Light’s Curvature: Gravitational Lensing and Circular Paths

I am drawn in every circle ever made, yet I hide in places you might not expect to find me. Where straight lines end, I begin—and nowhere is this more evident than in the cosmos itself, where massive objects curve spacetime so severely that light’s “straight” path becomes circular.

When Geodesics Draw Circles

Gravitational lensing reveals my cosmic signature. Near black holes, spacetime curves into such extreme geometry that light following geodesics—paths that are locally straight—bends back upon itself. The back of an accretion disk folds around the event horizon, creating bright rings where photons complete impossible journeys. Einstein rings form when source, lens, and observer align perfectly: my geometry manifest at astronomical scale, light paths becoming circumferences around massive lenses.

This isn’t light bending in the way you might imagine. Light doesn’t “know” it’s curving—it follows geodesics locally, just as freely falling objects follow what feels like straight motion through curved spacetime. Yet geodesics that begin parallel don’t stay parallel in positively curved space. They converge, meeting like meridians approaching a pole on a sphere. What appears as gravitational attraction emerges from geometry itself: paths that curve back toward each other, circles hiding in the fabric of space.

Periodicity’s Infinite Return

I appear wherever systems cycle back to their starting point. Analog clocks trace my presence twelve times daily, their hands implementing modular arithmetic through pure rotation. Add hours, and the answer wraps around—circular motion physically manifesting the mathematics of periodic return. The number line becomes a circle, and suddenly addition requires different rules: you must add angles, not just magnitudes.

Hippocampal theta rhythms oscillate at 4–12 Hz, neural populations coordinating through pacemaker cells in the medial septum. These aren’t arbitrary electrical fluctuations but precisely periodic signals, phases cycling through 360 degrees and returning to zero. Theta provides a memory clock—a circular temporal structure organizing neural computation during navigation and exploration. Phase, frequency, oscillation: wherever you find these words, you find me.

The Constructible Transcendent

Here is my paradox: I am irrational, transcendental, impossible to express as any finite polynomial’s root. My digits extend infinitely without pattern or end. Yet I am geometrically constructible—compass and straightedge can approximate me to arbitrary precision. Infinite complexity emerges from finite operations, just as composable transformations in deep networks stack simple operations recursively to create sophisticated patterns.

Gravitational lensing creating circular light paths. Theta rhythms’ periodic oscillations. Clock hands tracing modular arithmetic. Each reveals my presence in systems where curvature or periodicity emerges—where geometry curves back on itself or time cycles through phases. I am the ratio of circumference to diameter: 3.14159…, appearing wherever circles hide. And circles hide everywhere.