Epicycloids and Spirograph

What shape is drawn when a circle rolls around the outside of another?

Imagine a small circle rolling continuously along the outside of a larger circle.

If you fix a point on the small circle and let it move along with it, that point will trace a complex curve on the plane.

At first, you might just see a small, spinning circle. But as it continues to roll, a beautiful trajectory gradually emerges. It might resemble petals or a star, or even form intricate ripple and loop patterns.

Curves generated by this kind of circular motion belong to a fascinating family of curves known as cycloids. Two of the most classic examples are the epicycloid and the epitrochoid.

Starting with a simple circle

Let's set aside complex mathematics for a moment.

Imagine a pen moving around a circle; it simply draws a circle—nothing complicated.

Now, place that pen on a small, moving circle that is itself rolling along a larger circle.

The pen tip now undergoes two simultaneous motions: rotating with the large circle and rotating with the small circle itself.

When these two motions combine, the path traced by the pen tip is no longer a simple circle; complex curves begin to appear.

This is the most fascinating aspect of these mathematical shapes: complex trajectories can arise from a few very simple circular motions.

What is an epicycloid?

Let's start with the simplest case: a small circle rolling along the outside of a larger circle. If we track a specific point on the circumference of the small circle, the path it traces is called an epicycloid.

A key feature here is that the point lies on the edge of the rolling small circle, so its motion creates a characteristic pattern of sharp 'petals'.

Changing the size ratio between the two circles results in a different number of 'petals'—some patterns have few, others have many; some are very sharp, while others are more rounded.

Now, let's make a tiny adjustment: instead of placing the pen tip on the edge of the small circle, we position it inside the circle or at a different distance from its center.

The small circle still rolls along the outside of the large circle, but the pen's position has shifted, altering the resulting path. This more general type of curve is known as an epitrochoid.

A tiny shift can completely change the pattern

This is precisely what makes this page perfect for interactive exploration. Imagine keeping the sizes of both circles constant while simply moving the pen tip slightly closer to the center of the small circle. The curve might transform from sharp petals into smooth, rounded ripples. With further adjustments, you might even generate complex, looping structures.

That is why these curves are so fun to play with—you don't need to know the final outcome beforehand. You simply change a number and observe: 'What will it draw this time?'

Why do petal shapes appear?

That is a very interesting question.

When the small circle rolls along the large circle, it doesn't simply rotate around a fixed center. Two movements happen simultaneously: a large-scale orbital motion and the small circle's own rotation.

There is a crucial relationship between these two types of rotation: the ratio of the radii of the two circles.

If the ratio between the two circles is simple, the pen tip will return to its starting position after a few cycles. As the path repeats—one loop, two loops, three loops, and so on—it eventually forms a closed pattern.

The ratio of the radii directly determines how many petals or repeating structures the pattern has.

You can even visualize it as two gears: the large gear provides the broad circular motion, while the small gear rolls along it and rotates at the same time.

If the two gears have different numbers of teeth, their rotational speeds differ, causing the pen tip to trace a complex, periodic motion.

This is why many cycloid-like patterns look as if they were created by precision machinery; they are indeed directly linked to the mechanics of gears.

Why do some patterns close so quickly?

Suppose the ratio between the large and small circles is very simple. The pen tip might return to the starting point after just a few rotations, causing the pattern to close quickly.

However, if the ratio is more complex, the pen tip may need to complete many more rotations before returning to the original position, resulting in a pattern with greater detail.

So, when you see a highly intricate cycloid-like pattern, it is often simply the result of the specific ratio between the two circles.

From Circular Motion to Mathematical Art

The most fascinating aspect of these curves is that you don't need to 'draw' them directly.

You simply define the motion—rolling one circle, rotating another, and positioning the pen tip—and then set them in motion.

The curve emerges on its own. This reflects a concept shared by many works of mathematical art: simple rules, repeated continuously, giving rise to complex structures.

Sometimes, a tiny change can radically transform the entire pattern. That is the most intriguing part of mathematical drawing: you aren't choosing a static image; you are altering the rules that generate it.

Exploring the World of Cycloids

Epicycloids showcase the intricate paths created when a circle rolls along the outside of another. If you enjoy these mathematical patterns born of rotation and gear-like motion, you might also explore another classic rolling trajectory: the Spirograph. Here, the smaller circle moves inside the larger one rather than along its exterior. This shift in motion produces a pattern with a completely different style.

Starting with a simple circle—through rolling, rotating, and repetition—we can create countless complex and beautiful mathematical curves. Motion itself becomes the paintbrush.

You can also explore more amazing mathematical patterns or read more about epicycloids on Wikipedia.