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Non-Euclidean Hyperbolic Spirographs: When Rolling Circles Bend Through Curved Space

What happens when Spirograph rules are applied to curved space? The familiar hypotrochoids and epitrochoids warp into infinitely detailed radial webbing.

The gallery at tthtlc.github.io presents 62 explorations of rolling-circle curves—classic epitrochoids, hypotrochoids, and their variations. These are Euclidean constructions: a circle rolls along a flat track, a pen traces a curve, and the result is symmetric, predictable, and finite. But what if the track itself were curved? Not the gentle curvature of a circle, but the negative curvature of hyperbolic space?

This is the premise of hyperbolic spirographs.

The Euclidean Baseline

A traditional Spirograph works through a simple mechanism. A small toothed wheel rolls inside (or outside) a larger fixed ring. A pen placed through a hole in the moving wheel traces a curve as the wheel rotates. The resulting hypotrochoid or epitrochoid depends on three parameters: the radius of the fixed circle, the radius of the rolling circle, and the distance of the pen from the rolling circle’s center .

The mathematics is well understood. The parametric equations are trigonometric, and the curves close after a predictable number of rotations. They are beautiful, but bounded. Each curve exists within the finite region defined by the fixed circle.

Enter Hyperbolic Space

The Poincaré disk model represents the hyperbolic plane as the interior of a unit circle. In this model, hyperbolic lines are circular arcs that meet the boundary at right angles. Distances are distorted: as you approach the boundary circle, Euclidean distances shrink, but hyperbolic distances grow without limit .

This has a striking consequence. A shape near the center of the disk looks Euclidean. But as it moves outward, it must compress infinitely in the Euclidean representation to maintain its hyperbolic size. What begins as a simple loop near the center becomes ever more tightly packed as it approaches the boundary. The pattern never terminates; it approaches the edge asymptotically, creating the illusion of infinite detail.

M.C. Escher exploited this property in his Circle Limit prints, where fish or angels and devils repeat infinitely, each the same hyperbolic size, growing smaller and denser toward the boundary .

Spirographs in Curved Space

Now combine the two.

A hyperbolic spirograph would take the familiar rolling-circle construction and transplant it into the Poincaré disk. The rolling circle, instead of moving along a straight or circular track in Euclidean space, would roll along a hyperbolic geodesic—a circular arc orthogonal to the disk boundary.

The resulting curve would be a hyperbolic hypotrochoid or epitrochoid. Its properties would be distorted by the curvature.

Near the center of the disk, where hyperbolic geometry approximates Euclidean geometry, the curve would resemble a conventional Spirograph pattern. Loops would be smooth and evenly spaced. But as the rolling circle moved outward, the distortion would take hold. The loops would compress. The spacing would tighten. The curve would begin to accumulate near the boundary, creating a dense radial webbing that extends infinitely.

This is the visual signature of negative curvature: exponential expansion of detail rather than the bounded symmetry of Euclidean patterns.

The gallery’s Spirograph section includes hypotrochoids with rotation and offset controls, epitrochoids with scrollbar exploration, and animated rolling-circle visualizations. These are Euclidean, but they provide the raw material.

Consider what happens when you apply a hyperbolic transformation to these curves. A simple hypotrochoid, projected onto the Poincaré disk, becomes a curve that begins fine and tightly packed near the circular boundary and expands into sweeping geometric loops as it bends through the center. The loops that would close in Euclidean space now spiral outward, each iteration smaller in Euclidean terms but identical in hyperbolic terms.

The result is not a closed curve. It is a curve that approaches the boundary asymptotically, weaving an infinite radial pattern that echoes the symmetry of Escher’s tessellations but with the organic, rolling motion of a Spirograph.

Why This Matters

Hyperbolic spirographs sit at an intersection of mathematics, art, and computation.

Mathematically, they are a natural extension of roulette curves into non-Euclidean geometry. The parametric equations become more complex, involving hyperbolic trigonometric functions and Möbius transformations. The curves that result have properties that are not obvious from their Euclidean counterparts.

Visually, they produce patterns that are simultaneously familiar and alien. The rolling-circle motion is recognizable, but the curvature warps it into something that feels infinite and organic, like a fern unfurling or a fractal branching.

Computationally, they are tractable. The gallery demonstrates that interactive exploration of rolling-circle curves is possible with scrollbars and animation. Extending these tools to hyperbolic geometry requires only a change in the distance metric and line equations—a shift in the coordinate system, not a fundamental redesign.

The Infinite Webbing

Traditional Spirograph curves close. They have a definite beginning and end, a finite number of loops, a boundary defined by the fixed circle.

Hyperbolic spirographs do not close. They spiral outward, each loop a hyperbolic isometry of the last, each one smaller in Euclidean terms but identical in hyperbolic terms. The pattern never repeats exactly, never terminates, never reaches the edge. It just becomes finer and denser, an infinite radial webbing that bends through the center and accumulates at the boundary.

This is what happens when Spirograph rules meet curved space. The geometry warps. The curves stretch. And the result is a pattern that Escher might have drawn, if he had owned a Spirograph set and an understanding of hyperbolic transformations.