Into the Void: When Spirographs Meet Hyperbolic Space

We all remember the classic plastic Spirograph gears from childhood. You pin down a ring, put your pen in a smaller gear, and trace out mesmerizing, looping patterns—mathematically known as hypotrochoids and epitrochoids.

But what happens when those rules are applied to curved space? How does hyperbolic geometry warp classic geometry into infinitely detailed radial webbing?

Today, we’re diving into a fascinating variation from the realm of generative math graphics: Non-Euclidean Hyperbolic Spirographs.

The Concept: Escaping the Flatland

Traditional spirographs roll circles along straight or flat circular tracks on a standard, flat piece of paper (Euclidean space). In this variation, we take those exact same rotational equations but project them onto Poincaré disk models or hyperbolic planes.

If you’re familiar with M.C. Escher’s famous Circle Limit woodcuts (where bats or angels shrink infinitely as they approach the edge of a circle), you have seen a Poincaré disk. It maps an infinite, negatively curved space into a finite, flat circle.

When you map a spirograph’s “gears” into this space, straight lines become arcs, and distances compress exponentially the further you get from the center.

The Visual Profile

What does a hyperbolic spirograph actually look like?

Imagine taking the complex geometric loops above and bending them through a funhouse mirror of negative curvature.

The Core: Near the center of the disk, the space behaves relatively normally. The curves here expand into massive, sweeping geometric loops that cross over each other with wide, elegant arcs.
The Boundary: As the “gears” roll outward toward the edge of the disk, the hyperbolic metric takes over. Space compresses. The spirograph curves begin to pack tightly together, becoming infinitesimally fine.
The Web: The result is an illusion of depth and infinity—a massive central eye of geometry that fades into a microscopic, infinitely dense radial web at its borders.

The Math Behind the Magic

To code this, generative artists can’t just use standard (x, y) sine and cosine functions. They have to map standard Euclidean coordinates through what are known as Möbius transformations, which calculate distance differently. In hyperbolic space, moving one “inch” near the edge of the boundary circle covers far less visual space on our screens than moving one “inch” in the center.

By feeding standard epitrochoid equations through a hyperbolic metric, the resulting trace leaves behind one of the most stunning—and rarely seen—variations of the classic childhood toy.