Tuesday, November 11, 2014

Spirals and Numbers (Non Integers in this case...)

Got fascinated by the 'spirals' described in 'Seed Spirals' by Michael Naylor. The idea for the spirals stems from an attempt to simulate the spirals that can be seen in for instance a aunflower. Correct or not in this respect: the spirals (like almost any figure with sufficient symmetry)are aesthetically pleasing, and do reveal some number theoretic patterns.
Basic principle is to plot the 'seeds' at a step wise increasing intervals in polar coordinates where the angle is equivalent to the number of steps multiplied by a 'seed factor' and distance from the origin is proportional to the square root of the number of steps:
  • x is 'seed factor'
  • n is number of steps

This results for a value of 1/12 of the seed factor in the following steps:
Only the fractional part of the 'seed factor' x has any significance: any integer part of x will result in full rotations.
To make this a bit more colorful I added color and the color will shift in a number of 'color steps', the interactive version: jvdm.info/NumberTheory/spiral.html. In order to better understand the patterns experimented with some numbers. Rational numbers can be represented as an integer numerator and a non-zero integer denominator. The number of arms the spiral has for rational numbers is the greatest common divisor of the numerator and denominator.

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