Spirograph is a wonderful mathematical toy, but frustrations can arise when pens, gears, pins, and paper blot, slip, come loose, or tear.
This is a good job for a computer.
Parametric equations for a circle are
   x = r cos θ
   y = r sin θ
Parametric equations for an epicycle (single spirograph image) are
   x = r₁ cos θ + r₂ cos (k θ)
   y = r₁ sin θ + r₂ cos (k θ)
In most of the following, I let r₂ = r₁ / 2. Here we have images for k = -3, -2, -1, 0, 1, 2, 3, 4, 5 and θ goes from 0 to 2 π.
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And this is what we get when k = -5/2, -3/2, -1/2, 1/2, 3/2, 5/2, 7/2, 9/2 and θ goes from 0 to 4 π.

And now k = -2/3, -1/3, 1/3, 2/3, 4/3, 5/3, 7/3, 8/3 and θ goes from 0 to 6 π.

Let's go back to k = -5/2 and vary r₂/r₁. In these images r₂/r₁ = 1/4, 1/2, 3/4, 1, 5/4

Enough blue, here's big red and green.