5 etc, each number is sum of two immediately above it; i.e. the ratio of the two tile-counts approaches the golden ratio, which is irrational, hence yes, this tiling IS aperiodic. ********Fractal Tile #3:********* Let R be 1/sqrt(GoldenRatio) times 0.3319277206059928507595641669946106715942093770*i-0.9433048225750305942590992480932547489056220928 where the latter is the nonreal root of "Lehmer's polynomial" x^10+x^9-x^7-x^6-x^5-x^4-x^3+x+1 with |x|=1 and arg(x)=160.61417788097273775412563559435163395654262868 degrees. (I have not worked out the minimal polynomial of this R.) Now do the same construction as for Tile #2, just using this R instead of that R. Then T is tiled by T*R and T*R^2, two rescaled & rotated copies of itself. By recursing forever, it follows that the two tiles T and S=T*R together suffice to tile the plane. T is fractal in shape. Is this tiling necessarily aperiodic? Yes because it involves irrational angles of rotation... because it is known that the Lehmer root of unit norm is not a root of unity, we know arg(R) is irrational in degree measure. ********About Joerg Arndt*********** He clarified some of the things he'd said... JA's computer has found thousands of new tilings arising from "weird complex-radix number systems" in as-yet-unpublished work. He has several ways to describe each one. His counts range from thousands to hundreds of thousands depending exactly what you count. Probably some or all of the tilings I just invented, already were in Arndt's collection. Arndt also pointed out the "Rauzy fractal" https://en.wikipedia.org/wiki/Rauzy_fractal is also a tile of this ilk, 3 scaled copies of itself will tile the plane (also will tile itself) similarly in nature to my example #1 above. Wikipedia gives a picture. My examples #2 and #3 are "cooler than Penrose" in the sense they involve only 2 tiles, are inherently aperiodic, and only 1 tile shape. Warning: I have not drawn pictures of these tilings, I merely proved it works; so caveat emptor. Maybe somebody will draw pictures now?