At first glance, the picture above could be mistaken for a piece of abstract art you’d find in a gallery. What’s striking is that it isn’t artwork — it’s the outcome of a single rule in mathematics, applied endlessly.
Professor Robert Benedetto, a Professor of Mathematics at Amherst College, was the speaker for the Math Colloquium on September 25th. In his talk, “A Beginner’s Introduction to the Mandelbrot Set”, he guided his audience through the foundations of complex dynamics and how they lead to one of the most iconic objects in mathematics: the Mandelbrot set.
Benedetto began by unpacking the idea of complex dynamics, which explores what happens when you take a complex number (one with both a real and an imaginary part) and feed it into a mathematical function repeatedly. Each time the function is applied, the result becomes the input for the next step, a process called iteration. A single rule, followed over and over again, can generate patterns that are anything but simple. To make sense of these behaviors, mathematicians turn to the complex plane, a two-dimensional map where every point represents a complex number. On this plane, each iteration traces a path that shows how that point evolves under the function.
Zooming out from individual iterations, Benedetto revealed how they give rise to the Fatou and Julia sets. The Fatou set is the calm region of the complex plane, where points behave predictably and small changes in starting values do not matter. The Julia set, by contrast, represents the butterfly effect, where even the tiniest change can lead to vastly different outcomes.
Finally, Benedetto introduced the Mandelbrot set, which ties all of these ideas together. One can study the behavior of points under iteration for each value of the parameter c in the function fc(z)= z2 + c. In this setup, c is a fixed complex number that serves as the parameter defining the function, while z is the variable point in the complex plane whose path we follow under iteration of the function. Starting at the point z = 0, if the values of the function do not escape to infinity for a certain value of c, then that value of c is included in the Mandelbrot set. Plotting all such values of c in the complex plane produces the iconic Mandelbrot shape.
Benedetto concluded by sharing a few important facts about the Mandelbrot set. For example, the Mandelbrot set tells us which Julia sets—those generated by fc(z) for each value of c—are stable and connected, and which are fragmented and chaotic. If c is within the Mandelbrot set, its Julia set is connected; if c is outside, its Julia set is disconnected. Yet, despite decades of study, many mysteries remain. Mathematicians still do not know the exact area of the Mandelbrot set’s boundary.
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