Two smaller images depict quite different aspects of mathematics that had powerful utility and applications. This image shows the ancient geometrical proof of the Pythagorean theorem, that the area of the square of the hypotenuse of a right triangle is the sum of the areas of the squares of the adjacent sides. This proof was accomplished through nothing other than simple manipulation of tiles and its eternal truth grasped by the mind alone. Geometry dealt with continuous quantities, with no measurement or counting allowed. Yet, it provided a very practical and simple way to survey land, assess quantities, and support a host of practical activities. When the Academy dome was being designed, the theory of relativity in physics and the cosmological implications of the expanding universe enthralled the public. Both relied on an underlying grasp of alternative geometries that Euclid or Archimedes could not have imagined.