Website for enumerating all 1x1, 2x2, 3x3, and 4x4 rotationally-symmetric
color-change patterns, using the following tiling square units:
A: Full white square
B: Full black square
C: NW black, SE white
D: NE black, SW white
E: SE black, NW white
F: SW black, NE white
Dan Brown
is attempting to design every such pattern in origami,
seamlessly when possible.
1x1: There is only one 1x1 rotationally-symmetric color-change
pattern with these tiles, i.e., the uniform square up to inversion:
2x2: Because we only want to generate rotationally-symmetric
patterns, we only place tiles in the NW cell of the pattern,
specifically cell 0 in the 2 x 2 grid:
0+
++
We generate all 6 patterns, and check for duplicates up to
reflection and color inversion and list each group, one per row.
There are 3 such patterns.
3x3: Because we only want to generate rotationally-symmetric
patterns, we only place tiles in three cells of the pattern,
specifically 0, 1, 2 in the 3 x 3 grid, where cells 0,1 may be any tile,
while cell 2 must be either A or B:
01+
+2+
+++
We generate all 6x6x2 = 72 patterns, and check for duplicates up to
reflection and color inversion and list each group, one per row.
There are 22 such patterns.
4x4: Because we only want to generate rotationally-symmetric
patterns, we only place tiles in the NW quadrant of each pattern,
specifically in the four positions 0, 1, 2, 3 in the 4 x 4 grid:
01++
23++
++++
++++
We generate all 6^4 = 1296, and check for duplicates up to reflection and
color inversion and list each group, one per row. There are 354 such
patterns.