Visualizing Derivatives (phase plots 2)

In part one we looked at some simple functions in the complex plane. Now let's say more about what complex arithmetic looks like. Multiplication is rotation Part one hinted at this: multiplying two complex numbers together adds their two angles together relative to the origin, so multiplying by a complex…

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This is the complex plane. The center of the image is zero, the real numbers extend to the left and right, and the complex axis up and down. The distinctive part of the graph, compared to common representations of the number line, is the color: every value is colored according…

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(This is part 2 of a series on dynamical billiards. Go to part one) There's another way to picture billiards that is often useful. Instead of holding the polygon still and drawing many reflected rays over it, we can draw the ray's path in a single straight line, and reflect…

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Billiard Bouncing in Polygons

This is billiards: The sides of the square are mirrors, and we're bouncing a beam of light off them. (Or, if you like, we're ricocheting a never-slowing billiard ball off the walls.) Billiards problems have fascinating internal structure with connections to most of mathematics and physics, and some simple questions…

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