Let S be a spherical cap with distance taken along great circles. Show that we cannot find a distance preserving map from S to the plane.
Let O be the center of the cap and C its perimeter. Then C is a circle center O. Its image must also be a circle with the same radius R, since the distance between each point of C and O is preserved. The circumference of the circle is also preserved. But the circumference is not equal to 2π R.
15th Putnam 1955
© John Scholes
26 Nov 1999