Simon said today upon waking up: “If a Physics constant suddenly popped up in pure mathematics, that would be really weird. If there is more than one universe, that would mean either that mathematics only exists in this particular universe, or that one of the laws of physics in our universe would go through mathematics to all other universes in existence. So for example, the speed of light would be the same in all universes, or Planck’s constant”.
What a blissful day at Creative Coding Utrecht! Simon also got a chance to show a few of his projects in Processing to a cool and understanding audience!
Simon told me this morning that he has found the answer to a question that bothered him so much lately -why photons (and gluons) violate E = mc². “It’s because photons are not real particles, they are virtual! That means that you cannot directly detect them. You need a photon to bounce off of an object to detect it. And gluons you cannot detect at all!”
Photons also don’t satisfy Eistein’s more complicated and less well-known equation about energy momentum relation: E² = (c²) + p²c², he added enthusiastically.
Mom, did you know there’s a density limit? Density is mass divided by size! If an object reaches the density limit it will become a black hole. If you have an object that is not homogeneous it can be more than the density limit in some places and less than the density limit in other places, and then in some places it’ll be a black hole and in others not. And so the object will swallow up itself!
(exactly what we are talking about when the picture was taken)
This is the first part in a series of four videos that Simon wants to record about Infinities Driving You Mad. Don’t worry, you won’t go mad just yet! The first video is about cardinal numbers, enumerable infinite sets and Aleph Null. Simon also shows Georg Cantor’s proof of why real numbers are not enumerable and explains what Continuum Hypothesis is about.
Earlier Simon told me about the Continuum Hypothesis, that states that there’s no infinity between the size of the natural and the real numbers: “There is no proof for it. It’s what I like to call a superposition problem: the answer is both yes and no. We do know the answer but the answer is that we son’t know the answer. You can choose what you want the answer to be and the mathematics will still be consistent!”
Warning: The next part may make your mind overheat as Simon will hop over to ordinal numbers.
“Are you impressed?” – Simon asks, laughingly, and I can see it must be a pun. We are in bed, reading up on Newton’s laws of motion that talk of forces being “impressed” upon bodies.
Simon continues: “Newton’s mechanics says that the speed limit is infinite, which says that matter doesn’t exist, which says that Physics doesn’t exist, which says that Newton’s mechanics doesn’t exist. Newton’s mechanics contradicts itself!”
The book we are reading (17 Equations that Changed the World by Ian Stewart) goes on to describe how in Newton’s laws, calculus peeps out from behind the curtains and how the second law of motion specifies the relation between a body’s position, and the forces that act on it, in the form of a differential equation: second derivative of position = force/mass. To find the position, the book says, we have to solve this equation, defusing the position from its second derivative. “Do you get it?” – I ask, “Because I don’t think I do”. — “I’ll need a piece of paper for this”, – Simon quickly comes back dragging his oversized sketchbook. Then he quickly writes down the differential equation (where the x is the position) to explain to me what the second derivative is. And then he solves it:
Euphoric fun at MathsJam Antwerp @MathsJamAntwerp last night, where Simon solved two 2×2 Rubik’s Cube puzzles and one tricky maths problem, and simply enjoyed socialising with like-minded folks. In the video below he explains how he solved the Rubik’s Cube puzzles:
1. Solve the cube so that on every face the 4 colors are all different;
2. Solve the cube so that not only the 4 colors on every face are different but also every face has a different color combination.
After we stopped filming, Simon added that a cube like this has 8! times 3^8 possibilities in total, because the cube has 8 corners and every corner has three orientations.
Simon also talks about the Choose function and symmetries:
Simon showing his solution to university maths students.
Simon also solved a tough problem (one of several tough problems) that asked to sum up the digits in x, if x equals 1111…1111 (number with 100 ones) minus 222…222 (number with 50 twos):
Simon spent the rest of the time trying to prove the ‘cosine rule’, an equation similar to Pythagoras’s theorem, that defines the side c of any triangle if it’s opposite to angle C: c² = a² + b² – 2abcosC. He got stuck with the proof, but luckily, with so many university professors walking around, he got great help from one of them, who came up with an alternative proof using vectors!
Simon is enchanted by Gödel’s incompleteness theorem (that he has learned about from Numberphile) and keeps talking about it:
“There’re problems that we just can’t solve. But if we prove that we can’t prove them, then we prove them! We can’t prove that we can’t prove that we can’t prove, and so on… Quirky! Standard math doesn’t really accept that because the statement goes on forever: you’ll just never get to what we can’t prove. What follows from Gödel’s incompleteness theorem is that that statement is actually true!”
The same evening, Simon is also bothered about the lies pupils are told in school. He repeatedly quotes James Grime that it’s a big lie that mathematics is about numbers. — “What is mathematics about? Mathematics is actually about proving! But there’s one more lie that even professional mathematicians don’t know. It’s that it’s about logic. Actually, mathematicians are a lot more creative!”
Simon is greatly impressed by the fact that if the Charge-Parity-Time (CPT) Symmetry doesn’t hold, the whole special relativity theory would have to be reconsidered. Does general relativity rely on special relativity? — I ask him. — What is special relativity about?
— It’s about inertial reference frames, — Simon answers.
— And what is that?
— It’s about either something static or moving at a constant velocity, — he gives me this big smile. — General relativity applies to accelerating reference frames!
And I am thinking, what a vocabulary he’s got in English, he talks like a grown-up. It strikes me that he doesn’t talk like this in his native Dutch or Russian. In those languages he sounds much younger, less self-assured, and he makes quite a few grammar mistakes in Russian (his little sis correcting his case forms and genders). What if he wasn’t living this trilingual life? Was communication with the outside world easier for him then? Will it become easier outside the English language that he has embraced so wonderfully? I continue to hold on to the idea that multilingualism can be nothing but enriching in the long run.
During his piano lessons, Simon has been working on a diagram that would map all the possible chords on the piano. I gave him a huge roll of paper to draw on that he spread on the floor of his piano teacher’s studio. He said he wanted to create a network of chords and how you go to other chords. “Music is basically like finding a path through the network that you like, messing with but preserving the chords in the network. What do I mean by preserving? There’re 4 things that you can do to a chord to preserve it:
- move some of the notes in the chord by multiples of an octave (like 1 octave, 2 octaves, 3 octaves, etc);
- split up or mix some of the notes in the chord (taking one note and splitting it into two copies of the same note you start with but in different octaves or mixing the note from two octaves into one);
- get rid of some notes in the chord;
- there are some notes you can add to the chord and preserve it.
So far, Simon has been able to map C Major and A Minor tonalities. He got a little bit stuck, but is determined to continue.