# Category Archives: Simon’s sketch book

# Ackermann Function

Simon solving the Ackermann function (a function that cannot be de-recursed). It’s computable but the computer’s soon runs out of its computing power (see the last line of code below):

# Heat Equation Visualization

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

https://editor.p5js.org/simontiger/present/EaHr9886H

https://editor.p5js.org/simontiger/sketches/EaHr9886H

https://editor.p5js.org/simontiger/present/ruN8CQV77

https://editor.p5js.org/simontiger/sketches/ruN8CQV77

Inspired by 3Blue1Brown’s Differential Equations series.

# Doing Brilliant.org together with Neva

Simon teaching his sister Neva from the Mathematical Fundamentals course on Brilliant:

# Triangular, Square, Pentagonal, Hexagonal Numbers

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

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

# 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 Brilliant.org, and now Neva actually associates “fundamentals” with “fun”!

# 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:

https://www.youtube.com/watch?v=n7GYYerlQWs

https://www.youtube.com/watch?v=gi-TBlh44gY

# 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 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

isspecial 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)