Ever since I was a kid, I wanted to know:
How can you predict how many points (or “petals”) a spirograph pattern will have?
There IS a formula. You need to know the number of teeth on the wheel and the number of teeth on the ring.
The number of points (N) is equal to the Least Common Multiple (LCM) of the number of teeth on the ring (R) and the number of teeth on the wheel (W), divided by the number of teeth on the wheel.
In a spreadsheet, you would use “= LCM(R,W)/W”. Replace R and W with the coordinates of the respective cells.
In the case of the pattern above, the least common multiple of 96 and 45 is 1440. One way to find this is by prime factorization. Find the prime numbers which
For example, for LCM(96,45) we find:
Prime factorization of 96 = 2 * 2 * 2 * 2 * 2 * 3 = 25 * 31
Prime factorization of 45 = 3 * 3 * 5 = 32 * 51
Using the set of prime numbers from each set with the highest exponent value we take 25 * 32 * 51 = 1440, which is the LCM.
Divide the LCM by the wheel, so 1440/45 = 32, the number of points in the design above.