The function x²ⁿ + y²ⁿ = 1 is the unit-circle for n=1, but approaches the unit-square as n→∞pic.twitter.com/AVx8wt8XaY
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Here's the proof. When n is large, then x²ⁿ≈0 and y²ⁿ≈0 for all |x|,|y|<1. So everything not strictly equal to 1 will get mapped to 0, as n→∞. Therefore the the limiting solutions to x²ⁿ + y²ⁿ = 1 are (±1,y) with |y|<1 (x, ±1) with |x|<1 QED
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Well... What is /the/ unit square anyway...
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unit square for the max norm n(x,y) = max(|x|,|y|)
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Good enough.
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If one of variable becomes complex, then when 2n goes to infinity one got a cylinder?
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lp balls should be used in ads as tv screens!
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