It's a topic we touch upon from time to time as it relates to things like howstock pricesbehave or how the economy works, but emergence is much more than that. Quanta Magazine's short 3-and-a-half-minute video from their In Theory series provides a taste of what it is and where else it is seen:

Quanta's John Rennie explains the wonderful way complex order comes into being:

Nature is filled with such examples of complex behaviors that arise spontaneously from relatively simple elements. Researchers have even coined the term “emergence” to describe these puzzling manifestations of self-organization, which can seem, at first blush, inexplicable. Where does the extra injection of complex order suddenly come from?

Answers are starting to come into view. One is that these emergent phenomena can be understood only as collective behaviors — there is no way to make sense of them without looking at dozens, hundreds, thousands or more of the contributing elements en masse. These wholes are indeed greater than the sums of their parts.

Another is that even when the elements continue to follow the same rules of individual behavior, external considerations can change the collective outcome of their actions. For instance, ice doesn’t form at zero degrees Celsius because the water molecules suddenly become stickier to one another. Rather, the average kinetic energy of the molecules drops low enough for the repulsive and attractive forces among them to fall into a new, more springy balance. That liquid-to-solid transition is such a useful comparison for scientists studying emergence that they often characterize emergent phenomena as phase changes.

If you have a spare hour and 13 minutes, you can hear economists Don Boudreaux, Mike Munger, and Russ Roberts discuss how emergent orders arise in human activities via an EconTalk podcast!

The Navier-Stokes equations describing the flow of fluids are among the most useful math equations that have ever been developed. So much so that if a mathematician should someday demonstrate they can either demonstrate the equations will always work or can provide an example where they do not, they will win a $1 million prize from the Clay Mathematics Institute for solving one of its Millennium Problems.

In doing that, the eventual winner of the prize will be able to predictably explain how order, in the form of the smooth, laminar flow of an incompressible (contant density) fluid, transitions into disorderly turbulence, as illustrated in the following mesmerizing two-and-a-half minute video from the Lutetium Project:

And they will be also be able to accurately describe how some sense of order might spontaneously emerge from turbulent flow, as recently modeled for a recent paper using the Navier-Stokes equations by Florian Reetz, Tobias Kreilos and Tobias M. Schneider of the Ecole polytechnique fédérale de Lausanne (EPFL). The following image from EPFL shows the emergence of an orderly, layered structure within a turbulent flow from their modeling work.

Here's why the application of Navier-Stokes equations to reveal this pattern is such a big deal:

Though physicists had observed this phenomenon experimentally, they can now explain why this happens using fundamental fluid dynamics equations, bringing them a step closer to understanding why particles behave in this way. [The Biggest Unsolved Mysteries in Physics]

In the lab, when a fluid is placed in between two parallel plates that are moving in opposite directions from each other, its flow becomes turbulent. But after a little while, the turbulence starts to smooth out in a striped pattern. What results is a canvas of smooth and turbulent lines running at an angle to the flow (imagine slight wind-created waves in a river).

"You get structure and clear order out of the chaotic motion of turbulence," said senior author Tobias Schneider, an assistant professor in the school of engineering at the Swiss Federal Institute of Technology Lausanne. This "kind of weird and very obscure" behavior has "fascinated scientists for a long, long time."

Physicist Richard Feynman predicted that the explanation must be hidden in fundamental equations of fluid dynamics, called the Navier-Stokes equations.

But these equations are very difficult to solve and analyze, Schneider told Live Science. (Showing that the Navier-Stokes equations even have a smooth solution at every point for a 3D fluid is one of the $1 million Millennium Prize problems.) So up until this point, no one knew how the equations predicted this pattern-forming behaviors. Schneider and his team used a combination of methods, including computer simulations and theoretical calculations to find a set of "very special solutions" to these equations that mathematically describe each step of the transition from chaos to order.

While not the kind of mathematical proof that will win the million-dollar Millennium Prize, Reetz, Kreilos and Schneider have created a remarkable demonstration of the ability of Navier-Stokes equations to accurately describe some of the chaotic mechanics of fluids in motion that are observed in the real world.

As for the status of the Navier-Stokes Millennium Prize, the jury is still out on whether Tristan Buckmaster and Vlad Vicol's initial work demonstrating that the equations do not always generate unique solutions will claim the prize, which we have previously described as the biggest math story of 2017. They have since teamed with Maria Colombo on a new paper (preprint available via arXiv) in following up the topic.

References

Florian Reetz, Tobias Kreilos & Tobias M. Schneider. Exact invariant solution reveals the origin of self-organized oblique turbulent-laminar stripes. Nature Communications 10, 2277 (2019). DOI: 10.1038/s41467-019-10208-x. 23 May 2019.

The Lutetium Project. The transition to turbulence. [Online Video]. 15 November 2016.

If you've ever had to deal with a pair of earphones after they've become tangled, you know exactly what kind of mess they can make and what kind of pain they can be to untangle. Is there anything you can do about it?

Before we go any further, let's draw some lessons from science for how cords can almost spontaneously become tangled from the following video:

Now, let's get to the practical matter of finding out how likely your cords will become tangled. In the following tool, we've adapted the math developed by Dorian M. Raymer and Douglas E. Smith in their 2007 paper to calculate the probability that your cord/string/rope will become tangled, assuming that it is made of a medium-stiffness material, based upon its length. If you're reading this article on a site that republishes our RSS news feed, click here to access a working version of this tool!

If your cord has a relatively low probability of becoming knotted or tangled, say below 5 or 10%, you might not need to worry much about taking any special measures to keep it that way.

But, if you want to avoid the hassles that come from your cords becoming tangled, you might consider the suggestions from the video, using shorter, stiffer cords (if feasible) or getting smaller containers to store your longer cords (if not).

Meanwhile, if you're looking to learn more about knot theory, and yes, there is such a thing in maths, here's a quick introduction:

References

Raymer, Dorian M. and Smith, Douglas E. Spontaneous knotting of an agitated string. PNAS. October 16, 2007 104 (42) 16432-16437; https://doi.org/10.1073/pnas.0611320104. Note: For our tool, we corrected the L₀ parameter to be 1.025 after replicating the other parts of the authors' logistic function regression using the data presented in Figure 2, where our L₀ correction allowed us to replicate their reported probabilities for various cord lengths.

On 24 September 2018, mathematician Michael Atiyah announced that he believed that he had cracked the Riemann Hypothesis at the Heidelberg Laureate Forum, which if his proof holds, would represent a very big deal. Not just because the proof would come with a one million dollar price from solving one of the Clay Mathematical Institute's millennium problems, or because it describes the distribution of prime numbers, but also because it would automatically prove a lot of other mathematical hypotheses that rely on the Riemann hypothesis being valid for their contentions to hold.

For more information about what the Riemann Hypothesis is, we recommend viewing either Numberphile's 17-minute video or 3Blue1Brown's 22-minute long video on the topic, both of which are well done but offer different strengths in presentation. What we found interesting in Atiyah's announcement is that he claims the proof came about because of work he was doing (leaked here?) to analytically derive the fine structure constant from physics, which is fascinating in and of itself.

In the following video from the University of Nottingham's Sixty Symbols project, Laurence Eaves provides a blissfully short 5-minute long explanation of the significance of the fine structure constant, which is the measure of the strength of the electromagnetic force governing how electrically-charged elementary particles interact.

The three fundamental constants that are combined in the fine structure constant are:

If we include pi, there are technically four universal constants in the formulation. It's estimated to be very nearly equal to 1 divided by 137, where if it were exactly equal to that ratio, would really be extraordinary because the value 137 happens to also be a Pythagorean prime number. As it happens, the denominator in the fine structure constant is currently estimated to be 137.035999138..., so it's close, but not quite.

Did you ever want to get into quantum physics, but didn't know where to start?

A good place to begin understanding a complex, complicated thing like quantum mechanics is at the beginning, where believe it or not, it all started with, dare we say, a light bulb moment! Check out the following video from MinutePhysics on the origin of quantum mechanics, which is really all about how physics works on the tiniest scales possible.