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One of the most famous such theorems is the Brouwer fixed-point theorem, which says in its simplest form that any continuous function <math>f: D^2 \to D^2</math> has a fixed point.  This fact has a very intuitive description: if we take a sheet of graph paper (so we can label each point with coordinates), crumple it up, and set it on top of another sheet of graph paper, then there is some point on the crumpled up sheet which is directly above the corresponding point on the second sheet!  While this sounds like a simple fact, proving it rigorously relies on the techniques of [[algebraic topology]].

One of the most famous such theorems is the Brouwer fixed-point theorem, which says in its simplest form that any continuous function <math>f: D^2 \to D^2</math> has a fixed point.  This fact has a very intuitive description: if we take a sheet of graph paper (so we can label each point with coordinates), crumple it up, and set it on top of another sheet of graph paper, then there is some point on the crumpled up sheet which is directly above the corresponding point on the second sheet!  While this sounds like a simple fact, proving it rigorously relies on the techniques of [[algebraic topology]].

Another celebrated example of a fixed point theorem is known as Sharkovsky's theorem.  In addition to fixed points, some functions have points for which repeating the function several times yields the same result: for example, a value of x for which cos(cos(cos(x))) = x is called a 3-periodic point of the function cosine.  Sharkovsky's theorem is the surprising result that if a continuous function f has points of period 3, then it has points of all other periods!  For example, consider the quadratic <math>f(x) = -\frac{5}{2} x^2 + \frac{7}{2} +1</math> has f(0) = 1, f(f(0)) = f(1) = 2, f(f(f(0))) = f(f(1))=f(2) = 0.  So 0 is a point of period 3 of f(x).  Sharkovsky's theorem implies that there are points of every other period.  For example, there is some x such that f(f(f(f(f(f(f(f(x)))))))=x!  Solving this explicitly would involve solving a polynomial of degree 14, an extremely difficult task.  But Sharkovsky's theorem guarantees that there is '''some''' value of x satisfying this.

Another celebrated example of a fixed point theorem is known as Sharkovsky's theorem.  In addition to fixed points, some functions have points for which repeating the function several times yields the same result: for example, a value of x for which cos(cos(cos(x))) = x is called a 3-periodic point of the function cosine.  Sharkovsky's theorem is the surprising result that if a continuous function f has points of period 3, then it has points of all other periods!  For example, consider the quadratic <math>f(x) = -\frac{5}{2} x^2 + \frac{7}{2} +1</math> has f(0) = 1, f(f(0)) = f(1) = 2, f(f(f(0))) = f(f(1))=f(2) = 0.  So 0 is a point of period 3 of f(x).  Sharkovsky's theorem implies that there are points of every other period.  For example, there is some x such that f(f(f(f(f(f(f(f(x)))))))=x!  Solving this explicitly would involve solving a polynomial of degree 14, an extremely difficult task that is not guaranteed to have a solution expressible in terms of ordinary arithmetic operations and taking <math>n</math>th roots (cf. the [[Abel-Ruffini Theorem]]).  But Sharkovsky's theorem guarantees that there is '''some''' value of x satisfying this.

==== Mathematical Billiards ====

==== Mathematical Billiards ====