Thursday, April 5, 2012

phyllotaxis multiplication colouring pages

I thought I would try to make some printable colouring pages of phyllotaxis spirals - thinking that they could be coloured-in using multiplication-table / skip-counting rules to make patterns like the ones shown in the previous post. I've put a two-page pdf here - page one has a spiral with the numbers filled in (as above), and page two has one without numbers (for unfettered colouring).

When colouring these in, you might just shade the multiples of five - and get the picture below.

As you colour, you'll see the spiral pattern formed by the sequence 5, 10, 15, ...

You also can't avoid noticing a radial pattern as the dots seem to line out on spokes pointing out from the center. Take a closer look at that, and you'll see that adjacent numbers on the radial arms always have a difference of 55. Neato! this was something I didn't see when I generated the same pattern via software - using a pencil slows things down a bit and helps you notice things.

I'm wondering what relationships might be found in the patterns that other multiplicative colouring rules produce.

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Postscript: I shouldn't have been surprised! Any time you make a number spiral skip-counting by n with k spokes, the difference along the spokes is going to be nk - in this case, skip counting by 5 and generating 11 spokes gives a radial difference of 55.