A taxonomy of 4x4 patterns

A TAXONOMY OF 4 X 4 tiles of 1 bit per pixels, according to symmetries. What symmetries am I counting?

inverse of colors
horizontal flip
vertical flip
90 degree rotations.
wrap-around offsets of the pixels (2 and 1 pixels, up and down making this 4 total mutations)
the above in every possible combination
sorted by the unique results.

first up, there’s two patterns that yield only two unique results after undergoing all of the above. Checkerboard, and pure black/white.

for the purposes of encoding an image, and we like variable width codes, the best way to do this seems to be to stick with the maximum width being “a code”, with smaler widths secretly being one big width code representing semantically, two or more smaller codes, this might be equivalent to huffman?

file:///Volumes/dev/organised/4x4%20experiments/enumerate256b.html

how enumeration works

enumerate every 4x4 1bit plane (65536).
for each, apply the permutations: hflip, vflip, rotate90, invert, in each of the 16 possible combinations.
associate all the resulting combinations with a sequential index
1. 3a. optionally index “palindromic” patterns that match one or more permutations of themselves.
add all the resulting combinations to a set of 1-bit planes that will now be skipped and not tested.
call the resulting set BP, with each sequential index being associated with a related set of permutations. the canonical permutation for each item in BP is the one with the lowest numerical value.
produce a reverse lookup to find the canonical representation for any 16-bit number, plus the 4 bit hvri number needed to transform the canonical to the supplied.
for each {BPi,BPi,BPi} set:
For each resulting image, remap 0-7 in every permutation.
permute every combination of HVRI,HVRI,HVRI
take resulting set and apply a sequential index called 3BPi
and the resulting permutations to a list of {BPi,BPi,BPi} that will not be checked, skipped.

the one with the lowest numerical value. 6. produce a reverse lookup to find the canonical representation for any 16-bit number, plus the 4 bit hvri number needed to transform the canonical to the supplied. 7. for each {BPi,BPi,BPi} set: 8. For each resulting image, remap 0-7 in every permutation. 9, permute every combination of HVRI,HVRI,HVRI 10, take resulting set and apply a sequential index called 3BPi 11, and the resulting permutations to a list of {BPi,BPi,BPi} that will not be checked, skipped.

numbers to the right = 16 symmetry group ids.

ID calculated by the lowest hexidecimal numeric value of the tiles in the group, when a tiles’ bitpatterns are converted to numbers.

0000