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Your Universe of Digital Possibilities
The same law as The Metronome — θ̈ = −(g/l)·sin θ — with one more joint, which the law never mentions. One joint keeps time for four centuries; two joints forget their past in seconds. The plane on stage is the atlas of every possible release, coloured by the time until the second arm first flips over the top: an exact energy line carves out a smooth island where it can never happen, and everywhere else the answer is fractal — move a hair, wait a lifetime longer.
A mass on a pivot under gravity — the most familiar law on Earth, and the whole of it. One line, no forcing, no friction, no dice. The law says nothing about how many joints the mass hangs from; that silence is where the two tempers come from.
The one-line law, written out for two joints — no new physics entered, only bookkeeping. This is the whole of what each atlas pixel integrates (Δ = θ₁ − θ₂, RK4).
The two arms are nonlinearly coupled — the source of the unpredictability. We integrate both with RK4 so energy doesn’t drift.
Energy alone decides who may flip: released at rest inside this line, a double pendulum (equal masses, equal arms) can never send its second arm over the top — however long you wait. Outside it, the flip is allowed, and the fractal decides how long you wait.
This is the other temper. The Metronome’s phase space is a plane, so its orbits are fenced onto closed curves forever — a clock by construction. Hinge one more joint and the energy shell becomes three-dimensional: room to tangle, and tangle it does — The Divergence fans an ensemble of these same arms apart to price the forecast this instrument refuses to make. The λ here is the billiard’s stretch rate wearing gravity (The Billiard measures it per bounce); the atlas plane is a sibling of The Basin’s root map — colour every start by its destiny and the boundary turns out to be the point. The silence between the two tempers is one integer: the number of joints.