Modulo 12 Fibonacci Pattern

Contents

• The Unit cell.
• The Pattern.
• The Numbers.

The Unit Cell

The base 12 pattern includes those of order 2, 3, 4 and 6.

N = 2	N = 3	N = 4	N = 6

The illustration on the left shows the cells derived from earlier patterns, the one on the right, those that are new to 12. Notice the rotated 'T' shape next to the origin in the new pattern (and in the base 4 pattern above). This appears in every pattern (after 2) because the sequence starting (1, 1) always includes (0, 1), (1, 0) and (1, 2). It is also in the base 3 and 6 patterns, above, but is obscured by adjacent cells of the same colour.

From 2, 3, 4 & 6	New to 12	Combined

The next two patterns show the whole divided between Diagonal and Off Diagonal. The Diagonal pattern is build from all the sequences that include a diagonal cell - (0, 0), (1, 1), etc. - the Off Diagonal being the left overs. Notice that the left hand edge and botom row always contain the same colour sequence as the diagonal and are therefore always included entirely in the Diagonal pattern.

Diagonal Terms	Off Diagonal Terms

The Pattern

A number of unit cells are tiled to show the overall pattern.

The following pattern is generated by tessellating the Off Diagonal pattern without the first column and row.

The Numbers

The points are generated using an algorithm where the only input is the number of cells across the pattern. The resulting series for N=12 are shown below.

Seed	Sequence	No.
(0,0)	[0], [0], [0], [0],  ...	1
(1,1)	[1, 1, 2, 3, 5, 8, 1, 9, 10, 7, 5, 0, 5, 5, 10, 3, 1, 4, 5, 9, 2, 11, 1, 0], [1, 1, 2, 3, ...	24
(2,2)	[2, 2, 4, 6, 10, 4, 2, 6, 8, 2, 10, 0, 10, 10, 8, 6, 2, 8, 10, 6, 4, 10, 2, 0], [2, 2, 4, 6, ... Twice the base 6 pattern [1, 1, 2, 3, 5, 2, 1, 3, 4, 1, 5, 0, 5, 5, 4, 3, 1, 4, 5, 3, 2, 5, 1, 0], ...	24
(3,3)	[3, 3, 6, 9, 3, 0], [3, 3, 6, 9, 3, 0], ... Three times the base 4 pattern [1, 1, 2, 3, 1, 0], ...	6
(4,4)	[4, 4, 8, 0, 8, 8, 4, 0], [4, 4, 8, 0, 8, 8, 4, 0],  ... Four times the base 3 pattern [1, 1, 2, 0, 2, 2, 1, 0], ...	8
(6,6)	[6, 6, 0], [6, 6, 0], [6, 6, 0], [ 6, 6, 0], ... Six times the base 2 pattern [1, 1, 0], ...	3
(7,7)	[7, 7, 2, 9, 11, 8, 7, 3, 10, 1, 11, 0, 11, 11, 10, 9, 7, 4, 11, 3, 2, 5, 7, 0], [7, 7, 3, 10, ...	24
(9,9)	[9, 9, 6, 3, 9, 0], [9, 9, 6, 3, 9, 0], ... Three times the base 4 pattern [3, 3, 2, 1, 3, 0], ...	6
(2,1)	[2, 1, 3, 4, 7, 11, 6, 5, 11, 4, 3, 7, 10, 5, 3, 8, 11, 7, 6, 1, 7, 8, 3, 11], [2, 1, 3, 4, ...	24
(4,1)	[4, 1, 5, 6, 11, 5, 4, 9, 1, 10, 11, 9, 8, 5, 1, 6, 7, 1, 8, 9, 5, 2, 7, 9], 4, 1, 5, 6, ...	24
Total		144