If you’ve ever played with a Spirograph, you know how magical it feels to trace a pen inside a plastic gear and create perfectly symmetrical, blooming flower patterns. It looks like pure art, but every loop and petal is actually tracing out precise mathematical equations.

These shapes belong to a family of curves called roulettes. A roulette is simply the path drawn by a point (your pen) attached to a moving shape (the small gear) as it rolls along a fixed path (the big ring).

Rolling Inside: The Hypotrochoid

When you put your small gear inside the big ring, the shape you draw is called a hypotrochoid.

To understand the math, think about riding a bicycle inside a giant circular track. The small gear is your bicycle wheel. As the wheel rolls along the track, the center of the wheel travels in a smooth circle. But what about a reflector attached to your wheel spokes? That reflector is spinning around the wheel’s center, while the whole wheel travels around the big track.

The final pattern is determined by three things:

• The size of the big ring ($R$).
• The size of the small gear ($r$).
• Where you put your pen ($d$, the distance from the center of the small gear).

The ratio between the big ring and the small gear is what decides how many "petals" your drawing will have. If the gear’s teeth fit exactly into the ring a whole number of times, your drawing will close up neatly on its first trip around. If the numbers don't divide evenly, the gear has to make many laps before the pattern finally connects back to the start!

Rolling Outside: The Epitrochoid

What if you flip it around and roll the small gear on the outside of the fixed ring? This creates a pattern called an epitrochoid.

Think of the Teacups ride at an amusement park. The whole platform rotates in a big circle, and your individual teacup spins in its own smaller circle. If you tried to trace your exact path from above, you'd draw an epitrochoid. Because the rolling happens on the outside, the loops tend to point outward, creating star-like shapes instead of flower petals.

The Math Behind the Motion

Computers can draw these shapes instantly using trigonometry (sine and cosine). The equations just add up the two motions: the slow, wide circle of the gear moving around the ring, plus the fast, tight circle of the pen spinning around the gear’s center.

This is a perfect example of kinematics—the part of physics that studies how things move (their paths and speeds) without worrying about what is pushing or pulling them.

Try the Spirograph Experiment →