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Twisting the Plane with Complex Numbers
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A computer animation by Jason Schattman that shows how complex-valued functions warp & twist the plane in stunning—and sometimes violent—ways. For example, the function f(z) = z^2 shown at the start of the video bends each vertical line of the grid into a left-facing parabola, and each horizontal line into a right-facing parabola. The square-root function (shown at 1:58) literally rips the grid in two! But my favourites are the reciprocal function f(z) = 1/z (at 1:23), which turns the grid inside out, and f(z) = sin(z) (at 2:20), which...well I won't give it all away. Just watch and enjoy. :)
More detail for the mathematically inclined
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These animations are illustrations of conformal mapping. A conformal map of a complex-valued function f(z) is a graph on the complex plane that shows how each vertical and horizontal line gets transformed by the function. (The complex plane is the set of all complex numbers of the form a+bi, where i is the square root of -1.)
For each point z in the grid that is being mapped, the program computes f(z) (think of this as the final destination of point z). The program then animates the journey of that point between z and f(z) over time. This is done using linear interpolation. At time t, where t ranges from 0 at the start to 1 at the end, let g(z, t) be the interpolated point between the starting point z and the ending point f(z). The program computes point g using the formula g(z, t) = t(f(z)) + (1-t)z, so that g(z, 0) = z at the start, and g(z, 1) = f(z) at the end, and g(z, 0.5) would be halfway between z and f(z).
To see more of my mathematical animations, check out my videos on...
Cool mathy fact: Watch how the quartic function f(z) = z^4 maps the vertical lines of the grid directly on top of the horizontal lines! Can you see how this relates to the fact that i^4 = 1?
With the code, you can easily create your own complex-valued functions and make new animations with them.
More detail for the mathematically inclined
********************************************
These animations are illustrations of conformal mapping. A conformal map of a complex-valued function f(z) is a graph on the complex plane that shows how each vertical and horizontal line gets transformed by the function. (The complex plane is the set of all complex numbers of the form a+bi, where i is the square root of -1.)
For each point z in the grid that is being mapped, the program computes f(z) (think of this as the final destination of point z). The program then animates the journey of that point between z and f(z) over time. This is done using linear interpolation. At time t, where t ranges from 0 at the start to 1 at the end, let g(z, t) be the interpolated point between the starting point z and the ending point f(z). The program computes point g using the formula g(z, t) = t(f(z)) + (1-t)z, so that g(z, 0) = z at the start, and g(z, 1) = f(z) at the end, and g(z, 0.5) would be halfway between z and f(z).
To see more of my mathematical animations, check out my videos on...
Cool mathy fact: Watch how the quartic function f(z) = z^4 maps the vertical lines of the grid directly on top of the horizontal lines! Can you see how this relates to the fact that i^4 = 1?
With the code, you can easily create your own complex-valued functions and make new animations with them.
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