# The Spectral Beauty of Universality

Math, man and nature all converge in universality. This very complex pattern — that looks like a bar code — appears over an over again in complex climate models, the structure of the Internet and even in complicated computer models of dark matter. Mathematicians have been scratching their heads because they don’t know why the same pattern can be found in human social networks, the beauty of nature and in complex math. But now they know it’s out there, they are looking for it in everything from subatomic particles to the movement of buses in Mexico. To them, it is just another universal constant — one that separates randomness and order.

Universality can be viewed through spectral theory in math and stands between random numbers and periodic numbers. It’s a very complex set that borders on chaos but has a distinct pattern that appears all around us, if we only take the time to look. Scientists believe it can be found in all complex, correlated systems.

Here is what that complexity looks like.

Czech physicist Petr Seba first stumbled across universality in action in Cuernavaca, Mexico. There he noticed men handing slips of paper to bus drivers when they boarded the buses there. After getting to the bottom of the mysterious transactions and loading the thousands of paper slips into a computer, he found the universality pattern.

It turns out that the bus drivers were paying passengers to spy for them. They wrote down the time that the last bus left the stop and handed it to the driver. The driver would then slow down or speed up, depending on what the paper suggested. Since drivers are paid by the number of passengers a bus driver would slow down if another bus had left more recently, allowing more passengers to congregate at the next stop. And if a bus had left a longer ago he would speed up to get to the stop before another driver, thus maximizing his profits.When Seba crunched the numbers he found something he’d only seen in quantum physics. A pattern of universality emerged from the spacing between departures, likely caused by the interaction between drivers.

Seba tells *Simons Science News*, “I was thinking that something like this could come out, but I was really surprised that it comes exactly.”

If universality can be found in the interplay between Mexican bus drivers then it probably exists in other unlikely spots. Since it began showing up scientists have determined the pattern stems from an underlying connection to mathematics. Now universality is helping them to model complex systems from the Internet to Earth’s climate.

First discovered in the energy spectrum of the nucleus of a uranium atom in the 1950s all systems have a spectrum — a bar-code-like sequence that represents data such as energy levels in a complex element. The same distinctive pattern appears where the data seem haphazardly distributed, and yet neighboring lines repel one another, lending a degree of regularity to their spacing. So although the data in a complex system is different and the system seems random the pattern that the distribution of data forms is identical time and time again.

Yale University mathematician Van Vu says, “It seems to be a law of nature.” He and University of California Los Angeles skateboarding physicist Terence Tao have used math to prove that universality exists in a broad range of random matrices from radio stations to prime numbers.

For mathematics awareness month in 2010 Tao describes universality in the distribution of prime numbers. He says, “The prime numbers are distributed in an irregular fashion through the integers; but if one performs a spectral analysis on this distribution, one can discern certain long-term oscillations in this distribution.”

And on January 25 the first prime number in four years was discovered. Missouri mathematician Curtis cooper. This is Cooper’s and his university’s third prime number discovery. Amateur and professional mathematicians applauded the rare discovery, equating it with finding a diamond.

University of Tennessee at Martin math professor Chris Caldwell tells*New Scientist*, “For some reason people decide they like diamonds and so they have a value. People like these large primes and so they also have a value.”

Martin keeps a record of all the largest prime numbers. The latest entrant on that list is over 17 million digits long and if downloaded is a 22MB text file. Download it here. As an equation it looks like this 2 multiplied by itself 57,885,161 times minus 1, written mathematically as 2^{57,885,161}-1. It eclipses the last largest Mersenne prime which was discovered in 2008 and sat at just under 13 million digits.

It was discovered as part of the Great Internet Mersenne Prime Search (GIMPS) project. GIMPS uses the computing power of over 1,000 computers (mostly on university campuses) to find large prime numbers. GIMPS has discovered the largest 14 of the 48 so-far calculated Mersenne primes. A Mersenne prime is a prime number 2^{P}-1 where P is also a prime number. The GIMPS website outlines the first Mersenne primes as 3, 7, 31, and 127 corresponding to P = 2, 3, 5, and 7.

Just because spectral analysis shows the same universality pattern in the nuclei of atomic particles and in prime numbers it doesn’t mean that the primes are somehow nuclear-powered or that atomic physics is somehow driven by prime numbers. Tao says, “It is evidence that a single law for spectra is so universal that it is the natural end product of any number of different processes, whether it comes from nuclear physics, random matrix models, or number theory.”

In simple systems individual components or people can exert a great deal of influence over the whole system, thus changing its outcome and its spectral pattern. In truly complex systems, changing one component doesn’t alter the outcome or the spectral pattern.

In fact, Vu and Tao discovered that they could sub out one component at a time in a complex math matrix without disrupting the system as a whole. By the time the exercise was complete the system was entirely different — having substituted all the components over time — but the pattern of distribution over the whole matrix still reflected the same mysterious pattern of universality.

In large, complex systems, no one component dominates. Vu says, “It’s like if you have a room with a lot of people and they decide to do something, the personality of one person isn’t that important.”

University of Utah mathematician Ken Golden is applying this theory to sea ice in the Arctic in hopes of building a more robust climate model by understanding the role of sea ice. He discovered that sea ice when viewed from a helicopter shows a vast patchwork of interconnected melt ponds against a backdrop of white ice as far as the eye can see. A similar view can be seen when looking at bits of sea ice under a microscope.

At the Joint Mathematics Meeting in San Diego last month, the Arctic sea ice expert told the group the heat transfer and fluid flow in sea ice is the same at the microscopic level and at the macroscopic level. Golden says, “Fluid flow through sea ice governs or mediates very important processes that you need to understand in order to understand the climate system.” And he thinks he’s hit upon the approach he needs to incorporate sea ice into existing climate models.

And it’s the same approach that might help doctors determine if someone’s has osteoporosis. Golden and his graduate student Ben Murphy found the spectrum of a dense, healthy bone exhibits universality, while that of a more porous bone doesn’t. Murphy says, “It’s amazing that the same underlying mathematics describes both.”

That’s universality for ya.

Tao says, “The field is moving quickly, and in a few years we may have many more insights as to the nature of this still-mysterious universal law.”