WDS wrote: ********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. ****************<WDS Kids. You can't tell them anything these days. Here's Julian trash-talking to some of his fellow young delinquents: On 2015-06-19 09:55, Julian Ziegler Hunts wrote: Sure, but first I'd like to point out that he has not proved that any of these tile the plane, and I particularly do not believe that #1 does. He shows that each is the union of scaled copies of itself, but this does not suffice to prove that it tiles the plane—you would also need to prove that the tile has nonempty interior (the negligible-intersection part of being a tiling would follow). It is true that the scaling involved in the constructions would guarantee the tile has dimension 2, but only if the intersection between the scaled copies is negligible (being of smaller dimension would suffice, as would having measure zero if the whole has positive measure (which is not guaranteed by having dimension 2!)). And having dimension 2 does not guarantee that it has non-empty interior or will tile the plane. A simple counterexample is to remove all points with rational coordinates from the unit square; it is then the union of four non-intersecting half-scale copies of itself but does not tile the plane. Julian After In[2]:= SetAttributes[Root, Listable] In[3]:= Root[#^3 - # - 1, Range[3]] Out[3]= {Root[-1 - #1 + #1^3 &, 1], Root[-1 - #1 + #1^3 &, 2], Root[-1 - #1 + #1^3 &, 3]} In[4]:= N[%] Out[4]= {1.32471796, -0.662358979 - 0.562279512 I, -0.662358979 + 0.562279512 I} In[49]:= r = %3[[2]] Out[49]= Root[-1 - #1 + #1^3 &, 2] gosper.org/wdstile1.png (with indispensAble coaching from Julian) shows T/r separated by 2 from Tr + Tr^2 (where the actual separation appears to be 1/r) showing perhaps irremediable porosities and overwrites. --rwg