The ball, Bⁿ, is the subset of Rⁿ (this is n-dimensional real space) equal to the set of all real n-tuples (x₁,..,x_n) such that the square root of (x₁² + ... + x_n²) is less than or equal to one. Brouwer's fixed point theorem states that if f is any continuous function from Bⁿ to Bⁿ then there is some point x in Bⁿ with the property that f(x) = x. x is called a fixed point of f. For example, B² is just the unit disk in the plane. The function f((x,y)) = (x * cos (a) - y * sin (a), x* sin (a) + y* cos (a)) for some angle a, is the map which rotates the disk by the angle a. It has one fixed point at (0,0). The geometric shape of the ball is not...