The Freedom Of Going Back

As I discussed in a recent talk on Sir Isaac Newton, it may be of great importance to study original texts by scientists and mathematicians from the 17-19th centuries. My views found an echo recently in none other than one of the 2014 Fields Medalists - Manjul Bhargava, in this interview with Plus magazine. Bhargava clearly illustrates the utility of reading the masters, and of the importance of history in the sciences: 

Plus: I know when we get the opportunity to read original texts or very old translations of original texts, even though I’m not working on the mathematics I’m maybe writing about it, it’s still incredibly exciting, you feel the kind of nearness to the work. Is that what has led you to sometimes work on these old texts?

Manjul: Completely, that’s absolutely right. When one’s reading the text in the original, first of all it’s just very exciting from an historical point of view; you’re seeing it as it was discovered. Also from a mathematical point of view it’s very exciting to see how it was when it was discovered, because for example Gauss’ composition; it was discovered in a certain way, and then 200 years of modern mathematics was piled on top of it. It’s been reinterpreted numerous times in much more fancy mathematical jargon, with quick, elegant proofs in terms of this complicated mathematical jargon. That’s the way it’s taught now.

We sometimes then lose sight of how it was originally thought about.

Plus: Of the intuition perhaps as well?

Manjul: Yes exactly. So by going back to the original you can bypass the way of thinking that history has somehow decided to take, and by forgetting about that you can then take your own path. Sometimes you get too influenced by the way people have thought about something for 200 years, that if you learn it that way that’s the only way you know how to think. If you go back to the beginning, forget all that new stuff that happened, go back to the beginning. Think about it in a totally new way and develop your own path.

Plus: There’s a freedom, almost, in going back.

Manjul: There’s a freedom yes, the freedom of that ignorance in fact, and just knowing how something happened but not knowing anything else. Then paving your own path from there.

He goes on to say that this philosophy of education may be very important to schools as well. 

[ The image above is of a first edition Disquisitiones Arithmeticae (1801) by Carl Friedrich Gauss. ] 

Telescopes, Logarithms & Computers: A 400 Year Journey

1205HOFnapier_Fig.jpg (570×600)

First conceived as a technological aid to astronomers by Napier, the logarithmic tables were a kind of printable analog computer. The logarithms are now 400 years old, and this is a good time to ponder about their role in the history of science.

It was not until 1614 that Napier’s first work on this subject, Mirifia logarithmorum canonis descriptio (known as the Descriptio), was published. In addition to tables of logarithms the Descriptio also contains an account of the nature of logarithms and a number of examples explaining their use. The East India Company was so impressed by Napier’s Descriptio that it asked Edward Wright, a Cambridge mathematician and expert in navigation, to translate it into English for the benefit of the Company’s seafarers. From the very beginning of logarithms their utility to navigators has been of supreme importance in their development. (Graham Jagger, The Making of Logarithmic Tables)

The early tables by Napier, Briggs and others depended heavily on the book and printing press as a mechanical technology, and this device would later be fashioned into slide rules

While the tables certainly made calculation simpler, their own construction - to begin with, was extremely difficult.

Laying Down The Tables

The conceptual idea behind the tables was, for any calculation - replace the process of multiplication by addition, which was much simpler for a human being. This is possible because, recall the law of logarithms: log (a*b) = log(a) + log(b). You just compute the sum on the right and look up the tables to arrive at the value for a*b. 

Graham Jagger notes in the same essay, “Realising that astronomical and navigational calculations involved primarily trigonometrical functions, especially sines, Napier set out to construct a table by which multiplication of these sines could be replaced by addition: the tables in the Descriptio are of logarithms of sines. They consist of seven columns, and are semi-quadrantally arranged.”

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Because the early tables were far from perfect, subsequent innovators such as Henry Briggs heavily modified the architecture and design (or shall we say “user interface”?), and even theory - to simplify them further. Thus it was that a large body of new mathematics flowed simply from the problems involved in the very act of making logarithmic tables.  

How would the constructor of tables ensure that they contain no errors? Not unlike computer programmers, would he have to devise his own techniques of debugging and error-correction? 

Here is an example of the painstaking calculations involved in the process: 

To find the logarithm of 2, Briggs raised it to the tenth power, viz.1024, and extracted the square root of 1.024 forty-seven times, the result being 1.00000 00000 00000 16851 60570 53949 77. Multiplying the significant figures by 4342 … he obtained the logarithm of this quantity, viz. 0.00000 00000 00000 07318 55936 90623 9336, which multiplied by 2^47 gave 0.01029 99566 39811 95265 277444, the logarithm of 1.024, true to 17 or 18 places. Adding the characteristic 3, and dividing by 10, he found (since 2 is the tenth root of 1024) log 2 = .30102 99956 63981 195. Briggs calculated in a similar manner log 6, and thence deduced log 3. (Encyclopaedia Brittanica, 2013

A Fountain Of Discovery

What if a certain number was absent from the table and the averages of two values had to be taken? This becomes especially troublesome in the case of trigonometry, as this remark about William Oughtred’s 1657 tables will reveal: 

Following the tables there is an appendix of some ten pages which gives rules of interpolation in the trigonometrical tables for angles where linear interpolation is inappropriate; that is, near 0° for the logarithm of sines and tangents, and near 90° for the logarithm of tangents. In these regions the rates of change of the tabulated functions are large and highly non-linear.

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The design of logarithmic tables, hence directly propelled - some major advances such as the Lagrange three-point interpolation formula. H.H. Goldstine describes this eloquently: 

…the ideas of Napier and Briggs spread rapidly across Europe, and we shall see Kepler calculating his own tables as soon as he heard of Napier’s idea. From this point onwards the theory of finite differences was to be further developed with great artistry by such men as Newton, Euler, Gauss, Laplace, and Lagrange, among others. In fact we shall see that virtually all the great mathematicians of the seventeenth and eighteenth centuries had a hand in the subject.

Another fascinating aspect is Napier’s original description of logarithms in terms of the relative motion of points along a line, (as opposed to any kind of exponents or roots):

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For Napier, a logarithmic table is “a small table by the use of which we can obtain a knowledge of all geometrical dimensions and motions in space…" Rafael Villarreal-Calderon interprets this as saying: "In a calculus sense, Napier’s logarithms could be seen as measures of “instantaneous” velocities." 

This suggests that John Napier may have imagined his humble logarithms as something far more colossal that would bloom much later in the great minds of Newton and Leibniz: the calculus

References:

  1. The Construction of the Wonderful Canon of logarithms, by John Napier
  2. John Napier & the invention of logarithms, 1614 by E.W. Hobson
  3. Logarithms celebrate their 400th birthday, by Tom Siegfried
  4. Slide rule still rules, by Michelle Delio
  5. The making of the logarithm, by Marianne Freiberger
  6. Logarithms, the early history of a familiar function: by Kathleen M. Clark and Clemency Montelle
  7. Musical logarithms in the 17th century: Descartes, Mercator, Newton by Benjamin Wardhaugh 
  8. Chopping Logs: A look at the history and uses of logarithms (pdf), by Rafael Villarreal-Calderon
  9. Kepler, Napier & the third law: by Kevin Brown. 
  10. The image on the top is an example of Napier’s bones, described in his work Rabdologia (1617), and also re-designed later as a “promptuary”.Those who’re interested can make one at home very easily. 

These ideas are part of an ongoing workshop which is detailed hereIf you like this essay, you might want to join the Zetatrek expedition by registering in that workshop. Or you could just drop us some bitcoin love:  1PmjQT7uwSRtGNtkMrNvaa8pJSAmipLWva 

Answers To Queries About ZetaTrek #1

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I received a thoughtful mail from one of the people interested in our new workshop called “Clockwork To Chaos”. I felt that others might also wish to know the answers, and Jason Conklin ( @ninly ) consented to me sharing his queries and my response publicly. So here we go: 

—————————————

Jason: If I may ask a couple questions about ZetaTrek and the workshop group(s). This is a bit vague, but I’m wondering about things that I can’t really investigate, given your private-platform model. I’m not looking for specific answers (except to the last one).

Can you briefly characterize the variety of interests among the ZetaTrek participants and how they are pursued?

Rohit: Of the 70 people from 13 countries they come from various backgrounds - philosophy, art, advertising, physics, history of science, information technology, market research, film-makers, students, electronic music, computer art, poetry, games, writers, journalists, robotics, and medicine. Every year the expedition features one to three workshops which enable us to engage in all this through a strongly historical approach. 

For example, our mathematical expedition requires a deep understanding of the complex plane, which has historical roots in surveying, trigonometry, map-making, and geometry. So when we are discussing the science of the “Age of Discovery" we are exposed to a number of scientific cultures linked to that idea, and cultures in which it evolved over centuries. We could be learning about the complex plane without being highly aware of that learning process. This I think is very important, because any human mind which loves freedom also has a resistance to pedagogy. 

Jason: Clearly you’ve organized things around the grand goal of Riemann hypothesis research, and presumably the people involved support that — but given that you have encouraged non-academics and people without formal training, I am curious how “uninitiated” interests or backgrounds are encouraged or guided into the project, if at all.

Rohit: I design the modules along with an intern ( Ajinkya Kulkarni), and when necessary I take on the role of a teacher. But this is not always needed because most of the participants are learned, curious and harbor diverse interests. You could be over 50 in a profession and learning trigonometry from a 20 year old without feeling awkward. It’s more like a global village community than a classroom. 

Jason: Off the top of my head: do you have people especially interested in cryptography, alchemical or hermetic worldviews, or music? I have a particular interest in music and what’s called “xenharmonic” temperament theory, for example, and there’s been some talk about relevance of the zeta function in that community, too.

Rohit: I am personally interested in cryptography because it is based almost entirely on number theory, which is the very domain of the Riemann hypothesis and prime numbers. There are many others into music, naturally. 

As for alchemical and hermetic views we have Thony Christie, who is a very fine and widely read historian of mathematics too. And Tirolarn Krityakiern ( fictional name ) who runs the excellent site Kook Science. He’s also completely batshit crazy while being of dazzling intellect. Don’t let that stop you. 

Jason: Somewhat related, what about people who *do* have some training? I’m not a mathematician, but I am a (largely self-trained) engineer, and have about a two-year degree’s worth of math coursework under my belt (calculus, diff-eq, linear algebra), plus personal/hobbyist study. How calculus-introductory will the curriculum on Newton’s writing be? I don’t mind reviewing stuff at all, I’m just looking for some sense of what to expect and where I might fall in the mix.

Rohit: Since our approach is historical, even the trained or semi-trained minds would find the proceedings novel. From Newton’s corpus we are going to look very closely at the book on Fluxions, but we will also trace the entire history of infinite series ( which is what calculus is finally about ) from the Zeno’s paradox in ancient Greece to the Zeta function in the 19th century. The original post touches upon a lot of material - as in applications of infinite series to the real world, which really are too numerous to mention here.

As an exercise I would look at all the following great works of western mathematics and list the ones that do not use infinite series in any important way. The results could be surprising! 

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Jason: Finally (the concrete question): If the workshop starts and it’s clear to me within a week or two that the work or the crowd are “not for me” after all, is there a chance withdrawing and getting a refund? Don’t get me wrong: I don’t expect that, but since I can’t investigate the community/cooperative aspects (which are a part of what draw me toward the project), and the money is a factor, too (supporting a family here, etc.), I’d like to ask.

Thanks, and good luck in your search for participants!

Rohit: I’m afraid refunds aren’t possible. Because this is an international transfer and the whole process is not trivial. So you should join only when you are absolutely convinced of your interest in the project. I’d be happy to facilitate that to my maximum capacity. 

Besides, consider that this comes with a lifetime membership. All future workshops will also be free, and the community will keep on growing. You will eventually find people whose interests are closely aligned with yours. 
Thank you writing in, Jason - and I hope to see you join in the near future. 

Clockwork To Chaos: an online workshop (19 July-19 Oct 2014)

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This manuscript page from 1665 shows a 23-year old Isaac Newton calculating the area under a hyperbola ( the curve drawn on the top left of the page).

He calculates no less than 55 decimal places, meticulously adding values from each term of an infinite series. The series emerges naturally when the space under this curve is cut up into an infinite number of thin rectangular strips, and their areas are added up. Because Newton does not have a mechanical computer, his entire thought process (known by the archaic term quadrature) is completely visible on paper. 

One can imagine sitting on the shoulders of Sir Isaac Newton as he invented the symbolic machinery needed to describe his system of the universe. In the next 3 months we will try to experience exactly that, by studying Newton’s original masterworks, including the Method of Fluxions and his magnum opus the Principia.

What is there in thee, Man - that can be known? —
Dark fluxion, all unfixable by thought,
A phantom dim of past and future wrought

Newton’s own name for the full-blown architecture of calculus was the fluxions, a word that would feature almost a hundred years later in the poem above by Samuel T. Coleridge. 

Calculus is a language of movement and change, and underneath its facade lies the vast scaffold of infinite sums such as the one created by Newton. However, the historical origin of these ideas lies thousands of years ago in ancient Greece. 

The first infinite series were discovered in ancient Greece with Zeno’s paradox and Archimedes’ calculation of the area under a parabola. This proto-science lay largely dormant for centuries, with some important breakthroughs made by Nicole Oresme in 14th century France and his contemporary Madhava in India, for purposes of astronomy.

From Clockwork To Chaos

When Kepler began constructing his theory of orbital motion, and his Platonic vision of a universe in harmony - areas of shapes were indeed thought of as a sum of infinite lines. Volumes were similarly imagined as a collection of infinite discs. Naturally then, the summation of infinite series was always one of the most important and time-consuming tasks of any Renaissance mathematicus.

Here is a page from Bonaventura Cavalieri's Geometria indivisibilibus continuorum nova quadam ratione promota (Geometry, developed by a new method through the indivisibles of the continua, 1635): 

Cavalieri ms diagram 1653

After Isaac Newton described the laws of gravity ( not before having infinite series and fluxions firmly in his grasp ) the prevailing view of the universe was described famously by Pierre Simone, Marquis de Laplace in these famous words: 

We may regard the present state of the universe as the effect of its past and the cause of its future. An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes.

However, Laplace was to be proved wrong…..the problems of celestial mechanics did not yield so easily to the laws of Newton. Neither were the orbits of the moon, the planets and their gravitational effects on each other so predictable as to be some kind of clockwork

The solution to the motion of a two-body system, by Newton and his eighteenth century mathematical successors, is one of the triumphs of Newtonian mechanics.

In our own solar system, which has many more than two bodies, things are much more complicated.  The planets follow orbits that are almost, but not exactly, ellipses, the discrepancy being due to the fact that each planet has its own gravitational field, which influences – or perturbs – the motion of all the others. Consequently, the planets’ orbits are not exactly periodic: they return to a slightly different position, and their time of revolution about the sun varies slightly, from year to year. 

They needed a better mathematics to describe the perturbed mess that was our solar system. Passing through the deft hands of mathematicians like Barrow, Wallis, Gregory, Newton, Leibniz, Gauss, Euler, Laplace, d’Alembert, Clairaut & Lagrange….infinite series became one of the formidable weapons of mathematical physics.

After burying the all-knowing demon god of Laplace, these new methods yielded a completely new vision of the cosmos - the radical theme of which was a beautiful chaos. The damning shock of this new vision came from Henri Poincaré, who explained it thus: 

….imagine a small asteroid, moving back and forth between two larger bodies - call them planets A and B.  Given the right conditions, it is possible for the asteroid to alternate between the two planets, spending some of its time revolving around A, and some revolving around B, like a bee flitting back and forth between two flowers.  If we track which planet the asteroid goes around at each revolution, we will get a sequence of A’s and B’s which can look statistically like a sequence of random coin tosses.

The workshop will take the participants through this short journey from clockwork to chaos in the most interesting way possible. Throughout this period, we will be looking at original sources and manuscripts from history wherever accessible. 

Who can Join? Absolutely anyone. This course is for self-taught hobbyists, not for experts; and it is designed to require absolutely no prior knowledge of history, science or mathematics. Here is some feedback (on Twitter) from attendees of the most recent workshops. Apart from the existing members ( 66 from 13 countries ) we hope to enroll at least 75-80 new participants worldwide in this round. 

How does it work? The workshop will be conducted via the online ZetaTrek mailing list, which has been active for almost 3 years now. Participants will be guided through interactive modules according to a syllabus (always being updated). The duration of the workshop is roughly 3 months, extending from 19 July-19 October, 2014. The expected commitment is roughly 2-5 hours per week, depending on your enthusiasm.

This workshop is also the final 3 months (or, second semester) of The Age of Re:discovery online workshop. Our first semester was wide-ranging and diverse in ideas, cultures and images. In this second semester, we would like us to keep a sharp focus on Newton’s corpus of ideas and keep miscellany on the sidelines.  

For existing members of the Zetatrek expedition, this phase intends to bring a familiarity with the historical development of calculus, geometry and more importantly infinite series. We could then study Euler’s work on the zeta function ( because that too is an infinite series ) and thus by the end of October 2014, we can finally segue back into the original goal of Zetatrek - the Riemann Hypothesis.

Registrations: This is an independent platform without any institutional funding. Participants are expected to contribute a fee of $250 (approx. Rs. 15000) for the entire duration. All new participants will get access to our 3 year archives and a lifetime membership of the Zetatrek expedition. All future workshops will thus be free. 

You can pay using our online ticketing facility DoAttend, or Paypal ( the linked Gmail ID is “fadebox” ). Please contact me at the address above for any further queries or assistance.

Scholarships & Gifts: Since the fee may be too much for some people, we always create some free scholarships. This time, for every 10 people who register with the fee, we will offer one free scholarship. So if 200 people register, 20 others will be awarded a free seat. The details of applying will be announced later. You are also welcome to sponsor a scholarship for a friend, or send one as a gift. 

Convenor: Rohit Gupta (38, M) is an autodidact interested in the history of science and mathematics. In particular, interdisciplinary interactions such as between astronomy and geometry; or colonial science and its Oriental reception. Some of the previous workshops are listed here, along with a recent interview. His older projects have been featured at Wired and the BBC. Gupta also writes the blog Compasswallah, and tweets as @fadesingh. A complete CV is available on request.

Related Links: 

  1. The Zetatrekker’s Guide To The Galaxy, online documentation of the project is constantly updated, by Rohit Gupta & Ajinkya Kulkarni
  2. The secret writings of Isaac Newton, by Sarah Dry…author of The Newton Papers. 
  3. A video lecture on the historical development of infinite series by N.J. Wildberger. The whole series is worth watching. 

The Supreme Law of Unreason

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“I know of scarcely anything so apt to impress the imagination as the wonderful form of cosmic order expressed by the ‘Law of Frequency of Error.’ The law would have been personified by the Greeks and deified, if they had known of it. It reigns with serenity and in complete self-effacement, amidst the wildest confusion. The huger the mob, and the greater the apparent anarchy, the more perfect is its sway. It is the supreme law of Unreason. Whenever a large sample of chaotic elements are taken in hand and marshaled in the order of their magnitude, an unsuspected and most beautiful form of regularity proves to have been latent all along.”

Apart from its lucidity, this quote by Francis Galton is important because it appears in an article related to the Riemann Hypothesis, the central objective of the ZetaTrek expedition for citizen science. 

In 2011 when the expedition began, inspired by an essay Freeman Dyson wrote connecting prime numbers and quasicrystals - it already hinted at a startling array of similar patterns in nature. A remarkable instance of this has now been observed in the behaviour of a public transport system in Cuernavaca, Mexico: 

For centuries, probability theory has been used to model uncorrelated or weakly correlated systems. There is strong evidence that random matrix statistics plays a similar fundamental role for complicated correlated systems, among them the energy levels of the uranium nucleus, the zeros of the zeta function, and the spacing pattern of a decentralized bus system in the city of Cuernavaca, Mexico, which two physicists, Milan Krbálek and Petr Šeba, studied in the late 1990s. The bus system had no central authority or timetables to govern the arrivals and departures of the buses, which were individually owned by the drivers. In order to maximize their incomes, the drivers adjusted their speeds based on information obtained by bystanders about the departure times of the buses in front of them. Krbálek and Šeba recorded the actual departure times of the buses at various stops and found that the spacings between buses match the statistical behavior found in random matrix theory.

This behavior suggests that departure times that are close to each other will “repel each other” leading to optimum space. And this is exactly the process that unfolds in the Riemann Zeta zeroes. 

Montgomery had found that the statistical distribution of the zeros on the critical line of the Riemann zeta function has a certain property, now called Montgomery’s pair correlation conjecture. He explained that the zeros tend to repel between neighboring levels.

Further reading

  1. From Prime Numbers To Nuclear Physics And Beyond, by Kelly Devine Thomas
  2. In Mysterious Pattern, Math & Nature Converge, by Natalie Wolchover
  3. Living In A Galton Box, by Andy Tran 
  4. E pluribus unum: from complexity, universality, A brief tour of the mysteriously universal laws of mathematics and nature by Terence Tao. 

"रेखागणित" - कुमार अंबुज

अंतरिक्ष के असीम में 
क्षितिज के विस्तृत चाप पर 
वह शुरू होता है हमारी निगाह के कोण से 
और टिका रहता है अरबों बार छीली जा चुकी 
एक पेंसिल की नोक पर

जटिल विचारों की रेखाएँ काटती एक-दूसरे को जीवन में 
और कितना दुश्वार इस सीधे-सादे सच पर यकायक विश्वास कर पाना 
कि एक सरल रेखा में छिपा हुआ है एक सौ अस्सी डिग्री का कोण 
और यह कि वृत्त की असमाप्त गति में शामिल जीवन का पूरा चक्र 
अलग-अलग दिशाओं में जाने वाली रेखाओं में प्रकट 
असमान जिंदगियों के बीच की हर क्षण बढ़ती दूरी 
और वहीं कहीं छिपे वर्ग-संघर्ष के बीज 
समाज के स्वप्न में चीखता है स्वप्नद्रष्टा 
बनाओ समबाहु समकोणीय चतुर्भुज - 
शोषणमुक्त एक वर्ग !

याद करो सुदूर रह गए बचपन में 
सुथरा षटकोण बनाने का वह प्रयास अथक 
वह शंकु जिसको अलग अलग जगह से काटो तो मिलती आकृतियाँ नाना 
पायथागोरस की हर जगह उपयोगी वह मजेदार थ्योरम 
बस्ते में अभी तक बजता हुआ कम्पास बॉक्स 
और संबंधों का वह त्रिकोण ! 
यह रेखागणित है जिसमें सबसे आसान बनाना डिग्री का निशान 
और बहुत मुश्किल बना पाना 
एक झटके में चंद्रमा की कोई भी निर्दोष कला

हर बार एक छोटी-सी अधूरी रही इच्छा 
एक ऐसी सिद्धि 
जिसे हमेशा ही सिद्ध किया जाना शेष 
न्याय अन्याय की रेखाएँ चली जाती हैं समानांतर 
कभी न मिलने के लिए अटल 

व्यास, परिधि, त्रिज्या और पाई 
इन औजारों से भी मुमकिन नहीं नाप सकना 
जीवन के न्यून कोण और अधिक कोण के बीच की दुर्गम खाई 
एक बिंदु में छिपे अणु हजार 
और दबी-कुचली, टूटी-फूटी रेखाओं की पुकारों 
आहों से भरा यह नया ग्लोबल संसार 
वक्र रेखा की तरह अनिश्चित गति से भरा 
जिसके आगे रेखागणित भी अचंभित खड़ा 

हिरणी, सप्तऋषि, अलक्षित तारे 
नदी तट पर चंद्रमा बाँका 
रति की प्रणति, आकृतियाँ मिथुन और वह बाहुपाश 
यह लिपि प्राचीन, उड़ती चिड़ियाँ, हजारों विचार 
उत्तरायण-दक्षिणायन होते सूर्य देव 
पिरामिड, झूलती मीनार 
रोटी, कागज, बर्तन, 
हिलता हुआ हाथ और वह चितवन 
पहाड़ का नमस्कार, खजूर का कमर झुका कर हिलना 
दिशाएँ तमाम जिनके असंख्य कोण पसरे ब्रह्मांड में 
और यह अनंत में घूमती जरा-सी तिरछी पृथ्वी…

मैं यूक्लिड कहता हूँ देखो ! 
जिधर डालो निगाह 
उधर ही तैर रही है एक रेखागणितीय आकृति 
और वहीं कहीं छिपी हो सकती है 
जिंदगी को आसान कर सकने वाली प्रमेय। 

Our Invisible Culture: Van Gogh meets Riemann

Van Gogh meets Riemann 2

(A Van Gogh painting viewed through the Riemann-Zeta function. Here are some more from the same series. The following relevant excerpt is from a lecture by Lynn Arthur Steen, talking about the overwhelming but invisible role of mathematics in contemporary culture. ) 

The evidence for the revolution is all around us. High speed computers, fuel-efficient airplanes, world-wide communications are visible technological products of contemporary mathematics.

Less visible but more fundamental are deep insights such as gauge field theory, nonlinear dynamics, and computational complexity that provide unifying foundations for modern science. We live in a “minds-on” world created by the abstract theories of contemporary mathematics—yet hardly anyone really knows that mathematics even exists.

A well-known American mathematician reported an incident—all too typical—in which a new acquaintance asked his wife what her husband did. “He is a mathematician,” she said. The acquaintance, confused, responded after a slight pause: “Well, what else does he do?” Surely,  she thought, no one can keep busy just “being” a mathematician.

Tomorrow one of the Tome Symposia is devoted to C. P. Snow’s classic definition of the two  cultures—humanistic and scientific. One can, I believe, extend Snow’s metaphor from two cultures to three: humanists are to scientists as scientists are to mathematicians. Educated  laymen feel obliged to know something of Shakespeare and Mozart, of Eliot and Picasso; they are comfortable with only the most superficial (and usually distorted) acquaintance with the works of Darwin and Einstein, not to mention Pauling and Feynman. But they have not even heard of Gauss and von Neumann, or Hilbert and Poincare.

The only mathematics visible to our culture is the mathematics of the past—the completed masterpieces of Euclid and Newton, of geometry and calculus. Everyone knows that this mathematics exists, and that it is centuries old. No one knows that modern mathematics exists, much less that it is being continually created. Mathematics is indeed our invisible culture.

image

( A visualization of the Riemann-Zeta function…) 

How Things Distribute Themselves

How things distribute themselves in space or time or along some more abstract dimension is a question that comes up in all the sciences. An astronomer wants to know how galaxies are scattered around the universe; a biologist might study the distribution of genes along a strand of chromatin; a seismologist records the temporal pattern of earthquakes; a mathematician ponders the sprinkling of prime numbers among the integers. Here I shall consider only discrete, one-dimensional distributions, where the positions of items can be plotted along a line.

A more popular variant of the same idea exploring connections between prime numbers and quantum physics can be found here

Riecoin

Riecoin is a cryptocurrency based on the Riemann Hypothesis. 

Finding a prime number p takes O( log(p) ^ 4 ) work, while verifying if it’s prime requires O( log(p) ^ 3 ) using the Rabin-Miller test. That’s not much difference, so finding prime numbers is not practical for a PoW.

One possible solution for this is looking for prime number constellations (like twin prime numbers, or triplets, etc), that is n “consecutive” prime numbers. Consecutive in this case means that they are grouped as closely as possible minimizing the distance between the first and the last one. This takes O( log(p) ^ (n + 3) ), while verification still takes the cube of the log. This allows us to make the generation arbitrarily more difficult than the verification.

Difficulty can be adjusted by changing the length of the prime numbers. Riecoin currently uses constellations of size 6 (also known as prime sextuplets) which have the form p, p+4, p+6, p+10, p+12, p+16. So each block represents six prime numbers.