Hexaflexing

Simon has been into making various hexaflexagons, inspired by the Vihart channel. It was tough at first, but later the same day he didn’t need any help anymore and flexed away:

 

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Simon’s Archimedean Solids Project

https://youtu.be/dYBZXFW4bsg

Simon is working on a project that will involve constructing the Archimedean solids from paper pieces that he programs in Processing (Java) and prints out. In the previous video, Simon worked out the distance between two points to measure the side length of a pentagon that has the radius of 1 (i.e. the distance between its adjacent vertices if the distance from its center to its vertices is 1). He first made a mistake in his calculation and got a result that would be true for a hexagon, not a pentagon. He then corrected himself and got the value that he thought he could use in the Processing code, but as it turned out, the ratio between the radius and the side length was still not right. We recorded a whole new video full of calculations and playing with the code, and achieved pretty neat results after Simon used the new value in the code, but still not good enough, as Simon wanted to have his pentagons to have the side length of 40 (to match the triangles and the squares he’d already made). Simon later found a solution using a different formula for a polygon with n sides (from trigonometry, defining the radius as the side length over (2sin times 180/n)) and succeeded in getting exactly the pentagons he wanted, with the side equalling 40. See the result here:

https://www.youtube.com/watch?v=f4unEptU3Vs&t=1s

The winning formula:

If you are really into working out the calculations, feel free to check out our frantic attempts here:

https://youtu.be/TuVqc5A08es

Back in Shape

Simon prepared 100 2D shapes to make over 100 solids yesterday. He started with the easy one that he had built hundreds of times before, when he was much younger (like the Platonic Solids and some of the Archimedean Solids and anti-prisms), but then went on to less familiar categories, like elongated and gyroelongated cupolae and dipyramids! Never heard of a Gyrobifastigium? Take a look below!

Dodecahedron

Icosahedron

Cuboctahedron

Small Rhombicuboctahedron (by expanding a cube)

Icosidodecahedron

Simon didn’t build a snub cube (“is a real challenge and has two different versions that are mirror images of each other”). Nor did he make a truncated dodecahedron (as he has no decagons), nor a truncated icosahedron (doesn’t have 20 hexagons). “If you slice the corners off of an icosahedron, you get a truncated icosahedron also known as a… football!” The 62-sided rhombicosidodecahedron he had already made many times before, we’ll post an old photo later.

And then came the antiprisms:

A square antiprism – two squares connected with a band of equilateral triangles

A pentagonal antiprism

And the elongated shapes:

Pentagonal cupola (half a cantellated dodecahedron); there is no hexagonal cupola

Pentagonal rotunda (half of an icosidodecahedron)

Gyroelongated triangular pyramid

Gyroelongated square pyramid

“If you gyroelongate a pentagonal pyramid, it looks like an icosahedron, but isn’t quite that”:

Gyroelongated pentagonal pyramid

Elongated square dipyramid

Elongated triangular cupola

Gyroelongated triangular cupola

Gyrobifastigium (there it is, you found it!)

Square orthobicupola

Pentagonal orthobicupola (above) and its twisted variant – pentagonal gyrobicupola (below), looking like a UFO

The Paradox of the Mathematical Cone

Simon showed me an interesting paradox that’s difficult to wrap my mind around. If you slice a cone (at a random height), the section is a circle. The chopped-off part (a small cone) also has a circle as its base. Are those circles equal? They are the same, because they result from the same cross section. Hence the difference between them is 0. Now imagine slicing the cone an infinite number of times. “The difference between the circles will come up an infinite number of times: zero times infinity”, – Simon explained. “But zero times infinity has no value (or has any value, it’s indeterminate). Zero times infinity is the same as infinity minus infinity, which means that it can be whatever you want. Riemann’s rearrangement theory makes this true.”

The game of Loop

Simon has learned about a beautiful new game from Alex Bellos on Numberphile. The game is called Loop and resembles pool. The pictures below illustrate the layout on an elliptical game board/pool table. The black hole on the left side is the pool table pocket and the black ball with number 8 on it is the black ball. The white ball is the cuball. The colored balls are the only other balls used in the game. There is a lot of Geometry in this game.

Simon has explained how the pocket and the black ball are located exactly on the focal points of the ellipse, that is why if the black ball is hit (from whatever direction) it is always going to go towards the pocket. The winning strategy in the game would thus be to hit the cuball as if it comes from a focal point.

Simon writing the rules for stages 1 and 2 of the game:

The ball always bouncing at an identical angle:

Thus always hitting the second focal point if coming from the other one:

Configurations in a Rubik’s Square

Simon calculating how many configurations are possible in a Rubik’s Square – a flat puzzle he invented, resembling the Rubik’s Cube. He comes to the conclusion that there are only 384 configurations possible in a Rubik’s Square (including the rotations and reflections) and only 48 unique configurations. “For a Rubik’s Cube, it’s more or less 43.5 quintillion configurations!”

Think of a puzzle made of 9 wooden squares of some thickness. Every square (except the one in the middle) has one or two sides painted as shown below. The goal is to assemble the square to align the colors with on the edges. Flipping is allowed.

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.

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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!