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Spherical Geometry

After Simon read up on spherical geometry on Brlliant.org, he and Neva crafted some pretty colorful half-spheres. How’s that as an alternative to Easter eggs?

They also had fun looking for shortest routes across the Atlantic applying their knowledge of geodesics.

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Ancient Chinese Game Luk Tsut K’i Game in p5.js

Play Simon’s game: https://editor.p5js.org/simontiger/present/1pGeJY7c Code: https://editor.p5js.org/simontiger/sketches/1pGeJY7c

Simon learned this game on Brilliant.org at https://brilliant.org/practice/winning-moves/?chapter=competitive-games (Warning: this link will only work if you have a Premium Subscription to Brilliant). Brilliant describes the game as follows: “Luk tsut K’i is a board game from China in the time of Confucius. In medieval Europe, it went by the title Three Men’s Morris. This game is very similar to tic-tac-toe; the objective is for one player to get their three pieces all on the same line. If this occurs, that player wins”.

Simon and Neva playing the original paper version he made
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Fun crafty puzzles Simon did with Neva

Three boxes with fruit, all the three labels are misplaced. What is the minimum number of times one will have to sample a random piece of fruit from one of the boxes to know how to label all the three boxes correctly? From Mind Your Decisions.

Connect A and A’, B and B’, C and C’, D and D’ so that no lines intersect. (Neva added colors).

Dividing 11 coins among three people: “How many ways can you divide 11 coins to 3 people? How many ways are there if each person has to get at least 1 coin?” From Mind Your Decisions.

Solving a simple quadratic equation geometrically: the geometric interpretation of “completing the square”, a notion from deriving the quadratic formula. From Mind Your Decisions.

Which way do the arrows point? (Simon made this drawing in Microsoft Paint):

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Alternating series, a crafty solution.

What does this infinite sun converge to?
Cut the four L-shapes out…
Divide the central L-shape in four L-shapes and cut those out, too…
You can go on forever…
but it’s already clear at this step, that the sum converges to 2/3 (two of the three squares the original L-shape was made up of)

Simon learned this from an alternating series visualization by Think Twice.

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Area of a dodecagon without trigonometry

How do you find out the area of this dodecagon without using trig?
Rearrange the triangles to make…
Three squares! The area of the dodecagon with a radius r is equal to the area of three r-sided squares or 3r^2.
The formulas for other polygons. There seem to be no formulas for the heptagon, nonagon and hendecagon (without using trigonometry that is). Simon’s notes above also say that no polygon can possibly have an area equal to or larger than πr^2 (because that’s the area of a circle). A square is 2r^2, a dodecagon is 3r^2, and so no polygon is 4r^2.

Simon learned this from a visual mathematics video by Think Twice.

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Messing with the Periodic table

I want to mess with the Periodic Table to see what arrangements I can put it in.

This is called the Wide Arrangement. There are aso a few other arrangements, like the Left Step Wide (or Loop) arrangement, various 3D arrangements (like the ones where you make sure any consecutive numbers are next to each other and it looks like a layered cake).

Although it would be even nicer if we moved H and He over there where they obviously belong.

Spiral periodic table with no gaps between consecutive elements.
You can put it on the top of the Christmas Tree!
Mendeleev’s Flower

Simon learned this from a Minute Physics video.

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MathsJam Antwerp 18 December 2019

Simon had a wonderful time at MathsJam Antwerp again. One of the problems was something he was already familiar with – the puzzle about hanging a painting using two pegs so that it would definitely fall if one removes any of the two pegs. He explained the way to solve this problem in an abstract way (turning pegs into strings, using knot theory and compiling the algorithm). Later the same evening, he developed a new algorithm to solve the same problem for three pegs and successfully demonstrated the result on his own shoe laces. His solution was the most efficient/ elegant in the group and his enthusiasm was very catchy, the audience said.

In the video below, Simon at first fails to apply his solution correctly, but succeeds upon the second attempt:

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Approximating pi and e with Randomness

This has been one of Simon’s most ambitious (successful) projects so far and a beautiful grand finale of 2019, also marking his channel reaching 1K subscribers. The project – approximating Euler’s number (e) in a very weird way – is based upon a Putnam exam puzzle that Simon managed to prove:

The main part of the project was inspired by 3Blue1Brown Grant Sanderson’s guest appearance on Numberphile called Darts in Higher Dimensions, showing how one’s probable score would end up being e to the power of pi/4. Simon automated the game and used the visualization to approximate e. Below is the main video Approximating pi and e with Randomness. You can run the project online at: https://editor.p5js.org/simontiger/present/fNl0aoDtW

Code: https://editor.p5js.org/simontiger/sketches/fNl0aoDtW

The history and the math behind the project:

Simon’s proof od the math behind the project:

Simon has visualized this problem and proof at: https://editor.p5js.org/simontiger/present/2uMPZ8THW

Code: https://editor.p5js.org/simontiger/sketches/2uMPZ8THW