The math behind the graph is pretty simple. Basically, we are starting with 1 and multipying by our Multiplier, in the default case of this graph, 2. Then we draw a line from our starting number and the result. So a line from 1 to 2. Then we do the same for our result, 2, then repeat the process. If our result is more than 1 digit long, we sum the numbers until we get to a digit less than our Modulus and repeat the whole process again. The result are the pretty graphs you see above.
What's really neat is this whole process is essentially modular arithmetic using the Modulus. When each number is divided by our Modulus, the remainder is the same as the result of summing the digits of the number being divided! For a much better explanation, see the video in the About section of this page.
The length of each line is represented by its color. Red is the longest and blue is the shortest with yellows, greens and purples being somewhere in between.
Download the above graph as an SVG file for lasercutting. The SVG is just about 100mm in diameter. The inner lines are colored blue for engraving and the circle's outline is colored black for cutting.