Rosaces and multiplication tables

It turns out that representing multiplication tables on a circle can be quite beautiful ! The idea is to represent multiplication tables on a clock, where the origin is the multiplied number, and the destination the result modulo the table multiplier.

And when we sweep a range of multipliers and put all displays together in an amimated picture, the result is hypnotizing.

This Youtube video explains in much greater detail and fun the idea.

The implementation of this in Matlab is a great example of how to deal with a higher level language for fast graphics. The rosace code itself fits in 2 lines using complex numbers.

x = 78;
y = 854;
% Table of x modulo y, the trick is to represent it using the circle
% equation for complex numbers
in = [1:(y-1)].';
cir = @(x) exp(-1i .* x .* 2 .* pi + 1i .* pi / 2);
% Update the display, this will be way to slow if each line is an object
% so this uses a single vector
toplot = ([cir(in / y) cir(mod(in * x , y) ./ y )  in.*NaN].');
toplot = toplot(:);

It is then possible to display hundreds of lines in real time using two Matlab optimization tricks:

• create objects upfront and only update their properties at each iteration
• create a single high level Matlab line object by putting all segments in a long vector with a NaN between each to break the line.

The full git repository, with an interactive could be found here.