Spirograph Eye Candy – Desmos Graph

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I know I’ve blogged before (yes I did just use that as a verb) about Desmos, this great online calculator. But, the more I use it the more I am convinced that it is one of the greatest tools out there for both classroom demonstrations and outright mathematical investigations.

I was testing last Friday and decided I wanted to create a Desmos graph that generated a cycloid (the path a fixed point on a circle traces out as the circle rolls along a horizontal surface). For those of you who are new to Desmos, it can graph just about any 2-D curve and then make it dynamic (in this case by allowing me to dynamically change the domain of the parametric system so the curve is shown tracing itself out). Here’s the cycloid link. Just press the play button beside the a= parameter to allow the circle to roll and trace out the cycloid path:

Cycloid Graph on Desmos

So, then I got really ambitious and decided to try to make a spirograph. You know, those great plastic gears that you put a pencil in and then spin one circle around another. They never work that well these days when I by them for my kids, but that didn’t stop me from creating one. I botched the math a few times, but that just made the end product better when I finally worked out all of the right triangle trig. Here’s the sprirograph:

Spyrograph on Desmos

On the spirograph, hit the a= play button to watch the small inner circle move around the interior of the larger circle while the curve gets traced out. You can change the radii of the two circles using their sliders right above the a = slider. It’s super cool, right now it is set to generate the following:

Desmos Spyrograph

You may even make the inner circle larger than the outter one. There are wonderful effects. See what happens when the two radii are integer multiples of one another. See if any patterns emerge.

Or, just play around!!!