Modulo 8 Fibonacci Pattern

Contents

The Unit Cell

Whole Pattern Diagonal Terms Off Diagonal Terms
All the terms Generated from sequences seeded with (0, 0), (1, 1), etc. The remainder

The Pattern

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

The same pattern without the off diagonal terms

And the off diagonal terms (those generated by (2, 1) and (3, 1)) on their own (with the first row and column removed, as they are empty).

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=8 are shown below.

Seed Sequence No.
(0,0) [0], [0], [0], [0], ... 1
(1,1) [1, 1, 2, 3, 5, 0, 5, 5, 2, 7, 1, 0], [1, 1, 2, 3, ... 12
(2,2) [2, 2, 4, 6, 2, 0], [2, 2, 4, 6, 2, 0], ... 6
(3,3) [3, 3, 6, 1, 7, 0, 7, 7, 6, 5, 3, 0], [3, 3, 6, 1, ... 12
(4,4) [4, 0, 4], [4, 0, 4], [4, ... 3
(6,6) [6, 6, 4, 2, 6, 0], [6, 6, 4, 2, 6, 0],... 6
(2,1) [2, 1, 3, 4, 7, 3, 2, 5, 7, 4, 3, 7], [2, 1, 3, 4, ... 12
(3,1) [3, 1, 4, 5, 1, 6, 7, 5, 4, 1, 5, 6], [3, 1, 4, 5, ... 12
Total 64

The Reduced Pattern

Because 8 is a multiple of two and the series generated include multiples of those contained in hte smaller patterns. The 4 squares in the N=2 patter can be seen duplicated in the N=8 pattern: at (0, 0), (0, 4), (4, 0) and (4, 4). The 16 squares in the N=4 pattern can also be found: at (0,0), (2,0), (4,0), etc. The pattern in the fourth column shows the original pattern with these cells removed (transparent). You can also see the N=2 pattern in the N=4 one.

N = 8 N = 2 N = 4 Reduced 8