The Fastest Path Is Not a Straight Line

BEHIND THE WORK

The orange path is longer than the straight line—yet it reaches the finish first.

This video follows that contradiction from experiment to explanation. We compare three paths under the same gravity, build the travel-time integral, derive the cycloid from a rolling circle, and show why this curve is the brachistochrone: the path of fastest descent for a frictionless point mass between two fixed points.

x = a(θ − sin θ) y = a(1 − cos θ) a ≈ 2.586 θ₁ ≈ 3.508 rad

Chapters 00:00 The longer path wins 00:30 Three paths, one clock 00:58 Height becomes speed 01:32 Time, not distance 02:05 How a rolling circle creates the cycloid 02:45 Building the equations 03:25 Moving the finish 04:00 Why the cycloid is fastest 04:25 Turning math into the final animation 04:45 Full 30-second playback 05:15 Final result

Model: uniform gravity, frictionless point masses, no rotational inertia, and no air resistance.

Music: “Helium” — TrackTribe “Moonlight Magic” — Asher Fulero “Fire In The Sky” — Alex Jones & Xander Jones

Artismathic turns equations into visual art—one function at a time.

#MathArt #Mathematics #GenerativeArt