Spirograph mania!

According to RetroWow, among other sources, Spirograph was invented by Denys Fisher in 1965. It was first intended as a drafting tool, but was marketed as a toy. There were multiple versions, and there is a modern remake from Hasbro.

 We've tried out three versions. The new version didn't stay in the house. It was too difficult to use because the gears kept slipping underneath the outer template. The pocket version shown below works OK, but the designs are somewhat limited (no epitrochoids). The antique Kenner version is our favorite. This is getting a lot of use right now. It's great at 8 or 9 years old and up, but your six year old would have to be very adept with a pen to enjoy it for long.

Mom! Can I get out the Spirograph?

The old pens were dry, so I did get a wonderful set of Stabilos, which are working perfectly. The pens need to be narrow enough to fit through the holes in the gears. Felt tips are a bit softer than roller balls, so they don't seem to make holes in the paper as easily, although the ink sometimes runs a bit.

Once you do a few patterns, it's nice to be able to predict what the wheels will do. There is a handy chart on the inside of the box lid, but the math is rather fun, too. We're not quite up to common denominators and gear ratios yet, but predicting the number of points on a spirograph pattern will be a good tool when we get there:

• Outer wheel: 96 teeth
• Inner wheel: 60 teeth
• Step 1: Factor the number of teeth
  • 96 = 32 x 3 = 2 x 2 x 2 x 2 x 2 x 3
  • 60 = 20 x 3 = 5 x 2 x 3
• Step 2: Compare the numbers. Find factors that are the same.
• 96 = 32 x 3 = 2 x 2 x 2 x 2 x 2 x 3
• 60 = 10 x 6 = 5 x 2 x 2 x 3
• Step 3: Calculate the largest common denominator:
• = 2 x 2 x 3 = 12
• Step 4: Divide the number of teeth by the LCD to get the gear ratio:
• 96 / 12 = 8, 
• 60 / 12 = 5 
• for a gear ratio of 8/5

So it will take 5 round trips inside of the larger wheel to create a pattern with 8 points.

When the circle is rotated around the inside of the fixed circle, as in this pattern, the result is called a hypotrochoid. 'Hypo' is a commonly used Greek root for 'under' or 'inner' as in hypoglycaemia for low blood sugar and hypoxia for lack of oxygen. If the inner circle is fixed and the outer one is rotated, the pattern is an epitrochoid. I think of 'epi' as a Greek root meaning 'surface' or 'outer' as in the medical name for the outer skin - epidermis.

Of course these geometric forms have equations which can be used to describe them. And these equations have been implemented as interactive demos on the web. I don't find these as much fun as the physical drawing. Partly this is because the process of drawing a hypotrochoid is rather pleasant loopy-loop feeling. Also, though, the restriction to an integer number of teeth on the gears makes for a restriction on the variety of patterns. As in flowers, we don't really notice that number of petals is restricted to a multiple of 2, 3, or 5. We just find it pleasing.