Billiard-Ball Computer 2D: Conservative Logic Gate

Inputs (bits)

Launch speed 1.4

Ball radius 0.35

AND detector (collision)

no collision yet

Recovered bits downstream

A-out

B-out

Live readouts

Cycle

Collisions

Total KE

0.00

Balls in flight

How it works

This is a native 2D re-derivation of Fredkin & Toffoli's billiard-ball model — not a flattened camera view of a 3D board. Two equal-mass "bit" balls launch from the top corners along paths that cross at the centre. Whether they actually meet there is the AND of the two inputs, computed with the real equal-mass 2D elastic-collision law:

n̂ = (r_A − r_B) / |r_A − r_B|
v_A' = v_A − [(v_A − v_B)·n̂] n̂
v_B' = v_B + [(v_A − v_B)·n̂] n̂

Unlike a simple AND-only detector, this board adds two fixed circular reflectors further downstream, sitting exactly on each ball's original straight-line path. Working through the collision algebra for this symmetric geometry shows something remarkable: when both balls are present, the elastic collision exactly swaps their velocity components along the line of centres, so each ball leaves on the other ball's original heading — and therefore still reaches its own input's reflector. The result, verified numerically below: the reflector on A's line lights up whenever A = 1 (regardless of B), and B's reflector lights up whenever B = 1 (regardless of A) — the original bits are fully recoverable downstream even after the interaction. That is Fredkin's "conservative logic" made visible: the collision event itself reveals A AND B, yet nothing about A or B is destroyed by computing it.

A, B buttons — arm each input ball for the next cycle.
Fire cycle — launches the armed balls; Auto-clock cycles all four input combinations automatically.
Speed / radius — change flight time and collision cross-section without changing the logic.
Total KE stays flat through every collision and every reflector bounce — both interactions are elastic, so no energy is dissipated (no k_BT·ln 2 Landauer cost, unlike an irreversible bit erasure).

Ball A Ball B Fixed reflector