In 1999, while sitting at a bus stop in Cuernavaca, Mexico, a Czech physicist named Petr Šeba noticed young men handing slips of paper to the bus drivers in exchange for cash. It wasn’t organized crime, he learned, but another shadow trade: Each driver paid a “spy” to record when the bus ahead of his had departed the stop. If it had left recently, he would slow down, letting passengers accumulate at the next stop. If it had departed long ago, he sped up to keep other buses from passing him. This system maximized profits for the drivers. And it gave Šeba an idea.

“We felt here some kind of similarity with quantum chaotic systems,” explained Šeba’s co-author, Milan Krbálek, in an email.

After several failed attempts to talk to the spies himself, Šeba asked his student to explain to them that he wasn’t a tax collector, or a criminal — he was simply a “crazy” scientist willing to trade tequila for their data. The men handed over their used papers. When the researchers plotted thousands of bus departure times on a computer, their suspicions were confirmed: The interaction between drivers caused the spacing between departures to exhibit a distinctive pattern previously observed in quantum physics experiments.

“I was thinking that something like this could come out, but I was really surprised that it comes exactly,” Šeba said.

Subatomic particles have little to do with decentralized bus systems. But in the years since the odd coupling was discovered, the same pattern has turned up in other unrelated settings. Scientists now believe the widespread phenomenon, known as “universality,” stems from an underlying connection to mathematics, and it is helping them to model complex systems from the Internet to Earth’s climate.

The pattern was first discovered in nature in the 1950s in the energy spectrum of the uranium nucleus, a behemoth with hundreds of moving parts that quivers and stretches in infinitely many ways, producing an endless sequence of energy levels. In 1972, the number theorist Hugh Montgomery observed it in the zeros of the Riemann zeta function, a mathematical object closely related to the distribution of prime numbers. In 2000, Krbálek and Šeba reported it in the Cuernavaca bus system. And in recent years it has shown up in spectral measurements of composite materials, such as sea ice and human bones, and in signal dynamics of the Erdös–Rényi model, a simplified version of the Internet named for Paul Erdös and Alfréd Rényi.

Each of these systems has a spectrum — a sequence like a bar code representing data such as energy levels, zeta zeros, bus departure times or signal speeds. In all the spectra, the same distinctive pattern appears: The data seem haphazardly distributed, and yet neighboring lines repel one another, lending a degree of regularity to their spacing. This fine balance between chaos and order, which is defined by a precise formula, also appears in a purely mathematical setting: It defines the spacing between the eigenvalues, or solutions, of a vast matrix filled with random numbers.

“Why so many physical systems behave like random matrices is still a mystery,” said Horng-Tzer Yau, a mathematician at Harvard University. “But in the past three years, we have made a very important step in our understanding.”

By investigating the “universality” phenomenon in random matrices, researchers have developed a better sense of why it arises elsewhere — and how it can be used. In a flurry of recent papers, Yau and other mathematicians have characterized many new types of random matrices, which can conform to a variety of numerical distributions and symmetry rules. For example, the numbers filling a matrix’s rows and columns might be chosen from a bell curve of possible values, or they might simply be 1s and –1s. The top right and bottom left halves of the matrix might be mirror images of one another, or not. Time and again, regardless of their specific characteristics, the random matrices are found to exhibit that same chaotic yet regular pattern in the distribution of their eigenvalues. That’s why mathematicians call the phenomenon “universality.”

“It seems to be a law of nature,” said Van Vu, a mathematician at Yale University who, with Terence Tao of the University of California, Los Angeles, has proven universality for a broad class of random matrices.

https://www.quantamagazine.org/in-my...erge-20130205/

In the late 70s,

The Feigenbaum constant delta is a universal constant for functions approaching chaos via period doubling. It was discovered by Feigenbaum in 1975 (Feigenbaum 1979) while studying the fixed points of the iterated function

f(x)=1-mu|x|^r,

and characterizes the geometric approach of the bifurcation parameter to its limiting value as the parameter mu is increased for fixed x. The plot above is made by iterating equation (1) with r=2 several hundred times for a series of discrete but closely spaced values of mu, discarding the first hundred or so points before the iteration has settled down to its fixed points, and then plotting the points remaining.

So it became clear that chaos has an innate order.

The first Feigenbaum constant is the limiting ratio of each bifurcation interval to the next between every period doubling, of a one-parameter map

where f(x) is a function parameterized by the bifurcation parameter a.

where an are discrete values of a at the n-th period doubling.

https://en.wikipedia.org/wiki/Feigenbaum_constants

Along with this:

**Time’s Arrow Traced to Quantum Source**

A new theory explains the seemingly irreversible arrow of time while yielding insights into entropy, quantum computers, black holes, and the past-future divide.

https://www.quantamagazine.org/quant...-say-20140416/

The ideas of Discovery Institute Fellow Michael Denton, for a world "front loaded" for all the things we see around us, but as Denton writes at the beginning of his book

*Nature's Destiny*:

In large measure, therefore, the teleological argument presented here and the special creationist worldview are mutually exclusive accounts of the world. In the last analysis, evidence for one is evidence against the other. Put simply, the more convincing is the evidence for believing that the world is prefabricated to the end of life, that the design is built into the laws of nature, the less credible becomes the special creationist worldview.

(pages xvii-xviii)

It's getting harder to deny that the universe shows signs of teleology.

"I have no doubt that in reality the future will be vastly more surprising than anything I can imagine. Now my own suspicion is that the Universe is not only queerer than we suppose, but queerer than we can suppose."

J.B.S. Haldane.

Maybe not.

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