In 1889, while investigating the notorious problem of three gravitationally attracting bodies, Henri Poincaré discovered a tangled topological structure that we now recognize as the mathematical heart of chaotic motion. Poincaré would later write of his discovery, "One is struck by the complexity of this picture, which I do not even attempt to draw." Using the driven pendulum as an example and graphics as our primary tool, we will explore the topology of state space, where the trajectories of a system are visible as the streamlines of a flow. Saddle orbits within a flow are the keys to its topology and led Poincaré and his successors to understand that the trajectories of seemingly simple systems can be entwined in infinitely complex patterns. In 1960 Steve Smale introduced the horseshoe map as perhaps the simplest example of a state-space tangle and applied symbolic dynamics to demonstrate how chaos can be "as random as a series of coin flips.