Geometry

Abstract

Geometry is in many ways opposite or complementary to arithmetic. Arithmetic is discrete, static, computational, and logical; geometry is continuous, fluid, dynamic, and visual. The fundamental geometric quantities (length, area, and volume) are familiar to everyone but hard to define. And some “obvious” geometric facts are not even provable; they can be taken as axioms, but so can their opposites. In geometry, intuition runs ahead of logic. Our imagination leads us to conclusions via steps that “look right” but may not have a purely logical basis. A good example is the Pythagorean theorem, that the square on the hypotenuse of a right-angled triangle equals (in area) the sum of the squares on the other two sides. This theorem has been known since ancient times; was probably first noticed by someone playing with squares and triangles, perhaps as in Figure 2.1.