Take a thin sheet of paper, plastic, or rubber. Roll, crumple, stretch, or tear it. Sometimes the sheet can spring right back to its original form, as with a roll of paper, while other times it is permanently changed, as with torn plastic. Much can be learned from such everyday acts. The subtle mathematics of differential geometry is needed to make sense of the deformed sheets, and along the way it offers insights into issues such as the shapes of flowers and the speed of earthquakes.
Start the study of surfaces with Carl Friedrich Gauss. Chiefly known today for his abstract mathematics, Gauss devoted years of his life to practical pursuits, from contributing to the invention of the telegraph to mapping the Kingdom of Hanover. 1 His most famous geographical measurement was of a triangle formed by the shortest paths between the three mountain peaks shown in figure 1....
