* Warren D Smith <warren.wds@gmail.com> [Jun 15. 2015 07:58]:
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: [...]
For those "sparse" expansions, just use the greedy method.
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.
Again, greedy does it.
==========================================
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.]
http://jjj.de/tmp-rd/rd-p1m1-twindragon-sign-d012.png Note one has to be careful about the sign choices if one wants a region around zero covered (thus eliminating the need for signs in those expansions), see Knuth (but forgot exactly where).
NOW, consider a wider range of number systems where, like fibonacci system, we forbid certain substrings of digits.
Here are examples of "sparse" (non-adjacent forms, aka NAFs): http://jjj.de/tmp-rd/rd-naf-m1p1-d1.png <--= I really like this one Note sure about the details of the following (again, sparse): http://jjj.de/tmp-rd/rd-naf-w17.4p0ps2-terdragon-d012.png http://jjj.de/tmp-rd/rd-naf-m1p1-ver2-d012.png
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
I am afraid these get "interesting" already for degrees >= 3, and even without "sparse" or any such forbidden patterns.
[...]
Knuth has some sort of Fib-addressing for a tiling into pentagons. Best regards, jj