Issue 20
On the cover

On the cover: Spirographs

Ashleigh Wilcox, Jenny Power and Rachel Evans show you how to draw pretty pictures

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8 November 2024

Spirographs blur the line between geometry and art. When spirographs are mentioned, a first thought probably goes to the kids’ art toy—but who says it’s just for kids?! After this issue’s cover artist introduced me to spirographing at a Piscopia Initiative event, my first task was a trip to Smyths toys superstores to grab myself a kit. The shapes of a spirograph (hypotrochoids and epitrochoids—but we will get to that later) are created by using two circles. One circle is stationary; we will call this a ring because it has a large hole in the centre. Around the inside and outside of the ring are small teeth, or notches. The other circle we use moves and we will call it a cog. The cog is a smaller filled circle with small holes for your pen, and teeth around the outside. These teeth help with the rotations of the circle and give guidelines for where to start. In short, you take a ring, put your cog on the inside (or outside), line up the teeth, put your pen in the hole and go around and around the circle. Although it sounds simple, the creation of spirographs takes practice and is a slow art form.

About the artist

One person who got this kit as a child and never gave it up is Rachel Evans. She is better known in the community as Spirograph Girl. When I asked her about her inspirations for choosing spirographs as an art form, she said this: “I would say that as an artist I found spirograph but that’s completely untrue! Spirograph made me into an artist and it’s been an amazing ride, just going through one project after another. My process is very simple: I spirograph, colour in the pattern, then cut it out and stick it on. I also work digitally via the Inspiral app—which is what I used for this cover, I hope you like it!”

Hypotrochoids

Let’s get into the maths! Most of the spirographs on the cover are hypotrochoids. These are roulette curves which are created when a circle rolls around the inside of a fixed circle. Consider the diagram of a typical spirograph setup. We have the following variables:

  • $R$: the radius of the fixed ring.
  • $r$: the radius of the rotating cog.
  • $d$: the distance between the pen and the centre of the cog.

Schematic of a spirograph setup

The equations of the two circles are given by \begin{align*} && && \text{Ring: }& & x^2 + y^2 &= R^2 && &&\\ && && \text{Cog: }& & (x-(R-r))^2 +y^2 &= r^2. && && \end{align*} By examining the path the pen takes as the cog rotates around the ring, we get the following pair of parametric equations for the hypotrochoid: \begin{align*} x(t) &= (R-r)\cos(t) + d\cos\left({\frac{R-r}{r}t}\right) \\ y(t) &= (R-r)\sin(t)-d\sin\left({\frac{R-r}{r}t}\right). \end{align*} Note here that the parametric variable $t$ is not the polar angle, but the angle between the centre of the rotating cog and the horizontal axis. To complete the full pattern, this angle $t$ takes values from $0$ (when the pattern starts) to $ 2\pi \times \operatorname{LCM}(r,R)/{R} $ (when the pattern is complete). Here, $\operatorname{LCM}$ denotes the lowest common multiple between the radii of the cog and ring. Interestingly, we can use this to determine the number of spokes a spirograph pattern will have, and how many rotations of the cog it takes for the pattern to be complete! We have \begin{equation*} \text{number of rotations} = \frac{\operatorname{LCM}(r,R)}{R}, \end{equation*} and \begin{equation*} \text{number of spokes} = \frac{\operatorname{LCM}(r,R)}{r}. \end{equation*}

What we can notice is that only the radii of the two circles affect the number of spokes in your pattern. The pen location $d$ has no impact on this. The pen location only changes how `spiky’ your pattern is. When your pen is close to the centre of the cog, you will obtain very wide and short spokes, and when the pen is closer to the edge of the cog, you will get more tall and narrow spokes.

In particular it is the ratio between the two radii that is most important. Let’s call this ratio \begin{equation*} k = \frac{R}{r}. \end{equation*} We get different behaviour depending on the value of $k$. If $k$ is a rational number, given in its simplest form as $k=a/b$, the number of spokes is given by $a$ and the number of rotations are given by $b$. This is equivalent to the LCM formulation presented above. We get some interesting examples when $k$ is an integer (but we will look at those later). If $k$ is irrational the pattern is infinite and will never complete.

The cogs and rings of a spirograph usually have the number of teeth written on them

Now, before you all take out your rulers and attempt to measure the radii of your spirograph parts, there is an easier way to calculate these. Each spirograph part comes labelled with the number of teeth it has (the rings have two numbers for the inner and outer circles). We can rewrite the formulae in terms of the number of teeth on the rings and cogs instead. Let’s say that $N$ represents the number of teeth on the ring and $n$ represents the number of teeth on the cog. If the distance between the teeth is uniform across all parts, and this distance is, say, $S$, then the circumferences of the wheels are given by \begin{align*} L &= NS, & \mathcal{l} &= nS. \end{align*} However, we know that the circumference of the circle is given by \begin{align*} L &= 2\pi R, & \mathcal{l} &= 2\pi r. \end{align*} From simple manipulation, we obtain \begin{align*} R &= \frac{S}{2\pi} N, & r &= \frac{S}{2\pi} n, \end{align*} and end up with the much easier to work with formulae \begin{align*} \text{number of rotations} &= \frac{\operatorname{LCM}(n,N)}{N}, \\ \text{number of spokes} &= \frac{\operatorname{LCM}(n,N)}{n}. \end{align*}

Epitrochoid with $R=8$, $r=3$ and $d=2.5$

Epitrochoids

Another type of roulette curve created by a spirograph is an epitrochoid. This curve is created when circle rolls around the outside of a fixed circle. Following the path the pen takes as the cog rotates around the outside of the ring (as we did for hypotrochoids), we get the following pair of parametric equations for the epitrochoid: \begin{align*} x(t) &= (R+r)\cos(t)-d\cos\left({\frac{R+r}{r}t}\right) \\ y(t) &= (R+r)\sin(t)-d\sin\left({\frac{R+r}{r}t}\right). \end{align*}

Ellipse and rounded polygons with $d=r/2$. From left to right: $k=2$ (ellipse), $k=3$, $k=4$, $k=5$

Ellipses and rounded polygons

The first example we’ll look at is when the ratio of the ring to the cog is an integer, $R = kr$. The parametric equations are given by \begin{align*} x(t) &= (k-1)r\cos(t) + d\cos\left({(k-1)t}\right), \\ y(t) &= (k-1)r\sin(t)-d\sin\left({(k-1)t}\right). \end{align*}

Using the formulae above, the number of spokes the pattern will have is the integer $k$ and it will only take one rotation to complete the pattern! Additionally, if $k$ is even the pattern will be symmetric about the $y$-axis. A very special case is when $k=2$, as this produces an ellipse!

If you have a standard spirograph kit at home, you can see this behaviour with the 144/96 ring and the 48, 32, 24 cogs ($k= 2,3,4$ in these cases). Note that this only works if $d \neq 0$, ie your pen is not in at the centre of the cog. Regardless of the choice of $k$, for $d = 0$ you will always get a circle!

Hypocycloids with $R = kr$ and $d=r$. From left to right: $k=2$ (Tusi couple), $k=3$ (destroid), $k=4$ (astroid), $k=5$ (pentoid)

Hypocycloids

The second example we’ll look at is when the pen is on the circumference of the cog ($d=r$). This is not something we can draw with a physical spirograph kit unfortunately, but the resulting shapes are still very cool to look at. The equations for these are the exact same as for a hypotrochoid except with $d=r$. The spokes obtained in hypocycloids are what are known as cusps, sharp corners where the curve is not differentiable. A famous example of a hypocycloid is when \mbox{$R = 2r$}. This results in a 2-cusped hypocycloid which is just a line segment! This is often referred as a `Tusi couple’ named after Persian mathematician and astronomer Nasir al-Din al-Tusi (1201–1274): \begin{align*} x(t) &= (R-r)\cos(t) + r\cos\left({\frac{R-r}{r}t}\right), \\ y(t) &= (R-r)\sin(t)-r\sin\left({\frac{R-r}{r}t}\right). \end{align*}

Rosettes

Last, but certainly not least, we have my favourites—the rosettes. This example produces spirograph patterns that look like flowers. This occurs when we choose $d = R-r$. The parametrised equations for this curve are given by \begin{align*} x(t) &= (R-r) \left[\cos(t) + \cos\left({\frac{R-r}{r}t}\right)\right], \\ y(t) &= (R-r) \left[\sin(t)-\sin\left({\frac{R-r}{r}t}\right)\right]. \end{align*} However, for rosettes it is much more convenient to write the equations in polar form instead. After doing some clever trigonometry, we end up with the polar equation \begin{equation*} \rho = 2(R-r)\cos\left({\frac{R}{R-2r} \theta}\right), \end{equation*} where $\rho$ is the distance from the curve to the origin and $\theta$ is the polar angle. In this formulation, it is the ratio \begin{equation*} p = \left| \frac{R}{R-2r} \right| = \left| \frac{k}{k-2} \right| \end{equation*} that determines how many spokes (or `petals’) our flower will have. If $p$ is an integer we get a special pattern called a rose curve as it resembles a petalled flower. If $p$ is odd the flower will have $p$ petals and if $p$ is even the flower will have $2p$ petals. If $p$ is not an integer, you will get a rosette with the number of spokes given by the formulae introduced earlier for hypotrochoids.

Rosa curve, $p=2$

Rosa curve, $p=3$

Rosa curve, $p=5$

Rosa curve, $p=8$

 

 

$p=3/2$

$p=11/7$

$p=23/9$

$p=49/45$

Ashleigh is a PhD student and graduate teaching assistant at the University of Leicester. Her main mathematical interests are in number theory. She is passionate about outreach and inclusion in mathematics, volunteers as a STEM ambassador and is a representative for the Piscopia Initiative.
All articles by Ashleigh

Jenny is an applied maths PhD student at the University of Bath. She is passionate about science communication and runs an Instagram account where she talks about maths and documents her PhD experience.
All articles by Jenny

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