[1/2] Now that the ingenious @ChocoLinkage has posted linkages that compute both squares and cubes, I thought it would be fun to reverse-engineer the squaring one.
If you want to work through this in detail yourself, note that all lengths are multiples of 1/16.
The magic here…pic.twitter.com/K9xVNUuWSC
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[1/2] If anyone’s still bamboozled, the angle-doubling arises from pairs of non-convex quadrilaterals with pairs of equal sides. ABCD has side lengths 1/4,1/2,1/4,1/2. AFGB has side lengths 1/8,1/4,1/8,1/4. Because ∠ABG = ∠ABC, these quads are similar, and ∠BAF = ∠DAB.pic.twitter.com/eXF1SQaDfX
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[2/2] So ∠DAF=2∠DAB, and because AD=FJ, AF=DJ: ∠HDJ=2∠DAB Put ∠DAB=θ, ∠HDJ=2θ. The red & blue rhombi have sides 1/2, 1/4, so: AK=cos θ DH=cos(2θ)/2 Since AD=1/2, the identity (1+cos(2θ))/2 = (cos θ)^2 ⇒ AD+DH = (AK)^2. Other links impose some symmetries we assumed.
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I like the way the yellow and grey links force the red rhombus to sit symmetrically, with K lying on the horizontal line through AD, and the brown and grey links do the same for the blue rhombus and point H.
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Greg Egan is a national treasure
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That’s very kind of you — but it was
@ChocoLinkage who actually devised this! I’m just looking on, slack-jawed in amazement. - Još 1 odgovor
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Someone needs to 3D-print it.
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Maybe
@henryseg ? - Još 2 druga odgovora
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Is all this related to epicycles but on a restricted lattice? As inhttps://www.youtube.com/watch?v=QVuU2YCwHjw …
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