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Singing bottles, negative kelvins and resolving Zeno’s paradox

Picking up hiking keeps leading to beautiful conversations and thought experiments on the way. Yesterday, on our longest hike so far (over 8 km, partially in the sand), as we passed wild cows and Icelandic horses roaming free, it was a treat to hear Simon think out loud about all of these curious things:

Why his water bottle sings at a rapidly lower frequency after he takes a sip but why the frequency doesn’t change as dramatically when the bottle is half-empty? Is it because the frequency wavelength increases exponentially with tone?

Do ordered particles inside an infinite line of a laser beam mean laser has negative kelvin temperature? Or is its temperature undefined?

And as we got seriously off track as opposed to our original route, Simon began contemplating about objects catching up and whether they would ever catch up/ collide. He worked out a formula to calculate that time of collision as the difference in positions divided by the difference in speeds (what he called the algebraic approach) and the same formula changing the reference frame so that one of the objects appears stationary (relativistic approach).

As were watching a tiny duckling try to catch up with its siblings in the pond, Simon realized the catching up problem is actually the same as Zeno’s paradox (you know, the famous one about Achilles and the Tortoise). We continued talking about this the whole time while walking back and I even filmed a small part of our conversation as Simon explained how he would resolve the paradox:

The Zeno argument works, but now a more philosophical question arises: how do you define summing an infinite number of things?

Me: I though it was the difference between mathematics and physics, because in physics you can’t have an infinite number in between two other numbers.

Yes, in physics you can’t actually have this paradox, because at a certain point — the Planck length or Plank time — you can’t divide up space or time anymore. There’s just the smallest length possible or the smallest time possible. Even in math, if you have a sum like 1 + 1/2 + 1/4 + 1/8 +… you might think it’s slightly less than 2 because it never quite reaches there. If you make a list and if you chop up the sum at different positions then it gets close to 2 yet never quite reaches it. But the thing is, if you pick any number less than 2 as the answer, then there would always be more terms in that list that are closer to 2. So it can’t be less than 2, it must be 2.

The other objection is that the sum of infinitely many numbers must be infinite. And that’s also not true. Because if it were any more than 2, the sum would stop before it ever reached there. If you say the sum is 2 or higher, that means there must be terms of that list I just mentioned in that region. And there aren’t any.

So that was our intuitive explanation how you mathematically rigorously define adding infinitely many numbers together. And that actually resolves the paradox if you think about it, because if you do that you get the same answer as in the algebraic method and the relativistic method, that I mentioned earlier.

We’ll keep on hiking, even after the coronavirus crisis subsides and we can resume our usual summer activities again. It’s just so much fun to pick Simon’s brain in the wind.

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On June 22, at the sea coast in the south of The Netherlands, Simon got bitten by a tick. Climate change sucks. 54% of Dutch land is infested with Lyme-carrying ticks and the universe of this epidemiologically threatening bug keeps expanding. Contrary to what many people think, ticks live in the grass, not up in the trees. All Simon did was step out of the car and walk two meters in the grass.

In our particular case, I don’t think we should worry too much as I have managed to pull the damn thing from Simon’s leg within a few minutes, the most painful minutes in Simon’s life so far. A tiny part of the tick, however, remained under Simon’s skin and I just couldn’t get it out anymore, no matter all the instruments that a nearby hotel kindly provided. The local Dutch doctor’s assistant dismissed us, saying we “should just wait for another 24 hours”, without even having seen Simon! We then drove to the Belgian Knokke, less than half an hour away, where the amazing staff at the urgent help department at the hospital took us very seriously, no waiting time or any further ado, applied a local anesthetic and smoothly removed the rest of the tick with a needle. “You should have come here immediately”, the Belgian doctor told me. Well, I was sort of used to having the wait for hours at the Dutch emergency unit.

Simon doing math at the hospital, waiting for the anesthetic to get going

Two days later, I discovered that I had been bitten as well. Presumably while standing on my knees, removing Simon’s tick.

Hopefully, this was the end of this little nightmare for us. But it’s not the end of it for our planet. A growing number of countries is getting infected.

Neva got very inspired by this incident of how climate change has kicked our butts and devoted one of her future climate videos to this ticking bomb:

Be careful you all and check if you are in Holland for the current situation. Simon and Neva have reported both of our bites there. We view this as a memorable exercise in digital democracy: providing other people with valuable data and promoting communal trust. Isn’t this the way we should go about with COVID-19? Making the data public and open to everyone?

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Math To Go

Thanks to our usual summer hang-outs (Dutch beaches, local playgrounds and terraces) being closed, we have discovered that Simon doesn’t hate small hikes in the woods after all. At least not if he can continue thinking about math along the way.

Last week, we spent a good deal of our walk arguing whether math has been invented or discovered, juxtaposing a Plato-like ideal view on math to a more rational one, that I believe Stephen Wolfram shares, that math is an artifact. In the end, Simon brought us to a whole new level of abstract thinking, saying that, of course, math is made up, i.e. it has been invented, but just like everything that is made up it has also been discovered because the idea of anything that has been or will be invented already exists somewhere as information.

This week, Simon brought a sheet of paper to solve a puzzle he had seen on Euclidea — a wonderful learning environment for geometric constructions and proofs done the fun way (we believe, built by Russian developers).

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World Science Scholars Feature Simon’s visit to CERN in a newsletter. The current course is about neurons. Reading Stephen Wolfram.

Simon’s September visit to CERN has been featured in a World Science Scholars newsletter:

Here’s our update on the World Science Scholars program. Simon has finished the first bootcamp course on the theory and quantum mechanics by one of program’s founders, string theorist Professor Brian Greene and has taken part in three live sessions: with Professor Brian Greene, Professor Justin Khoury (dark matter research, alternatives to the inflationary paradigm, such as the Ekpyrotic Universe), and Professor Barry Barish (one of the leading experts in gravitational waves and particle detectors; won the Nobel Prize in Physics along with Rainer Weiss and Kip Thorne “for decisive contributions to the LIGO detector and the observation of gravitational waves”).

September 2019: Simon at a hotel room in Geneva taking pat in his first WSS live session, with Professor Brian Greene
September 2019: screenshot from Professor Brian Greene’s course module on quantum physics

At the moment, there isn’t much going on. Simon is following the second course offered by the program, at his own pace. It’s a course about neurology and neurological statistics by Professor Suzana Herculano-Houzel and is called “Big Brains, Small Brains: The Conundrum of Comparing Brains and Intelligence”. The course is compiled from Professor Herculano-Houzel’s presentations made at the World Science Festival so it doesn’t seem to have been recorded specifically for the scholars, like Professor Brian Greene’s course was.

Professor Herculano-Houzel has made “brain soup” (also called “isotropic fractionator”) out of dozens of animal species and has counted exactly how many neurons different brains are made of. Contrary to what Simon saw in Professor Greene’s course (mainly already familiar stuff as both relativity theory and quantum mechanics have been within his area of interest for quite some time), most of the material in this second course is very new to him. And possibly also less exciting. Although what helps is the mathematical way in which the data is presented. After all, the World Science Scholars program is about interdisciplinary themes that are intertwined with mathematical thinking.

Screenshots of the course’s quizzes. Simon has learned about scale invariance, the number of neurons in the human brain, allometric and isometric scaling relationships.

Another mathematical example: in Professor Herculano-Houzel’s course on brains we have witnessed nested patterns, as if they escaped from Stephen Wolfram’s book we’re reading now.

screenshot from the course by Professor Herculano-Houzel

Simon has also contributed to the discussion pages, trying out an experiment where paper surface represented cerebral cortex:

The top paper represents the cerebral cortex of a smaller animal. Cerebral cortex follows the same physical laws when folding is applied.

Simon: “Humans are not outliers because they’re outliers, they are outliers because there’s a hidden variable”.

screenshot from Professor Herculano-Houzel’s course: after colour has been added to the plot, the patterns reveal themselves

Simon is looking forward to Stephen Wolfram’s course (that he is recording for world science scholars) and, of course, to the live sessions with him. The information that Stephen Wolfram will be the next lecturer has stimulated Simon to dive deep into his writings (we are already nearly 400 pages through his “bible” A New Kind of Science) and sparked a renewed and more profound understanding of cellular automata and Turing machines and of ways to connect those to our observations in nature. I’m pretty sure this is just the beginning.

It’s amazing to observe how quickly Simon grasps the concepts described in A New Kind of Science; on several occasions he has tried to recreate the examples he read about the night before.

Simon playing around in Wolfram Mathematica, after reading about minor changes to the initial conditions of an idealised version of the kneading process
Simon working out a “study plan” for his Chinese lessons using a network system model he saw in Stephen Wolfram’s book “A New Kind of Science”
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Mind Mystery

Our visit to Mind Mystery, a place featuring a few famous optical illusions and math puzzles museum in the Dutch province of Limburg was really impressive.

by the entrance
this was my favourite effect, it felt like we were trapped in the Interstellar Tesseract (amazing how beautiful sphere can be shaped by a little mirrored tunnel with an LCD screen on the back wall, the trick is that the tunnel was not rectangular but trapezium-shaped, Simon explained)
and then Simon’s sister Neva lost touch with gravity
the only way to deal with that was to flip the whole building! and look there, at the back of the courtyard, isn’t that Penrose’s impossible triangel made possible?
if you look through the peephole, the triangle’s vertices seem to be touching
ahm… an Euler torus?
and we have finally tried a mirror maze
Simon playing with the Towers of Hanoi, applying a new algorithm; when we got home, Simon actually wrote a Python program that spits out the solution for up to 9 disks!
Upon coming home, Simon tried recreating one of the tricks he encountered at the museum, the 63-digit-number trick, and has discovered that the trick doesn’t always work!
no matter how exciting Mind Mystery was, this slide was ranked as the top activity by the kids, together with a similar slide in the neighbouring swimming pool (in the
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CERN Open Days September 14 – 15, 2019

The most important experience was actually simply to see how huge the Large Hadron Collider is. We totally didn’t expect the site of every experiment on the 27km ring to resemble an industrial town in its own right, scattered miles across a desert-like terrain with the Mont Blanc and the Jura mountains as the scenic back drop. It was a challenge to walk between the activities we had carefully planned in advance only to find out that some of the were full or required an hour of waiting in line. But the kids have withstood these challenges heroically and were rewarded with a few unforgettable impressions.

In front of the CMS experiment
A schematic of the LHC
It all begins with simple hydrogen protons…
the waiting
Magnet levitation above superconductive material used at CERN to create strong magnetic field to bend the path of the particles
Cloud chamber: we have actually seen energetic charged particles leave traces in the alcohol vapor in real time, in the form of a trail of ionized gas! What we saw were mainly alpha particles and electrons, we were told, judging by the character of the trail they left. Cloud chamber detectors used to play an important role in experimental physics, this is how the positron was discovered! Simon was a bit sad he didn’t get to actually build a cloud chamber as part of a workshop (they didn’t allow anyone younger than 12 to do the workshop), but he was lucky to get a personal tour at another site, where a couple of cloud chambers were available for exploration.
Our wonderful guide computer scientist George Salukvadze showing us around at DUNE, the Deep Underground Neutrino Experiment. George told us the detectors they are building will be employed at Fermilab in the U.S. Among other things, George has done the programming for the live website (screen with liquid Argon).
Playing the particle identity game
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Surrounded by the equations that changed the world

At the main entrance to CERN there is an impressive smooth curve of a memorial to the world’s most important equations and scientific discoveries:

Simon pointing to the Fourier transform function
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Supersymmetry: Why do we need the Future Circular Collider?

This is the text of the mini-lecture on Supersymmetry that CERN Research Physicist, CMS supersymmetry group convener and Deputy LHC Programme Coordinator Filip Moortgat kindly gave us during our visit to CERN and the Large Hadron Collider last week.

Filip Moortgat: Supersymmetry stands out among all the other Beyond Standard Model theories (like extra dimensions and so on). It’s particularly interesting because it answers multiple things at the same time. I would say that most other extensions of the Standard Model solve one problem but not five like supersymmetry.

The first problem: because it connects the internal property of a particle to spacetime, it actually opens a way of gravity entering the Standard Model. As you know, the main problem with the Standard Model is that gravity is not in there. So one of the major forces that we know exists is not in there. Nobody has succeeded to make gravity part of it in a way that is consistent. People hope that supersymmetry can do it, although we’re not there yet.

The second problem is called the hierarchy problem. What that means is that you have a base  mass for a particle and then you have corrections to it from all the other particles. What happens is that if you don’t have any other particles beyond the Standard Model particles you get corrections that become gigantic. What you need to do is tune the base mass and these corrections so that you get the mass that we measured for the Higgs Boson or for the w and z bosons. It’s like 10^31 minus 10^31 is a 100 type of tuning, and we find it unnatural. It’s ugly mathematics. In supersymmetry, you get automatic cancelation of these big corrections: You get a big one and then you get minus the big one (the same correction but with a minus in front of it), it cancels out and it’s pretty, it’s beautiful.

The third thing is dark matter, a big problem. 85 procent of the matter in the universe is dark matter (if you also include the energy in the universe, you get different numbers). And the lightest stable supersymmetry particle is actually a perfect candidate for dark matter, in the sense that it has all the properties and if you compute how much you expect it’s exactly what you observe in the universe. It works great. It doesn’t mean that it’s true, it would work great if you could find it.

And then there’re more technical arguments that make things  connect together in nicer ways than before. Normally, the electric symmetry is broken in the way that everything becomes zero. All the masses would be zero, the universe would just be floating particles that wouldn’t connect to each other, it would be very boring. But that’s not what happened. To show what actually happened you need to drive one mass squared term negative, which is kind of weird but that is what supersymmetry does automatically! Because the top quark mass is so heavy. Heavier than all the other quarks. For me it’s the most beautiful extension of the Standard Model that gives you a lot of solutions to problems in one go.

The problem is that we haven’t seen anything, yet! We have been looking for it for a long time and we have absolutely zero evidence. We now have reasons to believe that it’s not as light as we have originally thought, that it’s a little bit heavier. Which is not a problem. The LHC has a certain mass range, for supersymmetry it’s typically up to a couple of TeV. But it could be 10 TeV and then we couldn’t get there, we can only get up to 2 or 3 TeV. It could be factor 10 heavier than we think!

This why we are starting to discuss the planning of the Future Collider that will be able to go up the spectre of 10 TeV in mass, for supersymmetry and other theories. There’re several proposals, some of them are linear colliders, but my favourite one is a 100 km circular collider which will connect to the LHC, so that we have one more ring. That ring will actually go under the lake and that would be quite challenging, but in my opinion – although we don’t have any guarantee – we will then have a very good shot, at least in terms of supersymmetry. At the LHC we also have a good shot but don’t have enough reach that we need to really explore the supersymmetry. 

When we use conservation of energy and momentum at the collision point, what we do is we measure everybody, we sum it all up and what we need is we need to get the initial state. If something is lacking, then we know there’s something invisible going on. It could be neutrinos, or neutralinos, or it could be something else. So we have to look at the properties and the distributions to figure out exactly what we’re seeing. It’s not a direct detection but it’s a direct derivation if you want, from not seeing something, from lacking something, that we can still say it is consistent with neutralinos. 

How do you know if it’s neutrinos or neutralinos?

Neutrinos we know well by now so we know what to expect with neutrinos. Otherwise it could be neutralios but it could be something else. And then to actually prove that it’s neutralinos we have a long program of work. 

And is that mainly math?

No, it’s everything. It needs all the communities to work together, because we need to measure certain properties, distributions with the detector and we will need the theoretical ideas on how to connect these measurements to the properties of the particle. So we will need both the mathematical part and the experimental part. Translating the mathematics into the particle predictions, we will need all of that.