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):
Simon shows how to draw a segment that is Phi times longer than a unit segment. He learned from a video by James Grime how to draw the square root of 5, and worked out the rest on his own.
Simon came up with this Fibonacci function while taking a walk downtown:
f(0) = 0
f(1) = 1
f(n) = f(n-1)+f(n-2)
When we got home, he used the function to build a Fibonacci counter in p5.js:
You can play with Simon’s Fibonacci counter online at: https://alpha.editor.p5js.org/simontiger/sketches/Skhr3o8Gf
The idea about the Fibonacci function struck Simon when he was looking down at the cobbles under his feet. “Look, Mom! It’s a golden rectangle!”, he shouted:
He had read that golden ratio has a direct connection to the Fibonacci sequence. The same evening, he took out his compasses to draw a golden rectangle (this time not his own invention, but following the steps from his Murderous Math book):
If you turn the page, the smaller rectangle is a golden rectangle as well, and if you slice a square off of it, the remaining rectangle will also have the golden proportions. You can continue doing this infinitely. The sizes of the rectangles will exactly correspond to the numbers in the Fibonacci sequence, which makes these drawings an illustration to the sequence.