Sequence and Pearls

by Simon Whitechapel

The Fibonacci sequence has the formula Fn = Fn-2 + Fn-1, where F1 = 0 (or 1) and F2 = 1. Its first 50 terms are therefore:

As can be seen, the sequence increases without limit, but it generates these terms if the formula is modified to read Fn = (Fn-2 + Fn-1) mod 10, where "mod 10" (or "modulus 10") means the remainder left when the number is divided by 10: This kind of sequence is sometimes used to generate pseudo-random numbers, because without the formula the terms appear unpredictable until the sequence is extended long enough and begins to repeat: However, the apparent randomness also often disappears when each pair of consecutive terms is used to mark points on a 2-d grid, like this:

All of these grids contain distorted swastikas, and when the sequence is modified in other ways, producing other patterns, some of them contain the SS lightning-runes too. There's nothing significant about this: the swastika and lightning-runes are compelling but also very simple shapes, and they can easily be generated by simple formulae, like those used here, that generate shapes with rotational symmetry.

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