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π supremacy

Math Magician

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45,726 次观看 • 3 年前 •via X (Twitter)

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A double integral is the limit of a sum of prisms: cut [0, π]² into n × n squares of area ΔA = (π/n)² and raise each to the height of z = sin³x sin³y at its centre. The volume is (∫₀^π sin³x dx)² = (4/3)² = 16/9 ≈ 1.7777778. The prisms give 1.7602152 at n = 4, 1.7768141 at n = 8 and 1.7777776 at n = 64, always from below. Since sin³x = (3 sin x − sin 3x)/4, the h² errors of the two waves cancel, and the gap shrinks like 7π⁴/(180n⁴).

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A radius turns at a steady rate inside a regular hexagon, a decagon and a circle, and the height y = r(θ) sin θ of its tip is traced against time. For an n-gon of circumradius R with a vertex at θ = 0, r(θ) = R cos(π/n) / cos((θ mod 2π/n) − π/n). Each edge is one piece of the wave; the flat top edge is a plateau at R cos(π/n), 0.866R for n = 6, 0.951R for n = 10. As n grows the plateaus narrow and rise to R, and the wave tends to R sin θ.

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