Possible idea: re-express/re-examine that Minsky iteration and/or the Penrose tiling using numbers/coordinates written in "fibonacci number system" or "golden number system." Likely to be helpful. Fibonacci system: write integer n>=0 uniquely as n=SUM_k F[k] * d[k] using digits d[k]={0,1} and no two consecutive digits both 1. Here F[0]=1, F[1]=2, F[2]=3, F[4]=5, F[6]=8, etc, you may prefer to shift my subscripts. Example 27 = 1001001_F = 21+5+1 22 = 1000001_F = 21+1 ------------------------- 49 = 2001002_F unreduced 49 = 2001010_F still unreduced 49 = 10011010_F still unreduced 49 = 10100010_F = 34+13+2 Example 12 = 10101_F 10 = 10010_F ------------------------- 22 = 20111_F unreduced 22 = 21001_F still unreduced 22 = 110001_F still unreduced 22 =1000001_F Note, the fibonacci numbers always have form 10000...00_F and their doubles always have form 100100...00_F except that 2=10_F. Golden system: write real x>=0 uniquely as n=SUM_k g^k * d[k] using digits d[k]={0,1} and no two consecutive digits both 1, where g=1.61803... is the golden ratio g+1=g^2. ========================================== Meanwhile this suggests other possible applications & new questions. As I think Gosper already very familiar with, you can make weird fractalish tiles using funny number systems. For example complex numbers can be represented using radix=i-1 and digits={0,1}, and then the full set of complex numbers with 0s to the left of the decimal point, is a very weird-shaped but simply-connected object that tiles the complex plane. [Or digits={0,1,2,...X^2} and radix=i-X more generally, where X>0 is any fixed integer. Also, binary with radix=(+-1+-sqrt(-7))/2 or radix=+-sqrt(-2) with any fixed sign choices.] NOW, consider a wider range of number systems where, like fibonacci system, we forbid certain substrings of digits. If we use "binary" i.e. digit set {0,1}: If the radix R is some algebraic number with minimal polynomial equation P(R)=0 and P has degree<K with all coefficients either 0 or +-1, then we forbid the bit pattern arising from the positive coefficients of P, as a substring. (And demand P have more negative than positive coefficients, and have both signs.) [In the case of the golden ratio R=g, we have g^2=g+1.] We also demand that R satisfy a polynomial equation of the form 2*R^J=Q(R) where Q has all coefficients either 0 or 1. [In the case of the golden ratio R=g, we have 2*g^2=g^3+1.] These two demands try to assure uniqueness of the representation, and to assure you can add. We could also try to assure you can subtract by demanding a polynomial equation of form -R^J=S(R) exist where S has all coefficients either 0 or 1. I doubt these conditions really suffice. Probably more are needed. For example any polynomial multiple of P whose coeffs are 0 and +-1 only, yields possibly-additional forbidden substrings but I think you always get only a finite set of forbiddances in this way. Also, just because 2 or -1 are re-expressible does not necessarily suffice to make addition or subtraction be a finite process; and if it is an infinite process that might sometimes be ok. Also you might be able to add even without being able to express 2 in any finite-length way. For example, if 1 is expressible as 0.xxxxxx where xxxxx is some unique nonforbidden infinitely long string of 1s and 0s then maybe that means you can add. So the question of determining these number systems is a currently-confusing, but interesting topic. Anyhow, suppose one found such a "binary with forbidden substrings" number system with complex non-real radix R, and P(R)=0 with P monic and all non-leading coefficients -1 or 0. Let the set of numbers with only 0s to left of decimal point be called T. It seems to me that T usually will be fractally shaped and that with a finite set of tiles -- more precisely, a finite set of scalings of T by integer powers of |R| (the powers come from the locations of nonleading coefficients of P) -- you can tile the plane. Specifically, T will be tiled by a finite set of such scaled versions of itself, then recurse. If that is right, then we've got something kind of like the Penrose tiling in the sense that a finite set of tiles suffice, but it is even cooler than Penrose in the sense that they are all the "same" tile (just rescaled) and they have a very funny fractal shape. Also like Penrose, I think such a tiling will inherently be aperiodic in most cases (e.g. if irrational tile-count ratios). Also, in some cases these tilings should also yield, as an immediate side-benefit, interesting new planefilling curves. (I just said binary for simplicity, but the same ought to be true for other integers besides 2.) OK, so can examples be found? My first try: Let R^4=R+1 with R=-0.248126+1.03398*i and |R|=1.0633. Binary digits but forbid "11" and "101001" as substrings. (Are there any other forbiddances?) 1=0.0011(forbidden)=0.0001001001001...(allowed). Therefore it appears we can add and subtract. Tile T is tiled by T/R^2 (adjoin 10) and T/R^3 (adjoin 010). R*T is tiled by T/R and T/R^2. R^2*T is tiled by T and T/R. etc. Hence we apparently can tile the entire plane with just 3 tiles: T, T/R, and T/R^2. And this happens only aperiodically provided all the numbers that look irrational, are. And these tiles should have a fractal appearance. Caveat emptor: I have not actually drawn pictures confirming this, and I could have made errors. -- Warren D. Smith http://RangeVoting.org <-- add your endorsement (by clicking "endorse" as 1st step)