This has been an interesting arithmetic task Simon came up with from his Secrets of Sums book, aimed at practicing long division. You had to do the division and count how many times the 1s and the 2s appeared in your calculation, then use that number a sa secret code to decipher the second part of a rude joke (see below):
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:
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.
Simon programmed a presentation to explain why 28×28 is not the same as 20×20 + 8×8 geometrically. The code is quite complicated and involves some trigonometry and conditional statements: the grid is divided into different parts every time Simon clicks and depending on how many times he has already clicked. This is typical Simon – coming up with an inherently arduous and complex system to visualise the beauty of the world around him, even of the seemingly trivial things. By the way, the inspiration for the 28×28 grid came from Simon’s favourite math channel, 3Blue1Brown and its latest video on Neural Networks (the grid was used to explain computer vision).
Simon is doing quite a lot of sums in his head nowadays, looks like it’s a new trend. Today, while bathing in the fountain outside, he was calculating how long 1/16th of a minute lasted. And a couple days ago, while waiting for his appointment at the hospital, he calculated how long it would take someone to read a whole page of random numbers, taking an educated guess that one takes 4 seconds to read out one number and remembering Daniel Shiffman mentioned there were 100×5 numbers per page in his book.
Simon has been watching a lot of Siraj Raval’s videos on neural networks lately, brushing up his Python syntax and derivatives. He has even been trying the great Jupyter editor, where one can build one’s own neural network and install libraries with pretrained networks https://try.jupyter.org/
Just like with Danel Shiffman’s videos, the remarkable thing about Siraj’s (very challenging) courses is that they also touch upon so many subjects outside programming (like art and music and stock exchange) and are compiled with a sublime sense of humour.
Simon is a fan of the 3Blue1Brown channel and absolutely loved their video on solving the Towers of Hanoi puzzle with binary and ternary numbers. He practiced a lot with both. Eventually he developed his own, rhythmic, way to solve the puzzle:
Applying ternary numbers (solving the puzzle in 80 steps):
The video on 3Blue1Brown channel:
Applying binary numbers (solving the puzzle in 15 steps):
Simon watching 3Blue1Brown videos:
He also really liked the video on Visualizing the Riemann zeta function and yelled “Mom, look how beautiful this is!” the whole time:
Still from the 3Blue1Brown video that Simon found mesmerizing:
Later, Simon decided he was wrong with the math defining the vehicles intersecting each other and changed it to:
This is Simon’s explanation why he changed the formula:
From a lesson on importing images:
Circle Intersection (I like the design!)
Simon is not only doing Precalculus en Calculus, but also enjoys maths at all levels (see previous post about The Secrets of Sums). For example, we do some Dutch elementary school tests to make sure he can handle the arithmetic. Below is a beautiful example of how his mind works. The task was to find the number exactly between 36×36 and 38×38 on the number line. Simon immediately came up with a parabolic graph and said it couldn’t be 37×37, based on that graph. I think he can graph just about anything in the world 🙂
Simon pulled out his old Magformers Pythagoras set and this time around, he really nailed all the tasks independently. The set offers a variety of puzzles to “prove” the Pythagorean theorem and apply it to other shapes (even 3D!), as well as teaches several more tricks (such as the ratios between the areas of similar triangles or the areas of parallelograms).
Chinese square Proof:
Area of Parallelograms:
Applying Pythagorean theorem to other shapes:
Extended theorem by the Greek mathematician Pappus:
Areas of Similar Triangles:
More of Pythagorean theorem with various shapes:
The new favourite (Murderous) Math book is The Secrets of Sums by Kjartan Poskitt. It deals with simple arithmetic and Simon found it too easy when he first opened it, but then noticed that the book offers very funny and smart tricks to crack the sums, with loads of British humour.
Simon enjoys the Matrices section of the Precalculus on Khan Academy, scoring 100 percent in the quizzes:
And watching the 3 Blue 1 Brown channel: