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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Some more memories from Friesland. Binary Calculator.

A couple more images from our trip to Friesland. Simon’s binary calculator:

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Doing math at a restaurant where we were celebrating his friend’s birthday:

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The Best Shape for Train Wheels

Simon explains why train wheels are actually shaped like truncated cones. Inspired by a Numberphile video about stable rollers. The wooden slopes for the experiment Simon designed himself and his grandma (an ingenious craftswoman and woodworker, although a physician by profession) manufactured them for him.

The Pi Strip

Simon made a measuring tool to check the diameter of round objects: by wrapping the strip around them, he reads the Pi times the centimeters value, which basically gives him the diameter (as the circumference equals Pi times the diameter).

And here he is, measuring the diameters of Neva’s and Dad’s necks:

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 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:

Trinity Hall Prime Number

Simon saw this pattern in a Numberphile video featuring Tadashi Tokieda and recreated it in Excel, adding colours. There are 30 columns and 45 rows of digits in this picture, which means it is made of 1350 digits – the year that Trinity Hall (in Cambridge) was founded. the bottom is all zeros, apart from a few glitches. The glitches were necessary because the whole thing (reading from right to left, top to bottom) is also one number and it is a prime number!

Trinity Hall Pime Number 4 Jul 2018

Knots and Links

Simon explains: The Knot Atlas is a complete catalogue of all the possible knots and links and links with 3 or fewer components and 11 or fewer crossings. The number of crossings is the measure of how tight the knot or link is.

Simon making knots using pipe cleaners. the “Unknot”:

The Trefoil:

Figure 8 knot:

Cinquefoil (Mom helped a little with that one):

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And finally the three-twist knot:

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Tricks with paperclips and Knot Theory

Simon is pretty obsessed with Knot Theory at the moment (a mathematical theory that is widely used in advanced biology and chemistry, for example in handling tangled DNA).

He also learned a few tricks from one of his favourite teachers on Numberphile – Tadashi Tokieda – that probably also have something to do with Knot Theory. By folding a strip of paper in a certain way and placing rubber bands and paper clips on it and then pulling the ends of the paper strip, Simon gets the paper clips and the rubber bands linked together:

Making mathematical knots using rubber bands. A trefoil knot (the main prime knot):

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Simon says “it’s good for meditation”, too: