Murderous Maths

Rational Approximations for Phi

“If I get the next two digits right, I’ll be ecstatic!” Simon says, as he hurries on with a φ (Phi) approximation algorithm using Fibonacci numbers. He keeps dividing every following Fibonacci number by the previous one and eventually gets quite a good Phi approximation a precision of 6 digits! This experiment is inspired by Mathologer, who applied this algorithm to approximate both Pi and Phi and show how “wildly less irrational Pi is than Phi”, Simon says. Simon calculated more terms for Phi though.

Crafty, Geometry Joys, Physics, Simon makes gamez, Simon teaching

Simon made his own foam Rubik’s Cube

Simon saw this design in a video by Mathologer and adapted it slightly (Mathologer used glue and no screws). He had dreamt of making a cube like this for months, but the idea of crafting one from wood seemed too complicated. Today it occurred to him that he can make the design using his new woodlike foam and press iron screws into the foam to hold the magnets! On to the wooden model now!

Geometry Joys, Milestones, Murderous Maths, Notes on everyday life, Simon teaching, Simon's sketch book

Geometric Definition of e

The idea comes from a video by Mathologer. Simon sketches a geometric definition of the Euler’s number (e) using integrals. He messed up a little with the integral notation, but corrected it later (after we stopped filming). Please see the photos below:

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

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!

 

Coding, Java, Milestones, Murderous Maths, Simon's Own Code

Simon’s Times Tables Visualization

Times Tables Visualization 31 May 2018

Simon shows his Times Tables Visualization in Processing (Java) and talks about how it’s connected to Mandelbrot Set. See the code with the README on GitHub: https://github.com/simon-tiger/times_tables

View the full animation here:

 

Simon writes: This is a visualization for the times tables from 1 to 200 (including the in-between numbers that are multiples of .01). I used modular arithmetic to write the code:

0. Start with a circle with 200 points (I’ve chosen 200, your number could be anything, but we’ll use 200 in the instructions).

1. Label the points from 0-199, then from 200-399, then from 400-599, and so on (you’re labeling the same point several times).

2. We’ll first do the 2x table. 2×0=0, same thing so we don’t do anything. 2×1=2, so we connect 1 to 2. 2×2=4, so we connect 2 to 4, and so on.

3. 2×100=200, where’s the 200? It goes in a circle so 200 is where the 0 is, and now you can keep going.

4. Now you could keep going beyond 199, but actually, you’re going to get the same lines you already had!

5. You can now create separate images for the 2x table (which we’ve just done), the 3x table, the 4x table, the 5x table, and so on. You can even try in-between numbers (like 2.53) if you want.

In the program, you see an animated image at the left of the screen, and 4 static images (representing examples of times tables) to the right of that. They represent the 2x, 34x, 51x and 99x tables.

The idea of a times tables visualization comes from a video by Mathologer, but the code Simon wrote completely on his own.

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

Very irrational numbers expressed as their continued fractions

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:

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The Golden Ratio (Phi) and the Fibonacci numbers:

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Phi expressed as its continued fraction:

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

The irrationality of Pi and e

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:

Murderous Maths, Simon's sketch book

1 + 2 + 3 + 4 +… = – 1/12? Or does it break math?

Simon keeps coming back to this irresistible math problem about whether the infinite sum 1 + 2 + 3 + 4 +… converges to negative 1/12. He was quite disappointed to hear Mathologer explain that it’s highly contraversial and that many mathematicians don’t believe in the negative 1/12 solution, at least not as long as they use the “equal” sign. Then, about a week later, on May 7 he suddenly told me he is now completely convinced that “Numberphile’s proof (that 1 + 2 +3 + 4 +… does converge to -1/12) breaks math!”