What a blissful atmosphere at Maths Jam Antwerp yesterday, full of respect, encouragement and acceptance. It’s an international monthly meet-up taking place every second to last Tuesday of the month, simultaneously at many locations in the world, three hours of maths fun! This was Simon’s first time. He solved two difficult geometry problems and showed some of his current work to the math enthusiasts who attended. Was hopping and giggling all the way home.
Simon often drags his sketchbook to bed to “show me the beauty”, just before I would read a bedtime story to him and his sis. Last time he showed me a short proof of why there’re infinitely many primes. He assumed there were finitely many primes first… I think he learned that from James Grime:
Pi from Prime numbers:
Powers of 2:
Simon wanted to make an outside video this afternoon, about what he’s been thinking of a lot lately – continued fractions. In the video below, he looks for curious number patterns while writing irrational numbers Phi and square root of 2 as their continued fractions. Partially inspired by Mathologer’s videos.
Square root of 2 expressed as its continued fraction:
The Golden Ratio (Phi) and the Fibonacci numbers:
Phi expressed as its continued fraction:
Simon has been watching a lot of Mathologer’s videos lately, mainly about Euler’s Number (e) and Pi. He is fascinated by the proofs Mathologer presented of why each number is irrational. “Mom, the proof that e is irrational actually doesn’t require any Calculus and the proof that Pi is irrational does! While you would expect it to be the other way around, right? Because e is about Calculus!”
Here are some of Simon’s notes, inspired by Mathologer. Some facts about e:
Notes about the proof that Pi is irrational:
Notes about the proof that e is irrational:
Simon watching the Mathologer channel:
Simon and Neva make a 3D projection of a Hypertetrahedron – one of the regular solids in 4D – using straws. Simon looks up the formula for the center of the tetrahedron (radius of its circumscribed sphere) to measure the sides of the inside straws. To cut the exact length of the inside straws, he constructs a segment with the length of square root of six, divides it by 4 and multiplies the result by the original length of the straws.
Please also see our next and even cooler project – a 3D projection of a Hyperoctahedron:
The Hyperoctahedron came out to look very nice and four-dimensional. “It lands on the floor very nicely”, Simon says throwing it around – it is a very stable shape, made up of 16 tetrahedrons. Simon had to work out the centre of the triangle for this projection, which is easy to do for equilateral triangles.
The making of the Hyperoctahedron:
Measuring the center of the equilateral triangle:
Cutting the straws so that their length equals the distance between the vertex and the centre of the triangle:
The Hyperoctahedron is ready:
“I’m holding a four-dimensional shape in my hands!”
Simon: Mom, there’s no number that’s close to infinity.
I: There isn’t, is there? But even when I realise it’s true, I have trouble imagining this.
Simon: This is the way I imagine it: if you collapse an infinite length into some non-zero finite number, then all the numbers should be infinitesimally close together. That is, infinitely close together but not the same.
Simon reading from his favourite book by Murderous Maths – The Most Epic Book of Maths Ever, the chapter about the famous problem on filling a chessboard as a geometric series:
This is a Japanese version of the famous River Crossings Puzzle that Simon learned from the Scam School channel (yes, our little programming and math nerd actually watches Scam School, a channel dedicated to social engineering at the bar and in the street!)
The answer, a sequence of 17 moves:
Simon showing the classic River Crossings puzzle to friends
Math graphs for solving the simple and the more advanced River Crossings puzzles using minimum vertex covers and Alcuin Numbers (learned via Numberphile):
Simon trying to develop the remainder theorem further:
Imperial measurements system:
Clock-face Puzzle (guessing a number on the clock face after the player spells several numbers moving his finger clockwise with every finger, why it’s always 6):
Math behind card tricks:
and behind throwing dice: