(Roughly) Daily

Indeed. And as Jordana Cepelewicz reports, mathematicians are on the case…

Repetition doesn’t always have to be humdrum. In mathematics, it is a powerful force, capable of generating bewildering complexity.

Even after decades of study, mathematicians find themselves unable to answer questions about the repeated execution of very simple rules — the most basic “dynamical systems.” But in trying to do so, they have uncovered deep connections between those rules and other seemingly distant areas of math.

For example, the Mandelbrot set, which I wrote about last month [see also the almanac entry here], is a map of how a family of functions — described by the equation f(x) = x² + c — behaves as the value of c ranges over the so-called complex plane. (Unlike real numbers, which can be placed on a line, complex numbers have two components, which can be plotted on the x- and y-axes of a two-dimensional plane.)

No matter how much you zoom in on the Mandelbrot set, novel patterns always arise, without limit. “It’s completely mind-blowing to me, even now, that this very complex structure emerges from such simple rules,” said Matthew Baker of the Georgia Institute of Technology. “It’s one of the really surprising discoveries of the 20th century.”

The complexity of the Mandelbrot set emerges in part because it is defined in terms of numbers that are themselves, well, complex. But, perhaps surprisingly, that isn’t the whole story. Even when c is a straightforward real number like, say, –3/2, all sorts of strange phenomena can occur. Nobody knows what happens when you repeatedly apply the equation f(x) = x² – 3/2, using each output as the next input in a process known as iteration. If you start iterating from x = 0 (the “critical point” of a quadratic equation), it’s unclear whether you will produce a sequence that eventually converges toward a repeating cycle of values, or one that continues to endlessly bounce around in a chaotic pattern…

[Cepelewicz runs through mathemeticians’ efforts to understand– and find explanation, if not order– in the complexity, concluding with the “entropy bagel”…]

… Galois conjugates [see here] also paved the way to the discovery of a mysterious object dubbed the “entropy bagel,” a glowing fractal ring in the complex plane. Entropy is a measure of randomness; in this context, it measures how difficult it is to predict the sequence of numbers generated by iterating x² + c. In the last paper he wrote before he died in 2012, the renowned topologist William Thurston graphed the set of entropy values corresponding to almost a billion different real values of c — together with the Galois conjugates of those entropy values, which can be complex. The notion of entropy “is just on the real line, but somehow you can still see this shadow of the complex world,” Tiozzo said.

“You see that this is organizing itself into this incredible lacy fractal structure,” Koch said. “It’s so cool.” The entropy bagel is only one very complicated pattern that emerges from the iteration of real quadratic equations. “We’re still learning all these magical statements — little gems — about real quadratic polynomials,” she added. “You can always go back and be surprised by this thing you thought you knew extremely well.”…

Simple rules in simple settings continue to puzzle mathematicians, even as they devise intricate tools to analyze them: “‘Entropy Bagels’ and Other Complex Structures Emerge From Simple Rules,” from @jordanacep.bsky.social in @quantamagazine.bsky.social.

* Oliver Wendell Holmes

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As we untangle, we might spare a thought for cherished creator of chaos, Milton Supman (better known by his stage name, Soupy Sales); he died on this date in 2009. A comedian, actor, radio-television personality, and jazz aficionado, he is best remembered for his local and network children’s television series, Lunch with Soupy Sales (later titled The Soupy Sales Show), which ran from 1953–1966, a collection of comedy sketches frequently ending with Sales receiving a pie in the face, which became his trademark.

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