Mathematical Experiment 3

  1. Let X and Y be two given triangles of arbitrary shape. Construct a program that demonstrates interactively how the area of the set difference
  2. D(z) = X D(Y+z) = (X\(Y+z))È((Y+z)\X)
    between X and the translation
    Y+z = {y+z:yÎY}
    of Y by z varies according to the parameter z. This would be an experimental attempt to find the minimum value of D(z). Can you be slightly more ambitious and try to find the minimum area of XD(aA+z), where a runs over all complex scalars of absolute value one and z runs over all complex scalars? Still more generally, find the minimum area of X D(lA+z) as l and z run over all complex scalars.
  3. Let C be the cube with vertices at (±1,±1,±1). Make a movie showing how C is being observed from the spiral
  4. (2cos t,2sin t,t),tÎ [ -4,4]
    aiming at the origin (0,0,0).
  5. The following is a typical Ph.D. oral qualifying examination question:

  6. You are given a stack of pancakes of various sizes. The only operation you can perform on that stack is to use a spatula to lift the top n, for any n, pancakes from the stack and return them to the stack in reverse order (flip them back on upside-down). Can you sort (largest at the bottom) an arbitrary stack of pancakes using that one operation (you may use repeated instances of the operation, and may choose any n for each instance)? If so, how?

    Write a program to simulate the operations manually.