My current Twitter name highlights the fact that the function f(z)=1/(1-z) has the property that f(f(f(z))) = z. Here's a nice way to visualise that:pic.twitter.com/Rbw5RHeZ56
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Take the complex plane. Take a 3D sphere of radius √3/2 and place it with centre 1/2 in complex plane. Given any point z in complex plane there is a straight line joining it to "north pole" of sphere (which I've labelled ∞). Take point where line intersects sphere.
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This gives a map from plane to sphere-{∞} and vice versa. Consider this map: take z, map to sphere, rotate through angle 2π/3, project back from sphere to plane. The result is the point f(z).
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This gives a nice picture for f(f(f(z))) = z. f(f(f(z))) means project on sphere, rotate through 2π, project to plane. Clearly this is the identity. Note that the two fixed points of z=f(z) correspond to the "poles" of the rotation.
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There's a much bigger story here. Like why this particular sphere? But that's enough tweets for today.
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(Sorry about bad frame in animation where point jumps. It shouldn't. Not sure of reason.)
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BTW If you use the unit sphere at the origin and a rotation by π you'll get the function f(z)=-1/z.
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Now use the unit sphere at the origin. Every point in the plane corresponds to a point on the sphere. But the sphere has an extra point I called ∞. This sphere, including ∞, is called the Riemann sphere and you can now think of this is an extension to the complex plane.
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With this representation of the complex numbers, ∞ behaves a lot like you expect. So f(z)=-1/z corresponds to a rotation through π. It takes the south pole, representing 0, to ∞ and vice versa.
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For example, all of the functions f(z)=(az+b)/(cz+d), extended in the "obvious" way to include ∞, become nice continuous functions on the entire sphere. There are the Möbius transformations. There is an enormous world of geometry lurking here: https://en.wikipedia.org/wiki/M%C3%B6bius_transformation …
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If you like this and would like to make it rigorous, learn #ProjectiveGeometry! Founded by artists, studied by mathematicians, applied by physicists and internet security protocols. https://en.wikipedia.org/wiki/Flagellation_of_Christ_(Piero_della_Francesca) …
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