Fractal geometry is the natural geometry to describe shapes whose appearance is unchanged under magnification. As such, this is an efficient language for understanding shapes in nature. Chaotic dynamics is a way to analyze some kinds of behavior sensiti ve to tiny external effects. Many systems now thought to be chaotic were once believed to be random. One of the main themes of both fractal geometry and chaotic dynamics is that complicated appearances and behaviors can result from simple causes. These tools unpack some of the apparent complexity of the world, and yet, both also reveal other kinds of complexity. After developing the basic ideas of fractals and chaos, we present a new, visual method for quantifying the complexity of natural and mathema tical objects. We end by applying this to some of the oldest fractals - those produced by the geometric construction of inversion in circles.