matharium(@matharium):A pen is driven by two perpendicular oscillations, x = 3 cos(50t) and y = 3 sin(49t). Since 49t = 50t − t, each lap of the x-wave is an ellipse x = 3 cos u, y = 3 sin(u − t) whose phase lags by the slowly growing t. It starts as a circle, flattens onto the diagonal y = −x at t = π/2, reopens at π, flattens onto y = x at 3π/2 and closes at 2π. Any two frequencies one apart weave the same square: n laps against n − 1, one full turn of phase.

matharium

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matharium@matharium

2026.10.27 11:37

A pen is driven by two perpendicular oscillations, x = 3 cos(50t) and y = 3 sin(49t). Since 49t = 50t − t, each lap of the x-wave is an ellipse x = 3 cos u, y = 3 sin(u − t) whose phase lags by the slowly growing t. It starts as a circle, flattens onto the diagonal y = −x at t = π/2, reopens at π, flattens onto y = x at 3π/2 and closes at 2π. Any two frequencies one apart weave the same square: n laps against n − 1, one full turn of phase.