Brouwer's Theorem

Brouwer's Theorem, often called the Brouwer Fixed Point Theorem, states that any continuous function mapping a compact, convex set in a Euclidean space (like a solid ball or a cube) to itself must have at least one fixed point. In simple terms, if you gently move or rearrange such an object without tearing or creating gaps, there will always be at least one point that stays exactly where it is. This theorem underpins many concepts in mathematics, economics, and computer science, illustrating the inevitability of equilibrium states in systems that are continuous and bounded.