Category Archives: Genius

Emmy Noether

Source: Science News, Jun 2018

Noether divined a link between two important concepts in physics: conservation laws and symmetries. A conservation law — conservation of energy, for example — states that a particular quantity must remain constant. No matter how hard we try, energy can’t be created or destroyed. The certainty of energy conservation helps physicists solve many problems, from calculating the speed of a ball rolling down a hill to understanding the processes of nuclear fusion.

Symmetries describe changes that can be made without altering how an object looks or acts. A sphere is perfectly symmetric: Rotate it any direction and it appears the same. Likewise, symmetries pervade the laws of physics: Equations don’t change in different places in time or space.

Noether’s theorem proclaims that every such symmetry has an associated conservation law, and vice versa — for every conservation law, there’s an associated symmetry.

Conservation of energy is tied to the fact that physics is the same today as it was yesterday. Likewise, conservation of momentum, the theorem says, is associated with the fact that physics is the same here as it is anywhere else in the universe. These connections reveal a rhyme and reason behind properties of the universe that seemed arbitrary before that relationship was known.


James Clerk Maxwell

Source: Clerk Maxwell Foundation, June 2002

Paul Dirac & Abdus Salam – Thinking Geometrically and Algebraically

Michael Atiyah – Geometry & Algebra



Lord Kelvin: 2nd Wrangler (or Originality Intelligence)

Source: Clerk Maxwell Foundation, Jan 2008

The Tripos was not without its amusing anecdotes. One concerns Lord Kelvin. He was undoubtedly the best and most original mathematician of his year and thought he was a ‘dead cert’ for Senior Wrangler.  He said to one of the college servants on the day the Tripos results were published “Oh, just run down to the Senate House, will you, and see who is Second Wrangler”.

When the servant returned he said “You, Sir”.

Lord Kelvin had been beaten by Stephen Parkinson, later President of St. John’s College who, although not possessing great originality in mathematics was highly intelligent and had schooled himself to perfection in the executional skills of solving Tripos problems at speed .

Scientific Paper –> Dynamic Medium

Source: The Atlantic, Apr 2018

… the basic means of communicating scientific results hasn’t changed for 400 years. Papers may be posted online, but they’re still text and pictures on a page.

The Watts-Strogatz paper described its key findings the way most papers do, with text, pictures, and mathematical symbols. And like most papers, these findings were still hard to swallow, despite the lucid prose. The hardest parts were the ones that described procedures or algorithms, because these required the reader to “play computer” in their head, as Victor put it, that is, to strain to maintain a fragile mental picture of what was happening with each step of the algorithm.

Victor’s redesign interleaved the explanatory text with little interactive diagrams that illustrated each step. In his version, you could see the algorithm at work on an example. You could even control it yourself.

the whole problem of scientific communication in a nutshell: Scientific results today are as often as not found with the help of computers. That’s because the ideas are complex, dynamic, hard to grab ahold of in your mind’s eye. 

… to create an inflection point in the enterprise of science itself. 

In the mid-1600s, Gottfried Leibniz devised a notation for integrals and derivatives (the familiar ∫ and dx/dt) that made difficult ideas in calculus almost mechanical. Leibniz developed the sense that a similar notation applied more broadly could create an “algebra of thought.” Since then, logicians and linguists have lusted after a universal language that would eliminate ambiguity and turn complex problem-solving of all kinds into a kind of calculus.

 As practitioners in those fields become more literate with computation, Wolfram argues, they’ll vastly expand the range of what’s discoverable. The Mathematica notebook could be an accelerant for science because it could spawn a new kind of thinking.

To write a paper in a Mathematica notebook is to reveal your results and methods at the same time; the published paper and the work that begot it. Which shouldn’t just make it easier for readers to understand what you did—it should make it easier for them to replicate it (or not).

With millions of scientists worldwide producing incremental contributions, the only way to have those contributions add up to something significant is if others can reliably build on them. “That’s what having science presented as computational essays can achieve,” Wolfram said.

Pérez admired the way that Mathematica notebooks encouraged an exploratory style. “You would sketch something out—because that’s how you reason about a problem, that’s how you understand a problem.” Computational notebooks, he said, “bring that idea of live narrative out … You can think through the process, and you’re effectively using the computer, if you will, as a computational partner, and as a thinking partner.”

A federated effort, while more chaotic, might also be more robust—and the only way to win the trust of the scientific community.

It’ll be some time before computational notebooks replace PDFs in scientific journals, because that would mean changing the incentive structure of science itself. Until journals require scientists to submit notebooks, and until sharing your work and your data becomes the way to earn prestige, or funding, people will likely just keep doing what they’re doing.

When you improve the praxis of science, the dream is that you’ll improve its products, too. Leibniz’s notation, by making it easier to do calculus, expanded the space of what it was possible to think.

Einstein Thinks in 4 Dimensions

Source: Forbes, Dec 2016

Consider a famous shape, the Klein bottle, which is a bottle that loops into itself to create a shape with no inside and outside. You can obtain it by gluing together two Möbius strips along their edge.


Sadly, we can’t quite fit a Klein bottle in our 3-dimensional space, since it is not supposed to actually cut through itself. Like the ribbon above, the critical junction where it folds back in is meant to allow the two tubes to be completely separated.

But that’s quite easy to achieve using color, right? Here’s an example:

This uses the exact same approach as we had with the ribbon. The tube shifts into a different area in color space as it loops back in, and by the time it passes itself it is already very separated (green vs white). The Klein bottle sits very comfortably in 4-dimensional space, without any nasty self intersections.

Feynman Shares about Fermi

Source: American Institute of Physics, Jun 1966

Let’s start first, for example, with Fermi.

When I was at Los Alamos, I knew many of these great guys. Fermi was at Chicago all the time, but about half way through, somewhere, I don’t know exactly, he came to consult from time to time, to help out by consulting, at Los Alamos.

One of the early times, perhaps the first time, I don’t know, he was in a room, and we were supposed to be discussing some problem involving mixing uranium and hydrogen which I had been working on with my group. He wanted the results of it.

Now, for everybody else in the lab I was particularly good at, or I seemed to be good at, understanding the results of a calculation. So when somebody would make a calculation, I could see why it ought to be more or less like it was without actually calculating it by some physical reasoning. I was the expert, the guy they’d usually try to ask if he could see why it came out that way, and I was fairly successful.

As far as my own calculation for this hydrogen business, it was rather complicated for me to see why it came out the way it did. It was only to me the result of calculations I could give, rather than a real understanding.

So, Fermi was in this conference, this little group. It was in a small room, and there must have been eight or ten people or something like that, and he asked what about this stuff with hydrogen. He wanted me to report so he could think about it. So I thought it out.

I told the problem and he said, “Ah, let me see what I think might happen.” Then he started to give the kind of physical argument I’m always giving, and he went along; it would be this way and then it would be that way. I said, “No, you left out a feature. You see, there’s this extra complication.” “Oh, yes,” says he, and then he went a little bit further, and he worked it out and explained the results of my calculation, which I hadn’t previously understood.

It was very impressive to me, because he was able to do to me what I was able to do to somebody else. So he was just that much beyond me as I was beyond a lot of other people. So I remember that very distinctly, as a very clear thinking, clear physical thinking man.

Another story of this kind is that I wrote some kind of report on something, and when he was visiting — this was later, I know this was later — he was visiting, and he calls up on the telephone.

He says, “Hey, Feynman, I’ve been reading LADC l62” — whatever the number was -– “this article of yours,” and he says, “I wonder why you bother to print it, because couldn’t even a child see that this result has to come out this way?”

So I say, “Yes, if that child’s name was Enrico Fermi.” “No,” he says, “even an ordinary child.” It turned out at the end it wasn’t as obvious as he thought, but anyway, that’s the way we talked to each other. He’s a very nice fellow. I was often at his home, and his wife made us dinner — you know we had parties at his home and everything. I liked him very much.


How did you feel about him as to his relative position in physics? By this time, or even in Los Alamos, you began to know a lot of people. How did you regard him as to his relative standing in the world of physics?


One of the very great physicists. Yes. You must appreciate that I don’t put them in order. There’s no order, because their qualities are different, you know. I mean, each one has a way, and his was clarity of physical reasoning, which was his expertness, you see. So I don’t make an order. For instance, I couldn’t say which is the better physicist, Bethe or Fermi. That is to me an impossible assignment; Oppenheimer or Bethe, Pauli versus someone. They’re all very great guys. And Fermi to me was someone I loved to talk to, that I thought was marvelous, a very great physicist. Very great.

Post-2000 Nobel Prizes by Country

Source: Slate Star Codex,  May 2017