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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Romantic Mobius Origami

Inspired by the videos by Matt Parker

and James Grime:

 

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Simon also made the Borromean rings:

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And cubes (which Simon now uses to practice juggling!)

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Football

Wait a minute, is that Simon actually playing football? We never thought we would see that happen. All the credits go to Matt Parker. He is the one who got Simon interested in football, or rather – in the football as a geometrical object. Simon also gave us a lecture on the history of the football, its changing shape and aerodynamics.

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Dodecoration

Simon has been begging me to do this project together for days (fans of dodecahedrons that we are) and finally we made it – our own origami dodecoration! Simon learned this from a video by Matt Parker on Stand Up Math and looked up a dodecahedron graph on Google. In the video, Simon explains the main principles of how to make a dodecoration. The steps of how to fold your paper squares are below.

 

Make sure you have 30 origami paper squares, in 3 or 5 colours

Fold in two, valley fold down

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Fold again, so that the cross section looks like the letter “M”

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Make sure the  bottom of the “M” is facing you, then fold like below. Repeat for all the 30 squares.

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Now it’s time to start inserting the 30 pieces inside one another. The small triangular folds should fit together to make the vertices of the dodecahedron. Every vertex should be made up of three pieces of different colours.

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The dodecahedron graphs (the plan to follow while building):

Five colours

Three colours

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Calculating Pi by weighing circle and square

This experiment has been inspired by Matt Parker and his Stand Up Math channel.

As one of our Pi day activities, Simon attempted to calculate Pi by weighing a circle. In the video, you he first explains why this should work: the area of a circle with the radius r and the area of a square with a side of 2r can be expressed as Pi x r^2 and 4r^2 consecutively. This makes the ratio between the area of such a circle and the area of the square equal to Pi/4. In other words, Pi can be expressed as 4 times that ratio. But since both the circle and the square are made of the same material, their mass will also have the same ratio.

The result Simon got was pretty close, considering the low precision of our kitchen scales. As Simon’s math teacher correctly pointed out, the result would be much more precise if we had one thousand kids make their own circles and squares and weigh them, and then took the average of their outcomes.

Calculating Pi with a Pendulum

This video has been inspired by the wonderful Matt Parker and his video on the Stand Up Math channel:

Yesterday was Pi day and we are still celebrating! Simon experiments with calculating Pi with a physical thing, a pendulum. For the experiment, he cut a cord one fourth of the local gravity value (9.8m/s^2), that is 245 cm. One full swing of the cord makes Pi (measured in seconds)! Simon measures the time the pendulum makes 10 swings and divides that number by 10, to get the average duration of a swing.

The values Simon got were pretty close! The closest he got (not in this video, but later that day) was 3,128 sec., which is exactly the same value that Matt Parker got! What is the chance of that?

The formula is t = 2Pi times square root of l over g (where l is the length of the cord and g the local gravity).

Starring the cute 3Blue1Brown Pi. Here is some extra footage, with the 3Blue1Brown Pi riding the pendulum:

Wobbly Circles and the Center of Mass

What shape can roll well, other than a circle (wheel)? Two circles, attached together according to a formula involving a square root of two! Simon made these “wobbly circles” inspired by a Numberphile video where Matt Parker talks about how the ability to roll well (as in a wheel) is caused by the constant height of the center of mass (as opposed to a square wheel, whose center of mass goes zigzagging up and down). Wobbly wheels also have a stable height of their center of mass, hence they roll!

Simon also made a transparent version (with mom’s help):

 

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More Numberphile-inspired stuff! 

More Numberphile-inspired stuff! Simon has been studying Mersenne Primes (2^n – 1) and their relation to perfect numbers via the Numberphile channel and heard Matt Parker say no one has proved that there are no odd perfect numbers (that perfect numbers are always even). In the video below, Simon tries to prove why all perfect numbers are even. Here is Simon’s proof: When calculating the factors of a perfect number you start at 1 and you keep doubling, but when you reach one above a Mersenne prime, you switch to the Mersenne prime, and then keep doubling again. Once you double 1, you get 2, so 2 is ALWAYS a factor of any perfect number, which makes them all even (by definition, an even number is one divisible by 2):

 

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More topics Simon learned about from the Numberphile channel included:

The Stern-Brochot Sequence:

Stern-Brochot Sequence 16 Jan 2018

Prime Factors:

Prime Factors 16 Jan 2018

Checking Mersenne Primes using the Lucas-Lehmer Sequence. Simon’s destop could only calculate this far:

Checking Mersenne Primes 16 Jan 2018

The 10958 problem. Natural numbers from 0 to 11111 are written in terms of 1 to 9 in two different ways. The first one in increasing order of 1
to 9, and the second one in decreasing order. This is done by using the operations of addition, multiplication, subtraction, potentiation,
and division. In both the situations there are no missing numbers, except one, i.e., 10958 in the increasing case (Source). The foto below comes from the source paper, not typed by Simon, but is something he studied carefully:

10958 Problem 17 Jan 2018

Simon’s notes on the 10958 problem:

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The Magic Square (adding up the numbers on the sides, diagonals or corners always results in the number you picked; works for numbers between 21 and 65):

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Simon also got his little sis interested in the Magic Suare:

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And, of course, the Square-Sum problem, that we’ve already talked about in the previous post.

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Simon’s 3D version of the Square-Sum problem:

Square-Sum Problem 3D 17 Jan 2018

The Square-Sum Problem

Simon has become a full-blown Numberphile fan over the past couple of days. He had already watched two Matt Parker videos before, but it’s this week that he got seriously hooked on the channel, and it all started from the Square-Sum Problem video!

Simon recorded and edited two videos of his own (in OBS) trying to solve the Square-Sum Problem, manually and using JavaScript code:

 

 

Shadow of a 4D object

Simon built “the shadow of a 4D object” during math class, inspired by the Royal Institution’s video Four Dimensional Maths: Things to See and Hear in the Fourth Dimension with Matt Parker. Simon loved the video and watched it twice. We had come across similar thought experiments while reading a book by Jacob Perelman, a Russian mathematician, where the 4th dimension was visualized as the time dimension and the objects sliding along that 4th axis would appear and disappear in our 3D world just like 3D objects would appear as their cross sections if they were observed by 2D creatures. Here is how Simon visualized it. 

The first drawing is of a 3D object the way it actually looks when passing through a 2D world:

3D object in 2D world 1 Oct 2017

And this is what the inhabitants of the 2D world (unable to see in 3D) see – a sequence of sections of the 3D object. Similarly, we (unable to see in 4D) only see sequences of 3D sections of the 4D objects passing our world. Maybe, everything we see around us are such sections of much more complex objects as they are moving through time. “Maybe, we’re just 3D shadows of 4D objects”, says Simon.

3D object in 2D world 2 Oct 2017

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