A Quest to Solve One of Math's Great Puzzles

ZetaTrek / Zombal: Anatomy of an Expedition

The Riemann Hypothesis is a prime number conundrum that has perplexed mathematicians for over 150 years. It’s a problem so difficult, there’s a $1 million prize offered by The Clay Mathematics Institute of Cambridge, Massachusetts (CMI) to anyone who can solve it. In fact, the institute announced seven Prize Problems on May 24, 2000, but, so far only one prize has been awarded -- and famously and curiously declined! And, if you think solving it is a pursuit so esoteric that non-mathematicians need not apply, you may be in for a surprise.


Source/Creator: "keegan"

One undaunted, independent mathematician has embarked on solving the Reimann Hypothesis with a fresh, ambitious approach. Fadereu Le Fou launched the ZetaTrek project in 2011 as a way of tackling the problem by bringing together a crowd of both mathematicians and laypeople. In fact, anyone with an adventurous spirit, a desire to explore deep mysteries and a willingness to put pencil to paper is encouraged to join. One of the project’s goals is to bring fresh eyes to an old function, and Fadereu says artists are most welcome. Since one of our missions at Zombal is to solve problems through scientific collaboration, we decided to talk with Fadereu and learn more about how ZetaTrek might unlock the secrets of this fascinating riddle.

What made you embark on this project?

Since I was a kid, I’ve always yearned to be a part or instrument of something monumental. As I grew older, my short and sporadic interdisciplinary projects were getting tiresome and I wanted something with a longer vision. In addition, it had to be at the intersection of almost everything I had done in my life -- visual art, fiction, games, theatre, code, technology, music and mathematics. Hermann Hesse’s novel, The Glass Bead Game, describes such a thing -- a complete synthesis of human art and science, and I was constantly searching for it. Last year when I read Freeman Dyson’s essay Birds and Frogs, in which he suggests a strategy to attack the Riemann Hypothesis using quasicrystals, I found my inspiration and made the announcement of ZetaTrek.

By a sheer stroke of luck, the Nobel Prize for Chemistry in 2011 was given to Daniel Shechtman two days later for the discovery of quasicrystals. As a result, we are now sailing into the little-explored seas lying somewhere between two giant continents of number theory and physics.

Why do prime numbers intrigue mathematicians? And, why should non-mathematicians be intrigued by them as well?

Doing mathematics without knowing primes is analogous to doing chemistry without atoms. All integers are made of prime numbers, and this is the Fundamental Law of Arithmetic, the basis of almost every major mathematical idea since Euclid. Every digital device uses this fact, for example, in the guise of cryptography. If you look at the RSA algorithm, you realize that prime numbers are electronic money, in a very literal sense. So, is money important to non-mathematicians? What about codes by which space probes and military drones are controlled? This stuff is all around us in the form of electromagnetic signals. It is our new environment, it is part of the biosphere.

Tell us a little about more your background.

I live in India, and I’m 36 years old. People know me as Fadereu better than my real name, I think. I’ll only mention three of my later projects.

In the year 2004, when South Asia was struck by a tsunami and earthquake, I devised a toolchain to divert text messages from cell phones in the Sri Lankan disaster area directly to the web. At that time, Twitter was still 3 years or so away, but the project received worldwide coverage in the media. In the year 2007 I made Cellphabet, which was a way to send signals by the movement of your body, without touching the phone. This was noted by the BBC again. And then there was parab0xx (Wired blog, video), which enabled you to play music by touching projected images on a tabletop. Today, you can see cats doing that on an iPad, but that’s not using a 100-rupee Chinese webcam in India like I did. All of these were human-machine-interface ideas which would later become mainstream as commercial ventures.

Later I was drawn to the austerity of mathematics and decided to pursue it full-time. Or maybe I was forced into it by circumstance, I don’t know.

On your blog, you say everyone from artists to "polar bears with Polaroid cameras" should get involved with the project. Why do you feel non-mathematicians have a shot at solving the problem?

Georges Pierre Seurat, the pointillist painter, could see the world of quantum particles 50 years before physicists, and he invented the computer pixel 150 years before programmers. Picasso was painting the fourth dimension with Cubism, and Jackson Pollock foreshadowed the discovery of fractals and chaos in dynamics. The unfortunate chasm between the artist and scientist, which began widening after the Age of Reason (or Age of Enlightenment), is rapidly closing once again. A primary example of this is the new media and generative art movements, and to me an artist has a vision as enriching as a mathematician. Geometry too is an art, and there are people making origami cranes in 9 dimensions. I see no such boundaries.

I understand that you're attempting to complete the proof by classifying one-dimensional quasi-crystals. What are quasi-crystals, and why do you think they'll help you reach your end goal?

Any sequence of symbols can be used, like aaaabbbb, but for convenience we take numbers. Anything where a pattern repeats plainly is a crystal, to put it simply. The following decimal expansions of rational numbers are a type of crystal in one-dimension:

1/3 = 0.33333333333333......
1/7 = 0.142857142857142857....

This period is closely related to the denominator and some particular properties of primes. But some numbers cannot be expressed as ratios, we call them irrational numbers, such as the square root of 2. Among the irrational numbers we have two more types -- the algebraic numbers and the transcendentals (such as Pi). The type of algebraic numbers which correspond closely to quasicrystals is called Pisot-Vijayaraghavan, an example being the golden ratio (the Fibonacci sequence is therefore a quasicrystal in one dimension).

This is just one way of looking at how the properties of prime numbers are linked to quasicrystals. There are numerous unsolved problems in this area.

Your blog links to an article from The Economist titled Riemann's riddle, which gives a great overview of the hypothesis. The piece suggests that not all mathematicians think the problem can be solved. Will you know if your current approach (through quasi-crystals) is not going to give you the results you want? If so, will you continue trying to solve the problem from another angle?

Let me give you an example of how it works, there is a famous theorem of Gauss called the Law Of Quadratic Reciprocity. He called it his theorema aureum (or golden theorem) and provided six variants on a proof. Today we have catalogues of over 250 proofs of the same theorem, and yet -- why this law exists is one of the most mysterious facts of prime number theory. A proof may reveal very little about the pattern.

Similarly, the Riemann-Zeta function is a giant kaleidoscope, and I doubt that we can run out of different ways of looking at it anytime soon. Quasicrystals gave us an excellent starting point, and because of this approach I keep finding new insights every day.

You've put together a workshop called the ZetaTrek Basecamp, which gives all participants the knowledge they need to tackle the hypothesis. How is the workshop structured?

Basecamp is now over, and the expedition is in “cruise mode”. Until we achieve critical mass of say 75-100 people I am going to let it take its own course. One can’t decide everything and control too much in a project like this. It has its own evolutionary pattern, it is like an organism of some kind.

Is it too late to join the project? How much does it cost?

It’s never too late, the registration is always open. The fee is about USD 100 for a lifetime.

Many people may not have fond memories of high school math classes, for any number of reasons. Will novices feel comfortable joining?

It has to be a very particular kind of novice. This expedition plays the role of a bridge between different ways of doing science and art, but it is a certain kind of person who actively seeks this kind of experience. While schooling has its own issues, in hindsight I do not think that this expedition makes things any easier. The objective from schools is to get enough education for a job. In ZetaTrek, the task is nothing short of Herculean. It is so difficult -- seasoned mathematicians warn us, and I know what they mean -- that one wouldn’t even think of anything about this being easy. Despite knowing the terrible odds, across millennia of human history you find people doing exactly that sort of thing.

What has been the response since launching the project? Has it been attracting a good mix of mathematicians and non-mathematicians?

We have only 24 people so far, but from 8 different countries and extremely diverse backgrounds. Physics, biology, advertising, poetry, programmers, geology…so this is definitely the most interesting mix of people I have had the pleasure of being on a journey with.

You referenced Dyson’s Birds and Frogs essay as an inspiration. That piece categorizes researchers and scientists as one or the other, or sometimes both. He seems to be saying that you need “birds” to provide general guidance with a big picture view of things, while “frogs,” with a more limited view, are down on the ground pursuing some of these ideas in detail. Do you try to determine which category someone falls into when they join you?

Although Dyson’s essay is great, Birds and Frogs was probably an unnecessary metaphor. My approach oscillates between digging through the fine details, and when I get frustrated of that I try to look at the big picture. In both of these, the vision is often blurred. One can keep changing the tools. Sometimes the magnifying glass, sometimes the telescope, sometimes a helicopter. Sometimes you have to be a historian and look for something that was written 300 years ago by Gauss or Euler.

The project launched in October of 2011. Have there been any interesting discoveries thus far?

We haven’t made a breakthrough, but we have had some exciting clues, recently. If you see my post, The 3 Pills of Mobius, it includes a completely new way of visualizing the Mobius function, which is older than the Riemann Hypothesis but directly linked to it. The method we have used is rare, and the results are beautiful, but I can’t say we have a theorem yet.

I am personally becoming more interested in making number theory research more visual using plots and animated gifs. Mathematics has always been a kind of highly personal art, something that was always inside the head, but I want to make it visible.

You said that solving Riemann's Hypothesis was not the most important goal, but making mathematical discovery accessible to anyone with a desire to learn and explore. Are you suggesting that traditional ways of conducting research are impeding discovery?

I’m actually saying that a completely new species of researcher and discoverer has been made possible by the Internet -- the autonomous autodidact. And I am trying to attract a big enough group of that stuff, these are my kind of people. This is who I am. I know the problems they face because I have faced them myself. Anyone with an internet connection has access to literature, but there are other problems -- breaking the technical jargon, having the support of a community and finding motivation. Being part of an expedition like ZetaTrek fixes most of these problems. And then there is the question of earning money for this type of work. I have some idea of how to fix that too.

We’d definitely be interested to hear more about your ideas for earning money for this type of work. Do you think there are certain benefactors that will fund you purely for research reasons? Do you see people on your project picking up funding on the side because of exposure through ZetaTrek? Do you see Zombal fitting into the equation here?

One of the reasons why I’m in conversation with Zombal is because you’re creating a marketplace for exactly this kind of stuff. Mathematical research isn’t always about proving theorems. Sometime you have to write a clear exposition of a problem with its history, collect a lot of important results in one survey, or visualize a certain function...and so forth. These too are important things. I imagine putting up projects for ZetaTrek members and pay the winning bidder or team.

As for who will pay ZetaTrek for all this, I’ll be in a better position to say that a few more months down the line. We’ll be putting out some crucial research and organisations would like to sponsor this kind of work. This reminds me...you’ve probably heard of the Pentium FDIV Bug which was discovered by Thomas Nicely to calculate certain constellations of closely occurring prime numbers. The financial value of that discovery was not small (from Wikipedia): “On January 17, 1995, Intel announced a pre-tax charge of $475 million against earnings, ostensibly the total cost associated with replacement of the flawed processors.

That example shows there is real, practical value to mathematical research. What do you imagine to be the practical applications of the autonomous research you are doing through ZetaTrek?

Today number theory is almost ubiquitous in its applications apart from computer science and cryptography. I’m going to give you a very specific example of what I envision. Let us understand first that anything above a 100 people on this expedition is a worldwide network. More importantly, they all have access to a computer or phone. There is an experiment I want to do with such a network.

Install a small script on all these devices that logs ‘soft errors’ or single event upsets, which is sometimes a cosmic ray hitting your computer and flipping one bit. This algorithm will require number theory, I think. Now we compare the geographic location of the cosmic ray events to the known astronomical sources of high energy particles. And I should add that all these 100 computers are moving with the earth’s own rotation, while scanning the sky in all directions.

If the evidence on this 3D map of data corresponds closely to what we know from astrophysics, we have a completely new and cheap way of doing astrophysics - a distributed cosmic ray observatory. You don’t need NASA, you don’t need a billion dollars to make an atom smasher. You are listening directly to the cosmos, by converting these handheld devices into a vast telescopic array.

By the way, in 2008 Intel filed a patent for this but I think it was scareware and may not actually be practical. And they wanted to remove the flipped-bit errors, whereas we should be particularly interested in the errors themselves.

If you're able to solve the Riemann hypothesis, what will this mean for other areas of science? Will any long-standing techniques, in say cryptography need to be rethought?

Cryptography is the obvious area of impact, because all commerce and security depends today on codes like RSA, but I have a feeling physics too will be affected to no less a degree.

Further information about ZetaTrek can be found on Fadereu's blog, Kali & the Kaleidoscope, or you can follow him or contact him directly:

Twitter: @fadesingh
Email: "(((1/f)))" fadebox@gmail.com

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Comments (2)

  • sami said:
    Dear Sir i would like to assure you that i have full solution for Riemann hypothesis, i proved that all non trevial zeroes lie in critcal line and have real part equal to 1/2. i am looking forsome research group to work with . kindly advice ,what to do please. manythanks

    • Rushabh said:
      I am also working on this please let me know if you want to work as a group.

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