Aristotle's wheel paradox • If you roll a rigid wheel a distance of its circumference, each inner circle must also go that same distance wait..but how?pic.twitter.com/M5JU06Rccv
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Resolution: • Consider the 2 points 𝐵 & 𝑆 (big/small) • Each travels 2πR horizontally (R=rad. of B) • 𝑆's Cycloid path is *shorter* than 𝐵's ⇒ The closer 𝑆 is to the center, the shorter and more direct the path ∴ 𝑆 goes 2πR horizontally b.c it takes a shorter pathpic.twitter.com/UJyfSAuKJg
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I noticed after the fact that the main gif is somewhat misleading So I plotted the cycloids along the wheel, which makes the resolution to the paradox much easier to see.pic.twitter.com/uZ4gvYEHVF
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Also think its misleading. Circles of different radii cannot have the same circunference length.pic.twitter.com/GHXf6C6YrZ
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I think the paradox is independent of cicloids
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It is true that even the wikipedia takes the opportunity to exhibit a cycloid, but note that it is original work, against wikipedia own rules. Of course you can always recover the paradox for any function f(x) by discussing how its line integral betwen a, b differs from b-a
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