In keeping with the enthusiasm of the incoming total solar eclipse, I want to revive my presentation at the Astronomical Society of the Pacific annual meeting last year on this very topic *The eclipse that changed the picture of the universe*. Here is the abstract, and linked to it is its utube video (find in the widget area of the blog). I recorded the video after the talk, and so the discussion following the talk is missing in this video.

The Eclipse that Changed the Picture of the Universe

The distinguished total solar eclipse of May 29, 1919, gave new window to the universe. That eclipse truly stood as Einstein favoring cosmic phenomenon, authenticating his general theory of relativity; that the spacetime is conformed

viagravity, upending the hitherto upheld Newtonian picture—gravity as force between masses. The bending of light due to mass that the eclipse captured reformed our understanding: from spacetime dynamics to black holes to the recently detected gravitational waves.[Video]

My recent visit to Mathfest 2017 (Mathematical Association of America annual meeting) was interesting and inciting, and there will be opportunities to discuss the sessions in detail here. Following the meeting it occurred to me that there wasn’t a talk that addressed total solar eclipse. Surely a conversation of total solar eclipse in the context of mathematics would have been captivating in the spirit of all the current anticipation of the show of 21^{st} August. I could have brought up in my own talk. And yes, mathematics can very well be seen in the context. The dynamics of total solar eclipse lets us capture the mathematics of spacetime geometry; which we call Einstein’s general relativity in physics.

I have just uploaded my talk *Exposing general audience to the voice of mathematics*. Here is its abstract, and the video (find it in the widget area, just following the ASP talk).

Exposing general audience to the voice of mathematics

Under the theme of “Pursuit of Truth” at Saint Louis University I tried to shape up a TEDx talk on the subject of mathematics. From my perspective there isn’t a better subject to address reality than mathematics. Catching me off-guard, a facilitator in the rehearsal round frustratingly snapped for not to be able to follow anything. I scrambled to revamp the talk starting with plain and basic, such as squared and cubed number depictions, then moving to formulations of reality—first simpler of classical mechanics then more complex renderings, such as Dirac equation—to notice the audience cheerfully draw in into the farther intricacies of mathematics as detailed as the expressions in general relativity and quantum field. Foundational concepts and fitting analogies seems to be the key to garner enthusiasm [Video]

A few important links on the impending total solar eclipse: NASA; Being in the shadow; Great American Eclipse. And if you have some time before you embark on to glimpse this rare cosmic show, *Sun Moon Earth *by Tyler Nordgren would be worth a read.

Happy eclipse viewing!

Neeti.

]]>To prepare a talk for the upcoming MathFest, to be held in Chicago this year, I was ruminating over articulating a clean-cut yet telling narrative. Since the talk subject is on ways to effectively outreach mathematics to general audience, it should at least somewhat bring up core concepts of mathematics. Somehow allude to the essentiality of its graphical and revelatory power, compared to just an instrument to calculate. Meaning mixing in subtler forms of advanced math, even abstract ones. I am sensitive to oversimplifying anything (my take on popular writing). It’s like providing a forced picture—like peas and potato analogy of quantum and cosmic realms in *The Theory of Everything*—that is far from an actual picture, and importantly dampened down on beauty, and inspiration. The point of outreach is to convey the subject—its significance and elegance that lay in the eyes of those who swim in it—not recite a lullaby. And in my experience audience from all backgrounds, even without math ones, show true enthusiasm only when prompted into intricate and advanced forms of mathematics, yearning for the real sense. It’s there where the real message is, of what mathematics actually is about.

In my experience outreaching an advanced scientific field effectively rests on two basic elements. First, tell it the way it is, don’t soften it. That’s the hard part because all those elaborate labyrinthine equations with functionalities, symbols, and notations floating all over them is the very thing that makes some of us flee. And thus the second, present them correlatively as physical entity: Numbers to space, Algebra to geometry, Calculus to continual smooth change, Groups and matrices to potentiality of abstract objects, the list is endless, and that physics itself at the core is mathematics. All those preposterous looking equations are actually quite beautiful and insinuating if you understand that those terms are the pieces of the landscape. The tangled appearance of an equation, like Dirac’s, would dwindle away once one sees what a colossal argument the equation is making.

Persuasion in an outreach effort usually employs an object central to disseminating pronouncements of the subject. I have been thinking of having an actual physical object, and the top two in the list were tesseract and Calabi-Yau manifold. Tesseract represents four dimensional cube—Mathew McConaughey materializing in tesseract after he plunges into the black hole in the movie *Interstellar*, making tesseract currently an object of popular demand. Calabi-Yau manifold is a mathematical thing of a projective plane, surmising six dimensions. Both, thus, though may connect to reality in theoretical outlooks, cannot crystallize in our 3-D view. They are abstractions of mathematics, and stand to be significant (very) fully in their own right.

Having a real physical model in the talk, I thought, would be pedagogical, and a neat way to draw in enthusiasm. On simply googling tesseract I bumped into a 3-D printing enterprise shapeways, offering a model of tesseract (a beautiful one). (I didn’t look for Calabi-Yau model. Didn’t think it was possible to have a model of such an intricate complexity.) To my amazement, here they offered a Calabi-Yau 3-D printout as well, in different colors, snapshots, and sizes.

In conveying the actuality of mathematics with its ultra sophisticated developments, Calabi-Yau manifold can be an epitome that embodies conceptions of advanced algebra, cutting-edge geometry, mathematical abstractions, and advancements of modern physics all in one exhibit. And it is aesthetically pleasing as well. I got it from them.

Here is the snapshot of the 3-D printout (Itself a 3-D snapshot of 6-D object). It was also nice to exchange a few productive words with Rick Russell—at the Shapeways, who generated this 3-D printout with an expert eye for math and its models—on this very enchanting object. Hope the audience will like the object as much as I do.

The model emerges from the graphic that was originally rendered by A. Hanson, Indiana University, and it has done a phenomenal job in making its appearance from the nooks of abstract algebra articles, to academic and popular literature, to the explanations of modern physics. Somewhat surprised that it hasn’t shown up in the mainstream media, at least not yet.

Will be back soon,

Neeti.

]]>*All non-trivial zeros of the zeta function have real part one-half*

stated Bernhard Riemann in 1859, a German mathematician whose contributions to modern mathematics, and theoretical physics, is wide and deep—a commonly known one is in structuring the layout of Einstein’s theory of general relativity (spacetime conforms to gravity).

The relatively simple form of Riemann zeta function (in the above statement),

is an infinite series converging on its limit—a mathematical articulation worked out utilizing tools of analysis. This function with some clever number juggling, directed by Euler, transforms itself into a product (∏), that is, a series involving multiplication—as opposed to the above summation (the summation symbol ∑ we are familiar with)—over all primes, bringing the quirk of primes in the scope of palpable. Here we have the most significant milestone in connecting the nature of primes to the tapestry of all numbers (recall that at surface we don’t see a clear scheme in the distribution of prime numbers). The magic lies in the relationship of “product (∏)” to “summation (∑),” known as Euler product formula, with prime numbers coming into play. The above zeta function is then also this:

(*p*: prime, over all prime numbers)

Conceiving the dynamics of this function would then help grasp the inner nature of prime numbers, which Riemann did by the above hypothesis. Indeed visualizing the dynamic interplay not only involves seeing the structuring of prime product but also seeing it in the light of playing of the summation function, which involves perceiving through scrupulous analytics and advanced calculus.^{1}

Except for 1, the zeta function has values for both positive and negative numbers, and its value for every negative even number is a zero—but a trivial zero. (We will see what the zero of a function implies in a bit.) The availability of non-trivial zeros is the gripping point in the true portrayal of prime numbers, and it emerges from the zeta function only but under the guidance of complex field involving the above exponentiation with complex numbers (“*a* + *bi*” is a complex number, with *a* as real part and *bi* an imaginary where the standard *i* is taken to be √–1). The Riemann Hypothesis says that under the navigation of zeta function, the complex plane brings about a steadfast line that sits at a ½ real value, streaked all the way to infinity rendered by all non trivial zeros—known as the critical line (Figure 1). Infinitely many non-trivial zeros satisfy the Riemann hypothesis,^{2 }and the first ten trillion of them are seen to conform to the hypothesis.^{3}

The first few non-trivial zeros (known as Gram’s zeros) start approximately as:

½ + 14.134725i; ½ + 21.022040i; ½ + 25.010856i

See the ½ real in the complex plane with different “*i*”s. Important is to note that here all “*i*” comes to be an irrational number, that is expanding limitlessly without any pattern, but that’s another story, off from the point of this post.

Seeing the looming “½” takes exceedingly complex renderings like

and Riemann’s vision. Significant mathematical maneuvering and background would be required to even come close to how the non-trivial zeros align, but there it is. By it we have a hold of a crisp order executed by prime numbers—the very numbers that at the surface hover haphazardly (Figure 2). And this schematic is written in a regular numerical language right in front of our eyes. The root of the natural number landscape comes to be the tenacious halo of primes.

Lucid as it is, we haven’t seen the apex yet. In this deep-seated scope of a clear scheme the prime numbers take us further. Their fabric is stunningly indicatory one. It is here we see the dovetailing primes portending the coordination of the physical universe at its inmost depths.

To cut a lengthy and exceedingly labyrinthine story short, the mathematics that goes in describing quantum mechanical landscape constructs on advanced dosages of matrices—a group in an array that abides by certain set principles—algebra, and group theory. Mathematical operators, which underlie the rendering of matrices, are utilized to chart out the statistical mechanical territory of quantum landscape. Every matrix is stamped with a signature algebraic equation. An algebraic equation is like a prescription, realizing which one can decipher the nature of the object. At mathematical level this means finding its roots: incorporating what values in the equation do we get a zero. For example, for an expression *x*^{2 }– 3*x* – 4 (i. e. equation *x*^{2 }– 3*x* – 4 = 0) the roots come to be –1 and 4. Replacing *x* with either number annuls the expression, or makes it zero. The degree of the polynomial (algebraic) defines the number of zero(s) the polynomial has. Thus the squared ones, like in the above example, will have two zeros, or roots.

It is in these roots we merge the math and universe. For mathematical operators that go in describing quantum field these algebraic zeros are referred as eigenvalues—rings a bell? Indeed, it points to the eigenvalues of energy in quantum mechanical setup—that only certain values of energy are allowed.^{4,5}

It is here we have the natures unite. Some such specialized operators cast striking resemblance with the Riemann’s zeta function in a way that the operator’s eigenvalues coincide with the zeta function’s non trivial zeros. It is here that not only diverse mathematical branches meld but also mathematical and physical amalgamate (Figure 3), by the sharp correspondence of the quantum energy values (the eigenvalues) and the non-trivial zeros.

We now have prime numbers not only casing a universal principle of symmetry but also doing it in the well defined outlay of tactile quantum realm.^{5} Their symmetry isn’t on the surface but in the dynamical interplay—the aligning of zeta zeros—that the physical world at its roots dons.

The non-trivial zeros themselves fall in a pattern, and squeeze closer and closer, as we climb up the complex ladder of zeta function. The spacing of non-trivial zeros aligns with the spacing of the eigenvalues. The array of quantum eigenvalues constitutes the spectrum that the non-trivial zeros of zeta function bring forth. Then, the deep-hidden order of primes is the language of quantum depictions.

This was more than expected!

It’s even contemplated that the Riemann function itself can directly be prescribed by an operator which would model a physical system, i. e., a potency of seeing a physical system by the weave of Riemann operator—a physical system of semiclassical quantum chaos to be precise.^{4} Not chaotic chaos, but chaos of chaos theory which sees a crisp complexion in a rendering that at the surface appears completely erratic. The non-trivial zeta zeros of this operator would be eigenvalues of a semiclassical chaotic system.

The Riemann hypothesis not only substantiates the Prime Number Theorem, it exposes a stubborn structural identity to the prime numbers, and piece them in the all-embracing arena of symmetry. Indeed immense approximations are involved for us to see the diagrammatic of the hypothesis, but they are all with acute mathematical precision.

The nuance of the quantum world vindicates the hypothesis. Do we still need a proof!

The hypothesis isn’t proven or disproven yet,^{6} but it has incited a great deal of novelties and unified large swaths of mathematics and mathematical physics in the interim. The intricate interconnections that play out behind it is mesmerizingly suggestive, and offer deep insights of the natural structure that is both discrete and abstract at the same time.

Neeti.

**References**

- John Derbyshire,
*Prime Obsession*,*Bernhard Riemann and the Greatest Unsolved Problem in Mathematics*, A Plume Book, 2003 - H. Hardy (a British mathematician) in 1914 proved that infinitely many non-trivial zeros satisfy Riemann Hypothesis (or lie on the critical line): Sur Les zeros de la fonction ζ (s) de Riemann. French. In: Comptes Rendus de l’ Académie des Sciences 158 (1914), pp. 1012-14. Issn: 00014036.
- Gourdon (2004), The 10
^{13}First Zeros of the Riemann Zeta Function, and Zeros Computation at Very Large Height: http://numbers.computation.free.fr/Constants/Miscellaneous/zetazeros1e13-1e24.pdf

* For an overview (4, 5)*

4. Barry Cipra, *A Prime Case of Chaos*

5. Germán Sierra, *The Riemann zeros as spectrum and the Riemann hypothesis*

Underneath the uncoordinated display, the prime numbers incite well-structured tones—of mathematics and the universe in their finest resolutions.

For their unbreakability primes are viewed as atoms of mathematics—they construct all other numbers of the natural domain. But their appearance at the surface appears arbitrary, for the lack of a recognizable pattern in their structure or intermediary spacing. In the landscape of numbers, the prime numbers crop without any fabric of symmetry, which mathematics and the universe otherwise blatantly seize in their manifestation or flow. Starting from 2, 3, 5, 7, 11, 13, 17, 19, 23,…, Euclid of Alexandria around 300 BC showed that these asymmetric entities stretch to infinity—of which first 100 billion or so are crunched.

Neat schemes of reality often emerge in the territories elusive and outwardly inconsequential, and take subtler outlooks and deeper visualizations. The correspondence of antimatter, the underpinning of chaotic system, the essence of entropy, and the design in fractals of nature are some examples where principle plays underneath what seems a haphazard display. But, nowhere is this more obvious than in the instruction of prime numbers. It took both the magnetizing appeal of prime numbers and the sharp visionary intellect of the followers to stumble upon the spotless tone that underlie their superficial irregularity. In the abysmal subtleties of their materialization not only does reside a well-pressed systematic structure, its code is both mesmerizingly suggestive and hauntingly captivating.

Never get caught up with the deceptive lack of pattern—concept actually, in math or otherwise.

The number of primes up to a given max *N* is shown to be *N*/ ln *N* (ln: the natural log)* by a relatively analytical theorem known as Prime Number Theorem, which was proven independently by Jacques Hadamard and Charles de la Vallée in 1896 employing elaborate mathematical measures. The theorem implies that prime numbers thin out as we climb up the number ladder. The clarity of thinning though becomes apparent only at gigantic magnitudes, seen over logarithmic scales (as log function in the above formula suggests). This is slightly reflected at the onset: There are 25 primes to count 100, and 168 to 1,000 (instead of 250 if it were a regular distribution). Then there are 1,229 to 10,000, 9,592 to 100,000, and 78,492 up to a 1 million: the number of primes isn’t expanding proportionally. The tapering effect can be appreciated for large series of crunched primes at a site like primes.utm.edu. Albeit lightly, the Prime Number Theorem brings to light that underneath the mixed up display, the constitution of prime numbers and their mechanics appears to be a parameterized layout, but so far after centuries of effort a clear logic behind the mechanism remains obscure. But not, if we take the Riemann Hypothesis ^{1, 2, 3} to be not only authentic, but also natural.

The reason we aspiringly anticipate the involvement of design in occurrence and unfurling of prime numbers is the case of glorious Riemann Hypothesis (RH):

“All non-trivial zeros of the zeta function have real part one-half.”

Incredibly simple, isn’t it? The statement is more like a tip of the iceberg though (my thoughts on conveying its potential to general audience), with not only immense and consequential cues lurking under it, it takes up full range of elements from basic arithmetic functions, analysis, calculus, analytic number theory, advanced algebra, probability, statistics, and a fair share of visionary mathematical sense—tailored in place ^{1} by Carl Friedrich Gauss, Leonhard Euler, Lejeune Dirichlet, and indeed Bernhard Riemann, who was also the one to conceive this interpretation.

Granting the well-groomed and weighty diagrammatic this statement brings forth—so much as to make the hypothesis a self-evident truth—how its intricate circuitry plays challenges even the shrewdest of mathematicians.

But before we question what the prime numbers tell us about the real universe (is it even possible?) and how Riemann hypothesis connects to the field of prime numbers, we need to first delve a little into the articulation of this Riemann message itself, and I will be back with that shortly.

- John Derbyshire,
*Prime Obsession*,*Bernhard Riemann and the Greatest Unsolved Problem in Mathematics*, A Plume Book, 2003 - Marcus du Sautoy,
*The Music of the Primes*:*Searching to Solve the Greatest Mystery in Mathematics*, Harper Perennial, 2002 - Roland van der Veen and Jan van de Craats,
*The Riemann Hypothesis*, Mathematical Association of America, 2016

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* A tighter way of saying this is π (N) ≈ Li (N), where π is the Prime counting function (up to N), Li is logarithmic integral, ≈ is “tends to approximately equivalent” as N gets larger, that the ratio π (N)/ Li (N) tends to 1 as N gets bigger and bigger.

]]>As indicated, this post is on the dialect of mathematics in exposing the deep reality of the universe—physical and dynamic. Voice of mathematics helps us understand this world, starting with simple algorithmic functions, and ceaselessly unfolding to rather complex and abstract hidden truths.

Mathematics is the language we employ to make sense of the physical universe, so much so that to whatever level, and however precise, we see the reality of the universe only in the window of mathematics. Both blatantly outward and deeply abstract nature of reality is captured in the words of mathematics. Take for example distance travelled by time spent allow us to seize the sense of speed: 60 miles/hr. Or at a sophisticated level, picturing the structure of space-time in the illustrious *E* = *mc ^{2}*, snapped by Einstein. Mathematical principles automatically come into play as soon as we urge to define, give a structure, or establish a relation. It’s not always this plain however. Complex behaviors of reality that involve lots of variables unfold exceedingly intricate numerical systematizations, but the glancing of reality, however deep, stretches all the way in the form of mathematics. This is only part of the whole story though.

The play of mathematics inseparably connects to the dynamics of the universe, so much so that one seemingly reflects the nature of the other.^{1,2} The expressions of mathematics and universe recite in identical tone—of an order that in one way or another imprints a cast of symmetry. From simple arithmetic to complex advanced algebra, from clear-cut cosmos to intricate quantum realm, the dictum of symmetry always palpates. Mass against energy, space against time, time versus movement, matter for antimatter, weaving fractals, growth patterns of plants, weather and stock market are all shades of symmetry other than the blatant ones we encounter in nature everyday. Physical laws governing the continuum of space-time too embody the plan of symmetry. The examples are endless, seen in myriad both forthrightly plain and mysteriously abstract ways.^{3} More often than not, symmetry is first seen in the mathematical articulation, and then at the physical level. Architectural and dynamic voices of mathematics are instructional, surfacing often to the amazement of unsuspecting mathematicians. The discovery of antimatter is a renowned such example where it first became manifest in the mathematical rendering, which was then authenticated physically.

At the surface or in its deep roots, every way we turn to we find a code of sorts gaping at us to be translated in the language of symmetry, so much so that we have gotten used to lookout for a motif in anything mathematical, at times just for its pure inspiration. The quandary arises when the helter-skelter seems to be *the* code. We find ourselves in the landscape of prime numbers.

In its utterly plain definition of being cut only by one or itself, the prime numbers have mesmerizingly incited the toughest of conjectures and trickiest of equations. Both its structuring mechanism and sprouting in number landscape eludes any systematic order that mathematics otherwise forthrightly displays in all forms—from plain number depictions to its intricate advanced molds. Prime numbers elude a pattern. Should they? Steep advancements, and shrewd insights unfurl prevailing of a conformity—subtle (very!) but robust—in the landscape of prime numbers.

Underneath the uncoordinated display, the prime numbers incite well-structured tones—of mathematics and the universe in their finest resolutions.

To comprehend this (its logic and beauty) we will have to dig deeper into some of the advancements of mathematics, and of physics—the storyline of the following post.

Will be back shortly,

Neeti.

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References

- Mario Livio,
*Is God a Mathematician*, Simon and Schuster, 2010 - Neeti Sinha,
*Physical Laws of the Mathematical Universe: Who Are We?*Archway Publishing, 2016 - Dave Goldberg,
*The Universe in the Rearview Mirror, How Hidden Symmetries Shape Reality*, A Plume Book, 2014