The mesmerizing shapes that a Spirograph makes are called hypotrochoids, which have some interesting mathematical properties.
I decided to design my own hypotrochoid-drawing-machine toy as one of my first lasercutter projects at Ace Monster Toys. I'll describe the process here, and maybe it will inspire you to design your own!
• a spreadsheet (such as in Microsoft Excel, Open Office Calc, or in Google Docs)
• Inkscape (open source)
• a laser cutter
• 1/8" acrylic
• pencils or pens
Step 1: Choose some ring and gear sizes
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:
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.