Crafty, Geometry Joys, Math Tricks, Murderous Maths, Simon teaching, Simon's sketch book

Inscribed angle theorem

“It reveals itself once you complete the rectangle to find the centre. Because, of course, the diagonal passes through the centre once you inscribe a rectangle inside the circle, because of the symmetry”.
Tiling the quadrilaterals Simon has crafted applying the inscribed angle theorem.
Tiling the “shapes generated by the inscribed angle theorem”
“The theorem says that if you have a circle and just three random points on it, then you draw a path between te first point to the second, to the centre, to the third point and back to the first point”.
Coding, JavaScript, Murderous Maths, Physics, Simon teaching, Simon's Own Code, Simon's sketch book

Heat Equation Visualization

A visual solution to Fourier’s heat equation in p5. Play with the two versions online:

Inspired by 3Blue1Brown’s Differential Equations series.

Geometry Joys, Math Tricks, Murderous Maths, Simon teaching, Simon's sketch book

Triangular, Square, Pentagonal, Hexagonal Numbers

Applying one of his favorite materials – checkers – Simon showed me the tricks behind polygonal numbers. The numbers written in pen (above) correspond to the actual triangle number (red rod) and the row number (blue rod).
Square numbers
Pentagonal numbers
And the next pentagonal number
(Centered) Hexagonal numbers
Fragment of the next (centered) hexagonal number
The following morning I saw that Simon came up with these general formulae to construct square, pentagonal and hexagonal numbers using triangle numbers. The n stands for the index of the polygonal number. Later Simon told me that he had made a mistake in his formula for the hexagonal numbers: it should not be the ceiling function of (n-1)/2, but simply n-1, he said.

I asked Simon to show me how he’d come up with the formulae:

Here is a square number constructed of two triangle numbers (the 5th and the 4th, so the nth and the n-1st)
The working out of the same construction. In the axample above n equals 5, so the 5th square number is indeed 25.
The nth pentagonal number constructed using three triangle numbers: the nth triangle number, and two, n-1st triangle numbers.
The working out of the pentagonal number formula
The nth hexagonal number
The formula for calculating the nth hexagonal number from six n-1st triangle numbers plus 1. (Simon later corrected the (n+1) into (n-1)).
Coding, Computer Science, JavaScript, Simon teaching, Simon's Own Code, Simon's sketch book

Back to the sorting algorithms: Beadsort (and a short lecture about the generator function)

Link to the project:
Made my Beadsort step-by-step with a generator function! (Scroll down to see the “Next” button!) Code:
The video also contains a bonus tutorial about what a generator function is!
Math Riddles, Murderous Maths, Notes on everyday life, Simon teaching, Simon's sketch book, Together with sis

Teaching Mathematical Fundamentals

Simon loves challenging other people with math problems. Most often it’s his younger sister Neva who gets served a new portion of colourful riddles, but guests visiting our home also get their share, as do Simon’s Russian grandparents via FaceTime. Simon picks many of his teaching materials in the Mathematical Fundamentals course on, and now Neva actually associates “fundamentals” with “fun”!

Crafty, Geometry Joys, motor skills, Murderous Maths, Simon teaching, Simon's sketch book

A Square Triangle?

Simon explains what Gaussian formula is to check a shape’s curvature and shows how to make a triangle with three 90° angles. Or is it a square, since it’s a shape with all sides equal and all angles at 90°? He also says a few words about the curvature of the Universe we live in.

Almost everything he shares in this video Simon has learned from Cliff Stoll on Numberphile:

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

Trinagular birthday probabilities

“What is the chance that two people in a group of, say, 30 people would have their birthday on the same day?” I asked Simon as we were sitting on a bench by the river Schelde late last night, waiting for his Dad and sister to arrive by boat. The reason for this question was that one of the professors at Simon’s MathsJam club turned out to have celebrated his birthday exactly on the same day as I the week before. Besides I was afraid of Simon getting bored just sitting there, “enjoying the warm evening”. At first, I thought he didn’t hear my question and repeated myself a couple of times. Then I noticed he was so silent simply because he was completely immersed in the birthday problem.

Eventually, at that time already on Antwerp’s central square, Simon screamed with joy as he told me the formula he came up with involved triangle numbers! “It’s one minus 364/365 to the power of the 29th triangle number!” he shouted. “It’s a binomial coefficient, the choose function!”

Simon’s solution defining the probability of two people having the same birthday in a group of n people. The highlighted diagonal in the Pascal triangle are the triangle numbers. For example, 15 is the 5th triangle number. So in a group of 6 people, the probability would be 1 minus 364/365 tothe power of 15.
A few days later Simon told me his previous formula wouldn’t work for a group of 366 people and quickly came up with a simpler formula, without any triangle numbers.
Milestones, Murderous Maths, Music, Piano, Simon teaching, Simon's sketch book

Simon’s Fibonacci Music Pesano Periods

Simon writes:

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)

Simon’s original score