In 1982, Edward Fredkin and Tommaso Toffoli published a machine that computes with nothing but billiard balls and mirrors. No transistors, no wires, no electricity: identical hard spheres glide across a frictionless table, bounce elastically off each other and off fixed reflectors, and out of the resulting choreography comes any Boolean computation you care to name. It sounds like a joke, and it was deadly serious — the paper is called Conservative Logic, and its point was physical, not practical: if a purely Newtonian contraption can compute, and Newtonian mechanics is frictionless, deterministic and time-reversible, then the laws of physics place no floor under the energy cost of computation. Every collision can be undone; nothing is ever forgotten; no bit is ever erased. The billiard-ball computer is the concrete mechanical model that the abstract machinery of this course — reversible gates, conservative logic — was invented to describe.

Balls as bits

The encoding is austere. Fix a synchronised clock and a family of straight-line trajectories — think of them as the wires. At each clock tick, a given trajectory either carries a ball or it doesn't:

All balls are identical, move at the same speed (diagonally, in the canonical layout), and are launched in lock-step, so that balls whose paths cross arrive at the crossing at the same instant — if both are present. Logic happens at those crossings: whether the balls sail on or collide depends on whether both are there, and that dependence is the computation. Fixed mirrors (immovable reflectors) do the routing between collisions: they bend trajectories, delay a ball to keep it in phase with the clock, and shift paths sideways so the right crossings happen at the right times.

The interaction gate: logic from one collision

The whole computer rests on a single primitive, the interaction gate. Two trajectories, A and B, cross at a point. Two cases:

• one ball arrives alone — nothing blocks it, so it continues straight through the crossing on its original heading;
• both balls arrive — they collide elastically and deflect each other onto different outgoing paths, displaced from the straight-through ones.

So the gate has two input paths and four output paths, and which outputs carry balls is pure Boolean logic:

Read the columns: the deflected pair of paths each carry a ball exactly when both inputs were present — the collision computes AND (twice over). The straight-through paths carry A \wedge \neg B and \neg A \wedge B — "arrived alone". Step through the trajectories:

And notice the last column of the table: balls in = balls out, in every case. The collision never creates or destroys a ball; it only re-routes. This is conservative logic incarnate — the number of 1s is a conserved quantity, just as the Fredkin gate's algebra demands, because here a 1 is a physical object that Newton will not let you vaporise.

From one collision to any circuit

Is the interaction gate enough? Yes. Mirrors let you bend an output path onto any heading, delay it by any number of ticks (a zig-zag detour), and cross two paths without interaction (route them through the crossing at different times). With those wiring tools, Fredkin and Toffoli assembled interaction gates and mirrors into a full Fredkin gate — a controlled swap: the control ball's presence decides which of two paths the data balls exit on, and the construction routes every intermediate ball to an output, destroying nothing. We won't draw the full plumbing (it is an honest tangle of half a dozen collisions), but the assertion is what matters: billiard balls implement the Fredkin gate, and the Fredkin gate is universal for Boolean logic. Any circuit — an adder, a CPU, the computer you are reading this on — compiles down to a pattern of balls and mirrors on a big enough table.

And the whole assembly is reversible in the strongest possible sense: it is Newtonian mechanics. Film the table, play the film backwards, and every elastic collision is again a legal elastic collision. Reverse every ball's velocity at any instant and the machine conscientiously uncomputes, ball by ball, back to its input configuration. No state ever has two pasts — which, after four modules of this course, you will recognise as the signature of computation without erasure, and (via Landauer) computation with no lower bound on dissipated energy.

The catch: a computer balanced on a knife-edge

So why isn't your laptop a pool table? Because the billiard-ball computer is chaotically unstable. Hard-sphere collisions magnify errors: displace a ball's position by a hair and the error grows by a multiplicative factor at every collision — exponentially fast, the defining symptom of chaos. After a few dozen collisions, a misalignment of one part in 10^{15} has swollen into balls missing each other entirely, and the "computation" dissolves into billiard anarchy. Thermal jiggling, tiny mirror imperfections and ultimately quantum uncertainty all inject exactly such errors, and the model has no error correction — correction would mean discarding the error information, i.e. dissipation, the very thing the model was built to avoid.

Fredkin and Toffoli knew this perfectly well. The billiard-ball computer is a thought experiment, and what it proves is a statement about physics, not engineering: the laws of mechanics permit computation with zero dissipation — no theorem of physics forbids a computer that never erases and never heats up. Whether one can be engineered stably is a separate battle (adiabatic circuits and quantum computers are today's fronts). One sentence of theirs is worth carrying through the rest of this course: the energy cost of computing is a property of our devices, not of computation itself.