Computing π Through Physics

Two elastic blocks and a wall turn a mass ratio into a collision count. Watch the physical motion beside its mass-weighted momentum trace: energy fixes a circle, while the ratio fixes the angle between two reflection lines. The tally comes from the simulated impacts; the expected answer is a separate reference.

Replace the small block with a rolling body. The no-slip constraint folds rotation into m_eff = m + I/r². Keeping the heavy-to-effective-mass ratio fixed lets a disc, sphere or hoop keep the same digits. That connection between spin and a simple collision counter is the second step of the story.

From a block to a rolling circle, then a piston with an atom that rises and falls: the scene becomes busier, yet the extra motion never feeds back into the along-rail collision law. This is about what can change without changing the answer. The rise-and-fall response is stylized, not a resolved gas simulation.

The turning-point comparison runs the existing collision engines directly. Double launch speed, add rolling inertia while preserving the mass ratio, or lower restitution: compare the recorded momentum states and final impact counts. The distinction is visible before the collection branches out.

Imperfect machines break different assumptions. Restitution changes the bounce, an off-center load changes contact inertia, soft forces replace instantaneous impacts, and gravity acts between them. Each has an ideal limit and an observable source of drift. Loss can lengthen a count; inelastic collapse can destroy termination entirely.

These are controlled experiments, not general rigid-body solvers. For example, the Loaded Disc uses a quasi-static contact-inertia model. A shared interface separates the measured result, reference, bias source and failure state. More sampling does not repair a wrong model.

Mass weighting turns the two contact constraints into mirrors separated by α = arctan√(m/M). Draw that wedge in real space and replace the collisions with a ray. Reflect the room instead of the ray: the zig-zag unfolds into a straight line crossing copies of the same wedge.

The opening is constructed from a slope, so π is not put into the input angle. Finite Nα approximates a half-turn; cone and cylinder views refold the same developed path without inventing a separate counting mechanism.

An even simpler geometric interpretation gives another family: polygon perimeters bound the circle from inside and outside, while shape relaxation drives L²/(4A) toward its circular optimum. Length, area, and bounds take over from collision counting.

Three blocks with a wall have three coupled velocities. Constant energy gives a sphere cut by several reflection planes, rather than the original one-angle circle. There is no general rule that turns this collision tally into the digits of π. Removing the wall is special: total momentum supplies a second constraint, reducing the free three-body system to a planar circle.

The new optional 3D chamber-area counter uses a different observable. It samples independent velocity directions and counts states in the escape cone: 0 ≤ v₁ ≤ v₂ ≤ v₃. Mass scaling steers samples towards a thin chamber; a solid-angle Jacobian corrects their weights. The chamber’s analytically computed solid angle and weighted hit rate together estimate π.

This is Monte Carlo area measurement, not a claim that the deterministic impact path covers the sphere uniformly. The display keeps the collision experiment separate, reports sampling uncertainty, and remains a statistical estimate rather than an exact digit counter.

A square resistor lattice offers a very different route: solve its voltage field, measure a resistance and compare finite boundary conditions. Saturated signed potential colours and gold current paths make the field solve visible alongside the numerical bracket.

The liquid experiments extend the question to e. In the random-pour glass, a trial ends when uniform random increments first cross unit volume; the mean number of pours approaches e. In the draining vessel, an exponential flow law removes the same fraction in equal times, and changing its exponent makes that e readout drift.

Waves, chaotic maps, random matrices, Buffon crossings and integer sieves remain further routes. Their readouts distinguish exact constructions, statistical uncertainty, finite numerical error and deliberate model bias. The Galton board here uses independent fair branching, providing a useful comparison with Pipeline’s simulated bead contacts.