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Auditing a class at the university

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:

Faculty of Mathematics in Antwerp
Can you see Simon?
Simon extremely nervous on the first day
Several weeks later, much more relaxed
“”Now it’s actually getting interesting!”
“I know where this is going!”
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Nash Equilibrium

Simon explaining the Nash Equilibrium with a little game in p5.js. Play it yourself at: https://editor.p5js.org/simontiger/sketches/lfP4dKGCs
Inspired by TedEd video Why do competitors open their stores next to one another? by Jac de Haan.

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Proof Visualization. Warning: Mind-boggling!

Inspired by the Card Flipping Proof by Numberphile, Simon created his own version of this proof. He made a solitaire game and proved why it would be impossible to solve with an even number of orange-side-up circles. He drew all the shapes in Microsoft Paint, printed them out and spent something like two hours cutting them out, but it was worth it!

The colourful pieces in the lower row are a “key” to solve the solitaire puzzle. The objective is to remove all the circles. One can only remove a circle if it’s orange side up. Once a circle is removed, its neighbouring circles have to be flipped. Using the key, start with the yellow pieces, and move in the direction of the “grater than” sign (from smallest to largest).

If there’s an odd number of orange circles in the middle, then the end pieces are the same, both orange or both white. In both cases the total number of orange circles will also be odd. If there’s an even number of orange circles in the middle, then the ends have to be different (one orange, one white).

In the case of odd number of orange pieces, the ends have to match. In the case of an even number of orange pieces, you would have pieces that point the same way at both ends. “Now we’ve proven that to make this puzzle possible it has to have an odd number of orange pieces”, Simon says.

Why? Imagine a stick figure that always walks to the right, but always faces in the direction of the arrow (as in it can’t go backwards). It would flip every time it reaches an orange circle. Focusing on everything except the ends, if there are an odd number of orange circles, the puzzle pieces would face the other way. Which means that the end pieces are the same, and therefore the end circles are the same. If there are an even number of orange circles in the middle, the puzzle pieces would face the same way. Which means that the end pieces are different, and therefore the end circles are different.

Simon finds this sort of proof easy, but I felt like my brains are going to boil and dripple through my ears and nostrils. He patently exlained it to me several times and types the above explanation, too.

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MathsJam Antwerp 20 November 2019. A Blast and a Responsibility.

Today, Simon returned to a problem he first encountered at a MathsJam in summer: “Pick random numbers between 0 and 1, until the sum exceeds 1. What is the expected number of numbers you’ll pick?” Back in June, Simon already knew the answer was e, but his attempt to prove it didn’t quite work back then. Today, he managed to prove his answer!

The same proof in a more concise way:

At MathsJam last night, Simon was really eager to show his proof to Rudi Penne, a professor from the University of Antwerp who was sitting next to Simon last time he gave it a go back in June. Rudi kept Simon’s notes and told me he really admired the way Simon’s reasoning spans borders between subjects (the way Simon can start with combinatorics and jump to geometry), something that many students nurtured within the structured subject system are incapable of doing, Rudi said. Who needs borders?

Later the same evening, Simon had a blast demonstrating the proof to a similar problem to a larger grateful and patient audience, including Professor David Eelbode. The first proof was Simon’s own, the second problem (puzzle with a shrinking bullseye) and proof came from Grant Sanderson (3Blue1Brown) on Numberphile.

“Don’t allow any constraints to dull his excitement and motivation!” Rudi told me as Simon was waiting for us to leave. “That’s a huge responsibility you’ve got there, in front of the world”.

Murderous Maths, Notes on everyday life, Simon's sketch book

The beauty of the Cubic Formula

One of Simon’s most beloved sources of knowledge is the Welch Labs channel. Recently he has been rewatching the series about imaginary numbers and the history of their discovery. Did you know that came about because of the Cubic Formula?

The proof of the Cubic Formula is a bit longer than that of the Quadratic Formula (on the yellow sheet)
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Social encounters

Such a pleasant play date last week with another eager learner. Simon shared his GeoGebra skills and some geometrical paper tricks, among other things. It’s heartwarming to see Simon blossom socially, he is growingly attentive to younger kids and generally engaging with people of various ages, as long as they show interest in anything Simon has an understanding of.

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More Puzzles from Maths Is Fun

In an earlier post, I have mentioned that for many games he programs Simon got his inspiration from the site Maths Is Fun. Perhaps I should add that at our home, Maths Is Fun has become an endless source of fun word problems, too! The problem below has been our favourite this week:

Simon’s equations to solve the problem
Simon has developed a system to show the relation between the actual time a and time m that a mirrored clock would show: m = 12 – a
Another clock puzzle from Maths Is Fun
Simon’s solution
solving this during his evening tea

Some of the puzzles Simon likes to recreate with paper and scissors rather than program:

A version of Connect 4 but this time with the tables of multiplication! Every player is only allowed to move one paper triangle at a time (the triangles indicate which two numbers one can use to get the next product in the table). The one who colours four products in a row wins.
As the game progresses it gets trickier
For the jug puzzle game, Simon has developed a graph plotting the winning strategy (analogous to what he once saw Mathologer do for another game).
Double-sided numbers, sort of a two-dimensional cellular automaton. The objective is to get to a state when all the numbers would be one colour. The rule: if a cell changes its colour, its four neighbours (not diagonal) also change colour. There’re also other versions of this puzzle with more difficult initial conditions.
A number-guessing game based on binary representation. When he was 8 years old, Simon programmed a similar trick in Processing. He also developed the same sort of trick for base 3 numbers.

Simon and Neva have also especially liked the Tricky Puzzles section (puzzles containing jokes).

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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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Simon’s Cycle Formula

During Chinese lesson yesterday, Simon came up with what he calls his “Cycle formula” to calculate all the permutations of placing n numbers in a cyclical order (like on a clock face). He also proved the formula. Wait, Chinese lesson? Yes, I know, this guy manages to squeeze some math everywhere. His Chinese tutor loved it by the way. “Well, we’ve both learned something!” Simon exclaimed delightfully.

the formula is (n-1)!