# Doing Math Everywhere

Doing math everywhere. At the hotel:

At The Tower of London:

Trying to prove the Wallace Product at breakfast:

Coming up with a “spot the mistake proof” at London St.Pancras train station:

# The skyscraper that set things on fire

Inspired by Matt Parker’s video  about the uniquely shaped building at 20 Fenchurch Street in London, Simon was very excited to visit this address. In the video below, made on the pavement in front of the skyscraper, Simon shows the geometric proof (he learned from Matt) of why the building’s shape used to let it set things on fire on extremely sunny days.

# Mom is 100111! And she is 33! And 124!

It’s my birthday today and Simon has calculated my age in all the bases up to 20! In the video, he explains what my color is in hexadecimal (base 16), how can turn my age into 33 and why it’s cool to be the age of a Mersenne Prime (so that I start looking forward to turning 63). He also shows a cool way to generate Mersenne numbers, Fibonacci numbers and Lucas numbers.

My birthday in all the bases up to 20 and my colour:

In binary:

In base 5:

Simon’s present: a magic square adding up to 39 in all the rows and columns (and diagonals):

# Irrationality of Square Roots

Simon has started a little video series about the Irrationality of Square Roots.

In Part 0, Simon talks about what square root of 2 is and in Part 1, he presents an algebraic proof that root 2 is irrational. He learned this from Numberphile.

In Part 2, Simon presents a geometric proof that root 2 is irrational. Based on Mathologer’s videos.

Parts 3 and 4 following soon!

# Simon’s Spirograph

Simon saw a way to draw epitrochoids (gear rolling outside another gear) and hypotrochoids (gear rolling inside another gear) on VSauce: two equal circles rolling around each other form a cardiod (a heart-like shape in the Mandelbrot set), and if you take an outside circle twice as small as the inner circle, you’ll get a nefroid, if the radius of the outer circle is 1/3 of that of the inner circle, you’ll get a flower with 3 petals, if it’s 1/4 – a flower with 4 petals and so on. Basically, this is the way a spirograph works. “What if I take an irrational number?” Simon asked, all excited. The radius of the outer circle will not equal a half, or a quarter of the radius of the inner circle, but let the ratio be an irrational number. “Let’s take an easy one: 1/Phi!” Simon took his compasses and constructed the golden ratio, then subtracted 1 from it (as Phi – 1 equals 1/Phi). “I’m almost certain something beautiful is gonna pop up!”

The two circles with the ratio of 1/:

Constructing  – 1:

Cutting the circles out of cardboard:

The first 1 1/4 rolls around the inner circle sort of resembled a cardioid:

Several rolls further:

And further:

Simon worked out the diameter and the circumference of the flower:

Two days later, we also tried rolling a circle in a circle (the ratio was 1/2 this time):

“It’s going to be very anticlimactic”, Simon warned.

Just a straight line!!

Simon writes: “But, the experiment wasn’t over yet. We then tried designing a handle going on to the circle:

When we cranked the handle, such that the circle rolled, we were supposed to get an ellipse, but instead of that, we got something else boring, a perfect circle (although you could say a circle is a kind of ellipse)! Then I tried it on my own, and I got a not that boring ellipse.”

# A trick with Lucas and Fibonacci numbers

Simon came up with this trick today and had Neva solve his riddle: any Fibonacci number is equal to the sum of its surrounding Lucas numbers divided by 5. And a Lucas number is Phi to the n, rounded to the next integer:

Simon made a game out of this (the purple ones are the Lucas numbers and the red ones are Fibonacci numbers):

# Vsauce

Completely fascinated with VSauce videos on the theory of relativity (and why there is actually no gravity, and weight vs mass), Banach-Tarski Paradox (and other paradoxes, including those by Zeno) and counting pas infinity lately. The big questions, where math kisses Philosophy and Physics.

Last night, Simon was lecturing Dad about the Alpha Null (the number of cardinal numbers), various infinities and cardinal vs ordinal.

# Simon’s proof that Phi is not transcendental

Simon has come up with a proof that Phi (the Golden Ratio) is an algebraic number (not transcendental). He proves it by showing that Phi can be the solution to a polynomial equation (which would be impossible if it was a transcendental number). Indeed, if you simplify Simon’s polynomial further, you can get x squared minus x equals one, which describes one of Phi’s remarkable qualities: the square of Phi (an infinite irrational fraction) equals exactly Phi plus 1. In fact, Simon has talked about this in his previous video (expressing Fibonacci sequence using Lucas Numbers):