Spirograph Math

Figure out the math in the spirograph!

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This page explains some of the math that an elementary school child might learn from doing spirograph patterns. Now a kid doesn't play with spirographs in order to learn complex polar-coordinate formulas - nor do they care that the proper names for spirograph curves are epicycloids and hypotrochoids.

A kid plays with spirographs because it's fun.

• It's fun to make beautiful pictures with a spirograph toy, or spirograph program.
• It gives a kid a buzz when mom or dad or grandma or grandpa says Wow! You did that all by yourself??
• It makes a kid proud to see his or her spirograph design displayed on the classroom wall or fridge door.

In the meantime, though, the kid is actually picking up some math. Amazing, but true.

This page has a free spirograph applet for your kids to try. Or, if you plan to buy one, please read this first.

For those not familiar with spirographs, I've made a short movie illustrating the concept. Or, you can skip straight to the math.

When the child starts to use a spirograph, they'll go through a few stages.

• First, the child will explore, to see what is possible. Randomly chosen cogs or clicks. Pages and pages filled with random designs.
• Eventually, the child learns how to make the kind of spirograph swirls he or she likes. How to make them with few or many points. How to make them big or small.
• Then, he or she will combine individual spirograph curves together into complex illustrations.

In the course of this, they will learn the answer to this question.

If the wheel sizes are thus and so, what pattern will I get?

and that's where the math comes in.

Let me show you what I mean. The table below shows a whole bunch of spirograph patterns. I've chosen the fixed and moving circle sizes in a systematic way.

	Moving Radius 1	Moving Radius 2	Moving Radius 3	Moving Radius 4	Moving Radius 5
Fixed Radius 9
Fixed Radius 10
Fixed Radius 11
Fixed Radius 12

Here are some questions that your child may learn answers for as he or she plays

• Why, when the fixed circle size is 11, does the star always have 11 points? What other sizes are like that?
• Circle sizes 9 and 3 give a triangle. So do 12 and 4. What other circle sizes make triangles? Would 15 and 5 do it? What's the pattern?
• What sizes make a four-pointed star? An oval?

These questions all have mathematical answers. Your child will gain an intuitive feel for these answers. Perhaps this is enough to satisfy you - your kid is having fun, making nice spirograph artwork, and beefing up their math intuition.

On the other hand, you may want to bring the intuition out, to make it concrete. Here's a couple of ideas that may help do that.

• Ask the child or children to experiment, and try to find answers to the questions above.
• Make a table like the one above, but
  • with blank entries instead of pictures, and
  • with a lot more rows and columns.
  then ask the kids to fill it with answer to the question how many points on the star? Ask them what patterns they identified. Ask them to guess some answers before actually using the spirograph.

There's certainly more possible questions than just these. Let your creativity go wild! If you need to encourage your child to use a spirograph, suggest making cards for mother's day or father's day, or for birthdays or other celebrations...