Where is the Math in Spirographs?

In our own teaching practice, we don’t start with presenting the math theory when introducing the Spiro Inquiry activity to students. We find that letting students explore their ideas within the playground of complex math can be a more creative and generative experience. However, having these explanations available when students or educators become curious is always helpful. In this section we offer three explanations that help us understand the math below the surface of spirographs.

Relationships between Gear Numbers

The number of petals a Sprio drawing has has to do with the ratio relationship between the numbers of the teeth in each the inner and the outer gears. When the pen stops making new loops and starts re-tracing what’s already been drawn, this indicates that the inner gear and outer gear have reached the same place that they started. Some gear ratios take more laps before meeting at the point where they started than others. These figures representing the inner and outer gears as if they were flattened out into straight pieces can help visualize this phenomenon:


Periodicity in Music

Similar patterns can be found in the rhythms of music. For example the polyrhythms of sub-saharan African music interlace more than one different rhythm at once. One drummer might be playing every third beat (black dots), and another musician clapping on the first of every four beats (red dots), meaning they would only play on the same beat at the same time every four bars. Below is a simplified visualization of just two polyrhythms that might relate to the points of convergence of two spiro gears:


Hypotrochoid Patterns


The red curve is a hypotrochoid drawn as the smaller black circle (r) rolls around inside the larger blue circle (R). The distance of the pen from the center of the smaller black circle is represented as d. (parameters are R = 5, r = 3, d = 5). *Images borrowed from

One of the underlying math principles represented in the Spiro designs are called Hypotrochoid patterns (shown above). These are algorithmic graphs that Spiro allows young people to play with the mathematics of hypotrochoid graphs without needing to first understand the more abstract equations behind them. By experimenting with the different gear parameters and their resulting designs, honing their own aesthetic motivations, and following their mathematical intuitions, students are developing their own understanding of this graphing concept/phenomena.


While hypotrochoid curves and periodicity are not usually on K-12 math curriculum, least common multiples and greatest common factors are present in both math standards and Spiro. Seeing how these concepts are present in their play is helpful for atuning educators to see childrens’ reasoning about numerical relationships between the gears. This shows that introducing projects that involve complex concepts can reveal that kids are capable of more advanced thinking than their grade level might indicate. The Spiro Inquiry makes complex concepts accessible for children’s play and creation without walking them through the simplest ideas first.

Consider the mathematical sensemaking evident in the following exchange between two young people playing with the Spiro app:

“Why is yours so big and so slow?”
“Because it’s big. That’s why it’s slow.”

Examples of Student Work

These examples demonstrate the difference between designs intended to be printed or saved as digital files versus designs meant to be cut with digital fabrication tools such as into stickers or stencils.