It approaches π.
Each cross-section is a circle with area π/x². Adding those slices from 1 to infinity gives a total volume of exactly π cubic units.
V = π ∫₁∞ 1/x² dx = π
EXPERIMENT 001 · MATHEMATICS
Gabriel's
Horn
A shape with enough room to hold a finite amount of paint—but an infinite surface to paint. Rotate the model and extend its visible tail to explore the paradox.
ACTIVE EXPERIMENTS
Choose something to test.
Small research environments with explicit limits.
001 · MATHEMATICS
Gabriel's Horn
Explore a surface that is infinite outside and finite inside.
Open below → 002 · ACOUSTIC ML Drones Sensor LabTest passive drone-sound detection, simulations, and a local phone microphone experiment.
Open experiment ↗Drones processes microphone audio locally in the browser. Raw audio is not uploaded or retained.
INTERACTIVE MODEL
A finite view of an infinite horn.
Drag to rotate. Scroll or pinch to zoom.
y = 1 / x, rotated around the x-axis
Volume of visible model—
Surface of visible model—
THE IDEA
Infinite outside.
Finite inside.
Start with the curve y = 1/x for x ≥ 1, then rotate it around the x-axis. The resulting horn narrows forever but never completely closes.
It grows without limit.
The circumference shrinks too slowly. When every increasingly narrow ring is added, the horn's total surface area diverges to infinity.
A → ∞
The apparent paradox: mathematically, the horn can be filled with a finite volume of paint, yet coating its entire surface would require an unlimited amount. A physical horn can never be infinitely long or infinitely thin, so this result belongs to the ideal mathematical model.
MORE TO COME