A famous problem in approximation theory is to approximate the function \left|x\right| as accurately as possible on the interval -1 \le x \le 1 , measured by maximum absolute error, using either polynomials or rational functions of fixed degree.
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I have a confession to make: I have always found symbolic algebra more intuitive than geometric pictures. I think you’re supposed to feel the opposite way, and I greatly admire people who think and communicate in pictures, but for me, it’s usually a struggle.
Robert Vanderbei has written a beautiful series of articles and talks about a method for finding the radius of the earth based on a single photograph of a sunset over a large, calm lake.