Number Spirals. Introduction The well-known Ulam spiral and the variant developed by Robert Sacks, the Sacks spiral, show interesting geometric patterns in the positions of primes. This page explores a simple extension of these spirals to visualize the number of unique prime factors for each number and provides Python code for drawing them, along with some pre-rendered examples, in PostScript and PNG format.
Every image has (or should have!) The Python script used to generate it -- see the code section for details of what you need to run these. Python source code files are shown inside square brackets. Construction The layout of the Sacks spiral aligns the squares (1,4,9,16,... etc) along a straight line heading east from the center. Theta = sqrt(i) * 2 * pi r = sqrt(i), and thus its x,y co-ordinates are given by: x = -cos(sqrt(i)*2*pi)*sqrt(i) y = sin(sqrt(i)*2*pi)*sqrt(i) The result is a spiral like this (primes are colored bluish-green, squares olive-yellow): Prime Factor Spirals Python Code Scaling Up.
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