⚡ Brachistochrone & Tautochrone: The Curve of Fastest Descent

In 1696, Johann Bernoulli challenged the world to find the path of fastest descent between two points under gravity. The answer is not a straight line—it is a cycloid! Test the race in real-time and discover the "magic" Tautochrone property where balls released from different heights finish at the exact same millisecond.

Straight Line:
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Circular Arc:
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Cycloid (Winner!):
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Steep Ramp:
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💡 Why the cycloid wins: It trades extra distance for an early steep drop, maximizing speed early in the run where it matters most!