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Crazy Curves in the Complex Plane: Where Do They Come From?
Why rotations arise from the multiplication of complex numbers
“The shortest path between two truths in the real domain passes through the complex domain.” — Jacques Hadamard (1865–1963)
I was first introduced to the Mandelbrot set while reading the August 1985 issue of Scientific American. The cover story was a column by A. K. Dewdney called “Computer Recreations.” The graphics were enchanting. It suggested an infinite journey through an intricate and ever-curving 3-dimensional space. Here’s that article:
The story inspired me to go see Benoit Mandelbrot at a presentation he did with the composer Charles Wuorinen at the Guggenheim Museum in New York City in 1990. I had no way of knowing that a decade later, I would be working with Mandelbrot at Yale.
Even though the simple formula for generating the Mandelbrot set was clear, it was a mystery as to how it acquired its complex, curved, and spiraling contours. For reference, here is that formula:
