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
I have composed a piece of music based on the Fibonacci sequence, using modular arithmetic (I assigned numbers from 0-6, the remainders after ÷ by 7, to notes C-B, i.e. 1-C, 2-D, 3-E, 4-F, 5-G, 6-A, 0-B. Then I added harmonies to the left hand). I noticed that after 16 notes, the sequence comes back to where it started!
But what really amazed me, is:
> I tried the same with Lucas #s, and Double fibonacci #s, and it always came back to where it started! Not only that, but always with the same length of period as well! It’s amazing!!!!
So, when you see something like this, you try to go over to a whiteboard and prove it, right? This is exactly what I did. In the vid below, you can see my proof of why this happens. I also analyze it a bit more, by seeing what is special of the Fibonacci #s, and also try ÷ by different numbers, instead of 7.
Disclaimer: Numberphile has already done a musical piece based on the Fibonacci numbers and a discussion of Pesano periods. What’s specific to my video:
* Trying different fibonacci-style sequences * Proof * What’s then special about the Fibonacci #s * Making a table of different divisors * (And, mathematics-aside, doing my composition in a more mathematical way, by being more strict about the melody)
Walking home from the swimming pool (where he and Neva had been jumping into the water exactly 24 times, calling out all the permutations of 1,2,3 and 4), Simon suddenly stopped to tell me that some day, mathematics may become engulfed by computer science. Apparently, this was what he was thinking about the whole time he kept silent on the way. Once we got home I sat down to listen to the elaborate proof he had coined for his hypothesis. Here is comes, in his own words:
Someday mathematics may become computer science because most of mathematics uses simple equations and stuff like that, but computer science uses algorithms instead. And of course, algorithms are more powerful than equations. Let me just give you an example.
There’s this set of numbers called algebraic numbers, and there’s this set of numbers called computable numbers. The algebraic numbers are everything you can make with simple equations (finite polynomials), so not like trig numbers, which are actually infinite polynomials, just simple finite equations with arithmetic and power. Computable numbers, however, are a set of numbers that you can actually make with a finite algorithm. It may not represent a finite equation, but the rules for the equation have to be finite. So the algorithm that generates that equation has to be finite. It’s pretty easy to see that every algebraic number is by definition computable. Because the algorithm would just basically be the equation itself.
Is every computable number algebraic? Well, we can easily disprove that. It took very long to prove that Pi is not algebraic, that it is transcendental, as it’s called. But Pi is computable, of course, because, well, that’s how we know what Pi is, to 26 trillion decimal places. So there you go. That’s a number that is computable but not algebraic. So the Euler diagram now looks like this:
Now we look back at the beginning and we see that algebraic numbers have to do with equations and computable numbers have to do with algorithms. And because the set of all algebraic numbers is in the set of all computable numbers as we’ve just proved, the set of computable numbers will have more numbers than algebraic numbers. We have given just one example of how algorithms are more powerful than equations.
What about the mathematics that deals with numbers that are incomputable? – I asked.
Well, that’s set theory, a different branch of mathematics. I meant applied mathematics, the mathematics that has application.
Simon believes that he has found a mistake in one of the installations at the Technopolis science museum. Or at least that the background description of the exhibit lacks a crucial piece of info. The exhibit that allows to simultaneously roll three equal-weight balls down three differently shaped tracks, with the start and the end at identical height in all the three tracks, supposes that the ball in the steepest track reaches the end the quickest. The explanation on the exhibit says that it is because that ball accelerates the most. Simon has noticed, however, that the middle track highly resembles a cycloid and says a cycloid is known to be the fastest descent, also called the Brachistochrone Curve in mathematics and physics.
In Simon’s own words:
You need the track to be steep, because then it will accelerate more – that’s right. But it also has to be quite a short track, otherwise it takes long to get from A to B – which is not in the explanation. It’s not the steepest track, it’s the balance between the shortest track and the steepest track.
Galileo Galilei thought that it is the arc of a circle. But then, Johan Bernoulli took over, and proved that the cycloid is the fastest.
We’ve also made some slow motion footage of us using the exhibit (you can see that the cycloid is slightly faster, but as far as I can tell, it’s not precision-made, so it wasn’t the fastest track every time): https://www.youtube.com/watch?v=5Brub0FnpmQ
I hope that you could mention the brachistochrone/ cycloid in your exhibit explanation. I don’t think you can include the proof, because for such a general audience, it can’t fit on a single postcard!
Simon writes: I’ve built a giant project; a website / community project / platform for making algorithms! I’ve built in this video Bubble Sort, Selection Sort, Insertion Sort, Mergesort, Quicksort, Heapsort, Shell Sort and Radix Sort. So I’m done with the sorting part of the project. In the next video I’ll show you the making of the Pathfinding part of the project, and then, I’m going to put it on GitHub, and pass it on to the community, to put more algorithms on there, and even new types of algorithms!
Repl.it has published a cool interview with Simon! It was interesting how Simon struggling to answer some of the more general questions gave me another glimpse into his beautiful mind that doesn’t tolerate crude dimensionality reductions. The first question, “If you could sum yourself up in one sentence, how would you do it?” really upset him, because he said he just couldn’t figure out a way to sum himself up in one sentence. This is precisely the same reason why Simon has had trouble performing trivial oral English exam tasks, like picking some items from the list and saying why he liked or disliked them. The way he sees the world, some things are simply unfathomable, or in any case, extremely complex, too complex to imagine one can sum them up in one sentence or come up with the chain of causes and consequences of liking something on the spot. He often tells me he sees the patterns, the details. Seeing objects or events in such complexity may mean it feels inappropriate, irresponsible, plain wrong to Simon to reduce those objects and events to a short string of characters.
This made me reflect upon how Simon keeps shaking me awake. I used to find nothing wrong with playing the reductionist game and frankly, had I been asked to sum myself up in one sentence, I would have readily come up with something like “a Russian journalist and a home educator”. It’s thanks to Simon that I am waking up to see how inaccurate that is. I begin to see how many games that we play in our society are forcing us to zoom out too far, to generalize too much. How often don’t we just plug something in, pretending we can answer impossible questions about the hugely complicated world around us and inside us! Well, Simon often honestly tells me that he just doesn’t have the answer.
For that first question in the interview, I suggested Simon answer something like “it’s more difficult to sum myself up in one sentence than to prove that e is irrational”, to which he replied: “But Mom, to prove that e is irrational is easy! It’s hard to prove that Pi is irrational!”
I must add that at the same time, Simon has really enjoyed the fact that Repl.it has written a developer spotlight about him as well as the social interaction on Twitter that the piece has initiated. It gave him a tangible sensation of belonging to the programming community, of being accepted and appreciated, and inspired him to work on his new projects in Repl.it.