This blog is about Simon, a young gifted mathematician and programmer, who had to move from Amsterdam to Antwerp to be able to study at the level that fits his talent, i.e. homeschool. Visit https://simontiger.com

The bulk are his latest three videos with detailed proofs of pile-splitting magic:

In this video, I will show a pile splitting trick, and show a really elegant proof for it. So elegant, in fact, that you can extend it to a lot of other pile splitting tricks!

Also Simon’s recent counting-in-binary project got featured, an interactive Napier’s Checkerboard simulation he programmed in p5.js:

I’ve figured out how to do square-roots in binary on Napier’s Checkerboard!

Last weekend, Simon started this new school year’s first World Science Scholars course, A Beautiful Universe: Black Holes, String Theory, and the Laws of Nature as Mathematical Puzzles with Breakthrough Prize winner and Professor of Mathematics and Natural Philosophy at Harvard University, Cumrun Vafa.The first module of the course consisted of classic puzzles, some of which Simon already knew. In one of the puzzles, the scholars were asked to use Calculus to estimate how high a belt stretched around the Earth’s equator will rise above the Earth once the belt is pulled tight:

Simon says using Calculus for such open problems is still tricky for him, but he actually managed to answer this question using geometry:

Here is my 🌟 CALCULUS-FREE 🌟 solution to the last puzzle:

Unfortunately, he hasn’t got any feedback about his solution so far.

For another question, Simon said it’s a special case of the Borsuk–Ulam theorem:

Yet another question concerned dividing a circle into regions: what is the relationship between the number of intersection ponits and how many regions form:

So, in the answer choices of the previous question, the correct answer was “it grows by a combinatorial function”. I don’t like this, because some people might derive it a different way, when it’s a polynomial function instead of a combinatorial one. I do get that the answer choices are written this way, I just don’t think they’re the best they could be.

Simon generally dislikes the formal approach in the way the course is explained. He would like to see more general discussion about these fundamental mathematical concepts that help build an intuitive understanding rather than excessively relying on formulas and definitions. Today, following Module 2 of the course that dealt with how Ramanujan-Hardy’s partitions of integers in number theory are related to/ applied in string theory, Simon got very upset because he felt like Professor Vafa’s teaching style didn’t resonate with him. He missed the in-between steps in Professor Vafa’s explanations and didn’t understand why Professor Vafa was mostly talking in formulas. The level of math was pretty much what Simon is used to, but something just felt off.

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.

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 https://www.euclidea.xyz/ — a wonderful learning environment for geometric constructions and proofs done the fun way (we believe, built by Russian developers).

If you’re interested in why #covid-19 tracing apps are important and the most privacy-friendly way to implement them, please read this interactive essay by Nicky Case and play with the colorful simulations of all our possible futures. For Simon, this has been the entrance into the Nicky Case @ncasenmare universe (first recommended by 3Blue1Brown). Simon has been gulping down the playable essays on human networks and the spread of complex ideas, self-synchronization in nature, the shape of society and several other burning themes (like coming out and anxiety) and watching Nicky Case’s talks, like this one. Nicky is a self-made indie artist, programmer and writer making very edgy, very 21st century multimedia products that are both profound in content and have an engaging/interactive interface. It’s as if reading an informative piece is turned into a game. And that’s exactly what Nicky stands for: learning through play and messing about. Maybe that’s why Simon has embraced his works so eagerly, Nicky has proven to be one of those perfect matches for our self-directed learning style.

Thanks to the lock-down, Simon’s got new friends. For a little over a month now, he has been part of exciting daily discussions, challenging coding sessions and just playing together with his new gang (warning: playing always involves math). We’ve never seen him like this before, so drawn to socializing with his peers, even taking the lead in some meetings and initiating streams.

And then we realized: this is how social Simon is once he meets his tribe and can communicate in his language, at his level. Most of his new friends are in their late teens and early twenties. Most of them didn’t use to hang out together before the crisis, probably busy with college, commuting, etc. The extraordinary circumstances around covid-19 has freed up some extra online time for many talented young people, creating better chances to meet like-minded peers across the world. Finally, Simon has a group of friends he can really relate to, share what he is working on, ask for constructive help. And even though he is the youngest in the group, he is being treated as an equal. It’s beautiful to overhear his conversations and the laughter he shares with the guys (even though sometimes I wish he wasn’t listening to a physics lecture simultaneously, his speakers producing a whole cacophony of sound effects, but he likes it that way and seems to be able to process two incoming feeds at once).

Last week, Simon took part in a World Science Scholars workshop by Dr. Ruth Gotian, an internationally recognized mentorship expert. The workshop was about, you guessed it, how to go about finding a mentor. One of the things that struck me most in Dr. Gotian’s presentation was her mentioning the importance of ‘communities of practice’. I looked it up on Etienne Wenger’s site (the educational theorist who actually came up with the term in the 1990s):

A community of practice is a group of people who share a concern or a passion for something they do, and learn how to do it better as they interact regularly. This definition reflects the fundamentally social nature of human learning. It is very broad. It applies to a street gang, whose members learn how to survive in a hostile world, as well as a group of engineers who learn how to design better devices or a group of civil servants who seek to improve service to citizens. their interactions produce resources that affect their practice (whether they engage in actual practice together or separately).

It is through the process of sharing information and experiences with the group that members learn from each other, and have an opportunity to develop personally and professionally, Wenger wrote in 1991. But communities of practice isn’t a new thing. In fact, it’s the oldest way to acquire and imperfect one’s skills. John Dewey relied on this phenomenon in his principle of learning through occupation.

It has been almost spooky to observe this milestone in Simon’s development and learn the sociological term for it the same month, as if some cosmic puzzle has clicked together.

Of course, it would be a misrepresentation to say nothing of the internal conflict the new social reality unveiled in my mothering heart as I struggled to accept that Simon started skipping Stephen Wolfram’s livestreams in favour of coding together with his new friends. 👬Yet even those little episodes of friction we experienced have eventually led to us understand Simon better. We sat down for what turned into a very eye-opening talk, which involved Simon asking me to take down the framed Domain of Science posters we’d recently put up above his desktop and pointing to those infographics depicted on the posters that represented the areas of his greatest interest.

Simon simply guided us through the Doughnut of Knowledge, Map of Physics, Map of Computer Science and Map of Mathematics posters as if were on tour inside his head. And he made it clear to us that he seriously preferred pure mathematics, theoretical computer science and computer architecture and programming to applied mathematics (anything applied, really) and even computational physics, even though he genuinely enjoyed cosmology and Wolfram’s books.

“Mom, you always think that what you’re interested in is also what I’m interested in”, he told me openheartedly. It was at that moment it hit me he had grown up enough to gain a clearer vision of his path (or rather, his web). That I no longer needed to absolutely expose him to a broadest possible plethora of the arts and sciences within the doughnut of knowledge, but that from now on, I can trust him even more as he ventures upon his first independent steps in the direction he has chosen for himself, leaning back on me when necessary.

So far, in just one month, Simon has led a live covid-19 simulation stream, programming in JS as he got live feedback from his friends, cooperated on a 3D rendering engine in turtle (🤯), co-created Twitch overlays, participated in over a hundred Clashes of Code (compelling coding battles) and multiple code katas (programming exercises with a bow to the to the Japanese concept of kata in the martial arts).

Last month, ten young programmers including Simon formed a separate “Secret Editors’ Club Riding Every Train” group on Discord, uniting some “nice and active” people who met on The Coding Train channel (they also included Dan Shiffman in the group). Simon really enjoys long voice chats with the other secret editors, going down the rabbit holes of math proofs and computer algorithms. Last Tuesday, he was ecstatic recounting his 3-hour call with his new peer Maxim during which Simon managed to convince Maxim that 0.999… equals 1 by “presenting a written proof that involved Calculus”:

We even talked about infinity along the way, aleph null and stuff. There was a part where he almost won, because of the proof I showed him when we talked about infinities. I was almost stumped.

The guys have now inspired Simon to take part in the Spring Challenge 2020 on CodingGame.com, a whole new adventure. To us, the lockdown experience has felt like an extra oxygen valve gone open in our world, another wall gone down, another door swung open, all allowing Simon to breathe, move and see a new horizon.

What has been your silver lining during this COVID-19 crisis so far, in terms of self-directed learning? Simon is happy that Grant Sanderson, Stephen Wolfram and Brian Greene all have more time now to make frequent streams and tutorials. In fact, he can’t even follow all of them live as they often overlap!

Luckily, years of homeschooling have allowed us to develop a very flexible approach to daily routine, enabling us to embrace learning opportunities from across the Atlantic, that mostly present themselves in the evening hours. Our learning is circular, cyclical, not linear (we learn around the clock and Simon often returns to the topics he has already covered before but at a new level).

Brian Greene publishes daily videos called “Your Daily Equations” on the World Science Festival channel, and viewers can “order” which equation they want to discuss next. He also does a weekly live Q&A.

It’s funny how both Wolfram and Greene are Simon’s professors as part of the World Science Scholars program, but he seems to have gotten a better chance to engage with them personally now that we’re all stuck at home (through the live chat and comments) than during the official World Science Scholars sessions!

December was all about computer science and machine learning. Simon endlessly watched Welch Labs fantastic but freakishly challenging series Learning to See and even showed me all the 15 episodes, patiently explaining every concept as we went along (like underfitting and overfitting, recall, precision and accuracy, bias and variance). Below is the table of contents he made of the series:

While watching the series, he also calculated the solutions to some of the problems that Welch Labs presented, like the question about the number of possible rules (= grains of sand) for a simple ML problem if memorisation is applied. His answer was that the grains of sand would cover all land on earth:

Simon loved the historical/philosophical part of the course, too. Especially the juxtaposition of memorising vs. learning, the importance of learning to make assumptions, futility of bias-free learning, and the beautiful quotes from Richard Feynman!

I have since then found another Feynman quote that fits Simon’s learning style perfectly (and I believe is the recipe to anyone’s successful learning as opposed to teaching to the test): “Study hard what interests you the most in the most undisciplined, irreverent and original manner possible.” We have discussed the possibilities of continuing at the university again. I have also asked Simon how he sees himself applying his knowledge down the road, trying to understand what academic or career goals he may have set for himself, if any. Does he have a picture of himself in five years from now, where does he want to be by then? He got very upset, just like when asked to sum himself up in one sentence for an interview last spring. “Mom, I’m just having fun!”

Simon’s just finished auditing a class at the University of Antwerp. His first experience at the university came via a road less traveled. But then again, one may argue that we all walk the road less traveled because there’s no “normal pathway” that fits everyone.

Last spring, I shared a few videos of Simon studying at home and a couple of university professors in his MathsJam club mentioned he would probably enjoy a course in Complex Analysis (Calculus with complex numbers). I grabbed that opportunity and asked whether they would actually allow him to sit in the lectures.

Simon audited the course for one full semester (September to December), with me accompanying him to all the lectures to make sure he didn’t disturb anyone with his “youthful enthusiasm”. Before we arrived at the first lecture, I’d made it clear to Simon that we absolutely must remain silent in class. I wasn’t sure he would manage to control himself, for the main reason that had never managed to do so before, not even at the theatre. But then again, maybe at the theatre he sensed that the condition wasn’t as crucial. On our first day, I knew the professor was nervous about Simon possibly disturbing the class, I was nervous myself and I couldn’t believe how nervous Simon suddenly was. There was one thought nagging me: Have I spoiled it by my stern warning about keeping quiet?

Simon kept incredibly quiet. He didn’t even dare introduce himself. I had never seen him this way before. The professor was relieved, even elated. On my part, I was shocked by the high level of the course and whether Simon was too tense to tune in. The course turned out to be for college seniors; in Simon’s case, possibly a year or two too early. With Simon you never know. He always learns top down, and when I say “top” I mean Mount Everest top. “We try a couple more lectures and then see if it’s too much for you”, I told Simon.

The second and the third time, he was still quite nervous, but later he let go of most of that tension. Several times he got very bored, two hours felt like a long time for him to sit quietly. Still he said he didn’t intend to quit. And once, at the end of October, at the moment when I positively lost it and didn’t have any clue about what the professor was talking about anymore, he whispered in my ear: “Now it’s actually getting interesting!” During the break, he summed up the general idea about the zeta function and the professor said he understood it correctly.

I don’t like asking Simon how much he understands every time. I don’t think it’s a fair question to ask. We didn’t attend the practice section of the course because it didn’t match Simon’s schedule (the practice lesson started early in the morning and was impossible to combine with Simon’s late night classes from New York). Auditing a class doesn’t involve any compulsory attendance, Simon won’t be doing the exam. During the last several sessions, he was relaxed about being able to control the volume of his voice and sit quietly when necessary. It was at the uni that I heard him whisper for the first time! At the last lecture, he was treated to his favourite topic, the zeta function.

My general conclusion is that auditing a course has been a nice way to get exposed to what studying at the uni is like, even though we may have picked the wrong course in terms of difficulty level or in terms of what interests Simon at the moment (contrary to last spring, when he was all about calculus and complex numbers, he is currently investing most of his time into logic, computer science and computer electronics). He definitely still misses a lot of fundamental knowledge, especially in integral calculus, but by now I’m familiar with his learning style and know that he will come back to what he hasn’t dealt with properly when the time is ripe, at the new turn of the spiral, so to speak.

I know attending classes won’t be Simon’s primary source of knowledge as he learns best through self-study (mainly videos and books), but such experiences are definitely going to mean something both in terms of personal growth and mathematical thinking. “Do you want to audit a more fundamental calculus or integral calculus class here at the uni?” I asked him the other day. “No, of course not! I can just learn that on Brilliant!” he answered. “A course on sequences perhaps, as suggested by one of the professors?” – “No, I don’t want to”, – Simon replied.

Maybe we’ll be back at the uni at a later stage, with more practical discussion involved instead of passive listening, and in a subject/at a level he feels less timid to actively contribute to that discussion. What would also help is if there was a more official way to follow university courses for bright young minds like Simon. At the moment, it’s only possible as a personal favour or if I sign myself in and take Simon along, which contributed to Simon’s timidity and being afraid to feel present.

We’ll just be taking it one step at a time, grateful for the freedom that we have. My very special thanks go to Simon’s math professor who has a kind and courageous heart. He has also signed his newly published book for Simon: