Chapter 1

Round and Round Again

The first thing you might do when given a spirograph is revolving the little gear inside of the big gear (and trying not to cry when it slips out of place). Playing around, you might notice the variety of curves that can be produced:

It looks like changing the size of the fixed gear, the size of the rolling gear, and the position where the pen is placed changes its shape. So how can we describe this mathematically? The first step is to make some assumptions: we assume that the gears can be modeled as wheels with a specific circumference. The makers of the spirograph likely chose teeth to prevent slipping, but in our mathematical utopia, that is not a problem. Notice now we are describing a hypotrochoid, where the rolling circle is inside of the fixed circle.

Setting Up Our Diagram

diagram with circular track and the starting position of the tracing wheel

In this diagram, let:

be the point where the pen is (the tracing point)
be the center of the fixed, outer circle with radius
be the center of the moving circle with radius
be the distance between the center of the circle and point
be a fixed point on the fixed circle
be a fixed point on the movable circle
be the point of tangency between the fixed circle and movable circle

Right, let’s roll the movable circle inside the fixed circle an angle :

diagram with circular track, starting position of the tracing wheel, and ending position of the tracing wheel

Notice how all of the points that have been moved now have a subscript of 1. Our goal now is to describe the position of point given an angle . To do this, we can use parametric equations! If point is at the origin, all we need to do is find coordinates of point .

But for this we need some more information! Let’s draw a line parallel to going through point , and drop the heights from point and .

diagram with the ending position of the tracing wheel and triangles showing the position of points M and P

Now, let

be the point where the dropped height from meets
be the point where the dropped height from meets
be the point where the the line parallel to from meets
be the .

Finding the Pen

Now notice that the x-coordinate of is , while the y-coordinate is

Since makes a rectangle, then and .

Now, recognizing that and are both right triangles (since we all we did is drop the height), we can use trigonometry to find some of these lengths.

Length is simply the difference between the radius of the fixed wheel and the radius of the moving wheel.

Calculating

We’re getting so close! But what is ? There’s one fact that we have to use: the movable circle is rolling inside the fixed circle. That means that the distance travelled along the fixed circle should equal the distance travelled along the rotating circle.

diagram demonstrating that rolling requires the length travelled around the track equals the amount the tracing wheel rolls

Looking at the diagram, we can see that the above statement is equivalent to saying that:

To help solve this, let’s draw a line parallel to going through point and call the point where this line intersect circle point . Since lines and are both parallel, we know that (thanks Euclid’s Propisition 1.29).

Now we can substitute in for our parametric equations. Don’t forget your trig identities!

That’s it! That’s our final equation!!

Playing Around

I’d like to show you the 8th wonder of the modern world: Desmos.

In Chapter 3, we will break down this equation to better understand how it works!

Making Our Equations more Practical

Now this mathematical utopia is nice, but if you have a spirograph kit, nothing can beat drawing it out yourself. Many spirograph kits will tell you the number of teeth on every gear, as well as different numbers for all of the holes. Instead of needing to take out a ruler every time you want to model your spirograph, we can change our equations to fit these numbers. Here is a diagram of what a rotating gear for our spirograph kit looks like:

a model of one of the spirograph gears showing the spiral pattern of the holes

Notice how the holes are in a spiral with the numbers counting up as you get closer to the center. Even though their in a spiral, the only variable about these holes that matters is the distance to the center of the circle. This means we can also model these gears as:

an alternate diagram of the spirograph gear aligning the holes into a straight line

Where

is the total number of holes on the small gear
is the hole number chosen for the pen

Now we can substitute . Another thing to note, if we scale and by an equal amount, then the graph itself will get bigger but shape won’t change. This means that you can replace with the number of teeth of the outer wheel, and with the number of teeth on the rotating wheel and the equation will still work!

We can play with this in Desmos too!