Limiting to 4 Letters RLXY, (5*5-1)*2*2 = 96 variants: Swastika Warning: None of these curves are ads for FASCISM! https://github.com/bradklee/Docs/blob/master/TwoByTwoZFs.pdf ( Hint 7: The parity sequences are buried in there somewhere. ) For 3x3, see also: https://0x0.st/zw1o.png HA Ha HA HA hA hA Ha. --Brad On Sat, Oct 5, 2019 at 10:48 AM Brad Klee <bradklee@gmail.com> wrote:
1. A path goes through a unit square, starting and ending on a vertex. 2. The path is self-similar, with deflation of 1 square to 2X2. 3. The path goes through all four sub-squares, each exactly once. 4. Consecutive sub-squares are adjacent at a vertex on the path. 5. The curve cannot cross itself.
Q. Restricting to 1/4^n Z-function, how many paths are possible?
When inflating from 1 square to 4, there are 5 valid paths between adjacent corners, and 2 valid paths between opposite corners:
Approximating Polygons: https://0x0.st/zwup.png .
Considering that paths between adjacent corners have L&R variants, an upper limit for the total count is:
5^8*5^(12*2)*2^(8*2) ~ 10^27 .
I wanted to reduce the count to the number of distinct curves, but the space appears too large. Maybe a fun idea for a computer App: Allow user to choose templates until the set closes, then print.
It is possible to compute all (1/9^n)-sampled approximating polygons for 3x3 via backtracking algorithm, but unfortunately I am already late for a game of Go with one of my friends around town.
Cheers --Brad