Modulo 10 Fibonacci Pattern

Contents

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

The Unit Cell

The unit cell in the main pattern.	Cells derived from the prime factors (2 and 5). See the number section for details. Although these derived cells contribute 4 of the 6 colours to the pattern they only occupy 28% of the squares.	Cells not related to the prime factors. These fall into two series, the one based on (1,1) contributing 60 squares to the pattern and the one based on (2,1) contributing twelve.

The Pattern

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

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=10 are shown below.

These can be divided into two groups. Those starting with (0, 0), (2, 2), (4, 2) and (5, 5) are multiplied copies of series in the base 2 and base 5 patterns. (1, 1) and (2, 1) are new to this pattern. Base 2 series in orange, base 5 in blue. (0, 0) is in both.

Seed	Sequence	No.
(0,0)	[0], [0], [0], [0], ...	1
(1,1)	[1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5, 6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5, 1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1, 0], [1, 1, 2, 3, ...	60
(2,2)	[2, 2, 4, 6, 0, 6, 6, 2, 8, 0, 8, 8, 6, 4, 0, 4, 4, 8, 2, 0], [2, 2, 4, 6, ...	20
(5,5)	[5, 5, 0], [5, 5, 0], [5, ...	3
(2,1)	[2, 1, 3, 4, 7, 1, 8, 9, 7, 6, 3, 9], [2, 1, 3, 4, ...	12
(4,2)	[4, 2, 6, 8], [4, 2, 6, 8], ...	4
Total		100

The Reduced Pattern

This shows the pattern after the base two and base 5 cells have been de-emphasised.