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 cupolas and dipyramids! Never heard of a Gyrobifastigium? Take a look below!

# Just a feeling

“Mom, I have the feeling I’m the Ramanujan or Paul of the 21st century”.

# 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.”

# Our first MEL Chemistry set has arrived!

We have received our first MEL Chemistry box, something the kids were really impatient to start. And guess what, finally something to be proud of being a Russian from St.Petersburg – that’s where MEL Science kits are actually being made! It’s been a while since I have seen a “Made in Russia” on anything awesome.

The first two experiments we tried today were part of the Artificial Sea Set: Chemical Seaweed and Chemical Jellyfish. They both involved working with metal salts (sulphates) and watching them react with different solutions. The time lapse video above shows the seaweeds “growing”: “Metal salts gradually dissolve and react with the potassium hexacyanoferrate(II). Insoluble copper, iron and zinc compounds form. These don’t just precipitate out but form “bubbles” because of the **osmotic pressure**. The fancy chemical seaweed grows from these bubbles”.

It was fun to watch the metal salts change colours: iron turned bright blue and blue copper sulphate turned brownish red!

The funny little things in the petri dish are the “jellyfish” we made as a finishing touch to our artificial sea. We created theses by firing metal salt solutions into **sodium silicate** (liquid glass). “An **ion exchange reaction** occurs between the sodium silicate and the metal salts. As a result, insoluble metal silicates form. These resemble jellyfish!”

Metal salts starting to grow in potassium hexacyanoferrate:

Unboxing the first kit:

Busy with the experiment(s):

We also dived into the MEL Chemistry app that allows you to see all the molecules of the reagents involved in 3D.

# 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:

# Rubik’s Cube Moves

Simon is getting faster and faster with the cube. Order a speed cube for his upcoming birthday? So much for “poor fine motor skills”.

# Live Stream #18. Living Code, Chapter 6: Particle Systems. 99 Balls Game.

Simon says: “In this live session, I am continuing Chapter 6 of my “Living Code” Course. This is the 4th live stream that I’m attempting to do this”. It was a tough one again, many thanks to Nahuel José for helping Simon out with an error! In the end Simon did manage to finish the second video in Particle Systems, but got another error in his third video in this chapter, so please feel free to help out if you have a minute to look at his code: https://alpha.editor.p5js.org/simontiger/sketches/HJK_bEjCf

Simon also started working on a “99 Balls” game. The next stream will be in two weeks, on July 24!

# Ramanujan-converging

Today Simon learned that it was Euler who first came up with the idea that the infinite sum 1 + 2 + 3 + 4 +… converges to -1/12. Simon explained to me the other day that there are several ways of looking at an infinite sum. One way is looking at its partial sums and summing those up. Another way is averaging partial sums and see what their average converges to (or what the average of their averages converges to). “That’s called Cesaro Summation and it’s good for closely related sums like 1 – 1 + 1 – 1 + 1… but not for 1 + 2 + 3 + 4 +…”, Simon explained. “Then there is Ramanujan Summation – a Calculus way of looking at infinite sums using derivatives and gamma, etc. That is the only way 1 + 2 + 3 + 4 +… converges to -1/12. All possible infinite sums converge if you use Ramanujan Summation.”

“Simon, you don’t trust Ramanujan Summation, do you?” I asked.

“No. Only an infinitely small section of infinite sums converge using the standard method. Converging means it settles down. That’s what we call a fixed point. If an infinite sum doesn’t converge, it can either explode to infinity or it can have more than one fixed point or do something else weird like that. Sums that are not convergent are called divergent.”

“And Ramanujan had none of those, no divergent sums?”

“Yeah, he really made it to the extreme! It’s an infinitely large extreme. All infinite sums Ramanujan-converge.”

# 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!

# Vectary.com

Simon has discovered a great new graphing tool: Vectary.com! “Finally I have found something topological,” Simon says. “There is a branch in math called topology. It’s about deforming things. I like topology!” Below two examples of Simon’s graphing work playing with it.