I loved playing with the Spirograph drawing toy as a kid.
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!

You need:
  • 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
  • paper

Step 1: Choose Some Ring and Gear Sizes

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 taking the least common multiple (LCM) of the number of teeth on each of the two elements (ring and gear), and dividing it 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:


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.

Step 2: Design Your Toy

Start designing your toy in Inkscape.  Its gear module makes this easy.  

In Inkscape, to render gears:

  Extensions > Render > Gear

I found that a circular pitch of 12px and a pressure angle of 25 works fine.

For the final design, I rendered the gears in Inkscape and then exported them as an .svg to Illustrator, which I'm more comfortable using.  Ultimately you will probably need to export your design as .dxf to import into the laser cutter software.

Step 3: Cut Your Design

I used 1/8 inch (~3mm) acrylic for my final design.  I tried 1/16 inch also, but for the ring size I wanted, 1/16 inch was too thin; the gear kept popping out of the ring.
It was helpful to implement two layers in the cutting software: one to cut out the components and the other to etch the numbers of the rings and gears.  Having the numbers made it convenient to keep track of which gear/ring combo yielded which drawing when troubleshooting the design.  The numbers also reinforce the mathematical underpinnings while using it.  
Here's my final design as a .dxf.  You can also find it here on Thingiverse.  But I think you should design your own, it's fun!

Step 4: Draw Some Hypotrochoids!

Have fun! 
Thanks for th Excel-Formula! I can use that one for a COMPLETELY other field (work) :)
Just curious, for what?
deep inside a formula to check for correlations of problems with a product vs the used lot's of materials to build the product.
Yepp. :) I am an Investigation Engineer for a company which develops and builds medical-technical products.
Pretty cool maiden Instructable! Keep it up :-)
Thank you!
<p>So how do the number of teeth on the gear vs the outer ring line up on the excel sheet? I'm a little confused about it...</p>
Row &quot;1&quot; entries are the number of teeth on the outer ring.<br>Column &quot;A&quot; entries are the number of teeth on the inner gear.<br>The lookup result tells you the number of &quot;points&quot; on the resulting figure when the spirograph is used.<br>This table is useful to select gear sizes. I like sizes that give you a lot of different options. Fifty-four seems like a good outer ring tooth count because it can yield many different types of figures. Some tooth counts, like 71, which is prime, don't give you a lot of options.
<p>Thank you! So prime tooth counts on the outer ring wouldn't be super useful? I'm planning on making 2-4 outer rings and a bunch of inner rings that fit into them. Any suggestions for tooth counts, or just trial and error?</p>
<p>Not trial and error! Have another look at Step 1. The table is there to help you choose. My suggestion is to choose some outer ring and inner gear tooth numbers that give you the highest number of unique combinations across a wide range of resulting number of points. Prime numbers will always give you the least number of combinations.</p>
<p>What diameter did you make the holes for the pens/pencils?</p>
<p>I suggest trial and error. But you can also refer to the .dxf attached in step 3. The holes in there work with most pens an pencils, but I do remember that they were too small for some ball point pens.</p>
<p>We just laser cutted it! The kids will love it! Thanks for the idea</p>
<p>Nice! I like your organic design.</p>
<p>Cool! The design reminds me of an etch-a-sketch! I loved those things!</p>
<p>So nice! I&acute;m about to do it thanks!</p>
<p>I loved these things when I was a kid! It never occurred to me that I now have the power to create my own! &lt;3</p>
Great Job !! Brings back childhood memories of my playtime. And Thanks for Inkscape, I'm going to install it later today.
oh nostalgia, i haven't seen one of these in ages.

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