005 / ORDER FROM CHAOS
One shared rhythm
Each metronome starts in its own rhythm. A freely moving platform connects them — no conductor, no hidden commands.
Can you stop them synchronizing?
Make the platform heavier or increase the frequency spread. The only link between rhythms is the reaction of their shared support. Synchronization is not guaranteed — that is what makes it interesting.
What is inside the model?
Each pendulum pushes the shared board; its acceleration affects every other pendulum. Bodies are fixed to the board: they contribute only translational mass, not pendulum rotational inertia.
A = M + N·m_body + Σmᵢ L = ½A·Ẋ² + Σ[mᵢlᵢ cosθᵢ·Ẋθ̇ᵢ + ½mᵢlᵢ²θ̇ᵢ² − mᵢglᵢ(1 − cosθᵢ)] cᵢ = mᵢlᵢ cosθᵢ, dᵢ = mᵢlᵢ² Qᵢ = [μ(1 − (θᵢ/α)²) − b] θ̇ᵢ, Qₓ = −BẊ Bᵢ = Qᵢ − mᵢglᵢ sinθᵢ Ẍ = [Σmᵢlᵢ sinθᵢ·θ̇ᵢ² + Qₓ − ΣcᵢBᵢ/dᵢ] / [A − Σcᵢ²/dᵢ] θ̈ᵢ = (Bᵢ − cᵢẌ) / dᵢ φᵢ = atan2(−θ̇ᵢ/√(g/lᵢ), θᵢ), Rₖ = |Σexp(ikφᵢ)|/N
R1 measures phase alignment, not stable synchronization: 100% means matching phases at this instant. Two opposite groups have low R1 and high R2. The circle shows individual phases; the line is their mean vector.
The escapement is approximated by smooth van der Pol drive. α sets the drive angle scale, not a target amplitude. The blade and slider illustrate an effective pendulum, not a full compound-balance model.
RK4, fixed 1/240 s step. Conservation error is meaningful only without drive and damping. Ticks follow real zero crossings, without quantization. Rollers: X/2 and angle X/(2R). Platform motion is not amplified.