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
Simon has just graphed this to show how the Lorentz factor or gamma ( on the y axis) is dependent on the speed of the object (the x axis). The 100 on the x axis is the speed of light. You can see how the speed makes virtually no difference to the Lorentz factor (of relativistic time and mass) until the speed of the object reaches about 85 percent of the speed of light. At around 90 percent of the speed of light the Lorentz factor reaches 2 (which means that time is twice as slow by then and the relativistic mass doubles), and at 99 percent the factor is already 7. For 100 percent or the speed of light itself, the Lorentz factor equals infinity, Simon explained.
Simon explains why the proof that root 4 is irrational is false and shows a couple more related theorems (he came up with) generalizing the relationship between the exponent and the factor of a number.
We were reading “17 Equations that Changed the World” late last night, the chapter about the wave equation. Simon got all excited about timbres (shapes of sound waves), that are essentially sine waves. He said he knew an alternative way to look for the fundamental frequency (the sin x wave):
“The smallest number that’s divisible by all the numbers in a sequence is the product of all those numbers divided by the greatest common divisor/factor of all those numbers. That’s the Chinese remainder theorem (or rather, a generalisation of it).
If you took a rational frequency and an irrational one and made them into waves, the waves would never ever ever meet, except for one point. So sometimes there’s no fundamental frequency. Because we need at least two points where the waves meet up to define a fundamental frequency.
Sadly, this happens most of the time. In fact, not even most. 100 percent of the time there’s no fundamental frequency. Technically, it’s an infinitely small chance that any number you come up with at random is rational! But fortunately for us, we can approximate the fundamental frequency here: use the two points that are closest to the waves meeting to get an approximate fundamental frequency. And it always works!
This is incredible! We’ve found a connection between a discrete problem, of what’s the smallest number that divides all of the numbers in a given sequence, to a continuous problem, of what is the fundamental frequency of a combination of sine waves. In other words, we found a discrete solution to a continuous problem!”
Simon, what does discrete mean?
“I’ll give you an example. The natural numbers, even though they are infinite, they are still discrete, because there are gaps between them. And a number in between those gaps is not a natural number anymore. A continuous thing however is for example like the real numbers. There’re no gaps. Because if there were gaps, any number in between those gaps was another real number”.
Simon has made an enormous poster from his earlier animated version of the Times Tables Visualization! Simon is hoping to present this project at the Processing Community Day in Amsterdam in January 2019. The poster is already being printed!
Simon writes: This is a visualization for the times tables from 1 to 200.
Start with a circle with 200 points. Label the points from 0-199, then from 200-399, then from 400-599, and so on (you’re labeling the same point several times).
We’ll first do the 2x table. 2×1=2, so we connect 1 to 2. 2×2=4, so we connect 2 to 4, and so on.
2×100=200, where’s the 200? It goes in a circle so 200 is where the 0 is, and now you can keep going. Now you could keep going beyond 199, but actually, you’re going to get the same lines you already had!
For the code in Processing, I mapped the two numbers I wanted to connect up (call them i), which are in between 0 and 200, to a range between 0 and 2π. That gave me a fixed radius (I used 75px) and an angle (call it θ). Then I converted those to x and y by multiplying the radius by cos(θ) for x, and the radius by sin(θ) for y. That gave me a coordinate for each point (and even in between points, so you can do the in between times tables as well!) Then I connect up those coordinates with a line. Now I just do this over and over again, until all points are connected to something.
Unfortunately, Processing can only create and draw on a window that is smaller than a screen. So instead of programming a single 2000px x 4000px poster, I programmed 8 1000px x 1000px pieces. Then I just spliced them together.