Design a laser-cut Spirograph-like toy

The number of teeth on the outer ring and the inner gear determine the number of "points" on the resulting full drawing. When setting out to design a spirograph, I first fiddled around with a few different size rings and gears.  I found that, for a given ring and gear size, the number of "points" can be determined by dividing the least common multiple (LCM) of the number of teeth on each of the two elements (ring and gear), divided by the number of teeth on the gear.*  

Using this equation you can make a spreadsheet with columns as ring sizes and rows as gear sizes to help you choose the numbers of teeth for the elements of your toy.

Excel has a function for least common multiple, and thus it's easy to write a formula this purpose:

=LCM(B$1,$A2)/$A2

where B1 contains the number of teeth in the outer ring and A2 contains the number of teeth on the gear; see the screenshot.  

...Or just download the Excel file below.

As you might expect, for rings with a prime number of teeth, the number of points is always that number.  For gears which have a prime number of teeth, they result in designs with a number of points equal to the number of teeth in the ring.

* Perhaps this value has another name, but I'm no mathematician; please let me know if there's a more simple name.

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