One of the marvels of mathematics is the way in which an austere equation can unexpectedly blossom into an appealing geometric shape when represented graphically.
![](https://i0.wp.com/www.sciencenews.org/wp-content/uploads/2002/02/1001.jpg?resize=130%2C150&ssl=1)
![](https://i0.wp.com/www.sciencenews.org/wp-content/uploads/2002/02/1002.jpg?resize=150%2C150&ssl=1)
![](https://i0.wp.com/www.sciencenews.org/wp-content/uploads/2002/02/1003.jpg?resize=150%2C81&ssl=1)
![](https://i0.wp.com/www.sciencenews.org/wp-content/uploads/2002/02/1004.jpg?resize=150%2C135&ssl=1)
I can still recall experiencing such a sense of wonder when, decades ago, I first learned about polar coordinates in a high-school trigonometry class. In those days, there were no graphing calculators (or even pocket calculators), and all plotting was done by hand, point by point, on sheets of graph paper.