To generate typical spirograph-like patterns (aka mathematical roulettes), you can trace the path of a circle rolling around inside a bigger circle. What if, instead, you rolled one sphere inside another sphere? That generates a three-dimensional pattern that I call a spherograph. This code generates spherographs.

It's easy to draw a circle. Loop over the values from 0 to 2 * π (there are infinitely many, so maybe use an interval). This produces an angle in raidans. Your x-coordinate is cosine of the angle and your y-coordinate is sine. Then draw a pixel at that location. Something like this in (pseudo-code):

for (angle = 0.0; angle < 2 * Math.PI; angle += 0.01)
{
    x = cos(angle);
    y = sin(angle);
    draw(x, y);
}

If you rotate the canvas while drawing the circle (and the centers of rotation are a little offset) then you'll get a basic spirograph. But how to rotate the canvas? The math isn't too hard, but it's trivial with a matrix math library (I used gl-matrix, which is very nice). But while we're rotating on the Z-plane, why not also rotate along X or Y? Then rotate for the camera and we're done.