Simon is trying to write a program for Sphere Morphing in Processing, first making a test code in p5.js (available here: https://alpha.editor.p5js.org/simontiger/sketches/S1zcwevkz)
In the video below, Simon is explaining the challenge using Magformers triangles:
Unfortunately, the test code doesn’t quite work yet: Simon is getting three infinite triangles around the circle.
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 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:
Oops, the Magformers are back in our life. I thought that Simon was over Magformers (which he built with excessively when he was six), but he has picked them up again and taken them to a new level. He seems to be using Magformers to illustrate his increasingly philosophical thoughts in the pauses he takes between lessons and programming. Yesterday, he was quite disturbed after building with the mirror piece for a while and said: “What if two mirrors reflect each other? Would that stop time?” He added: “Just for safety, I’m going to put the mirror in the box. Never, never ever put two mirrors opposite to each other!”
He hasn’t made it interactive yet though. This was the original plan but he got stuck.
Simon hasn’t put this project online yet, as he wanted to make it more interactive and hasn’t managed to do that so far.
This entry can be viewed as an extensive English practice: Simon wrote both the title and the entry (I have corrected one sentence) and recorded an English-language video (you’ll notice he had a problem with the word “vertice” 🙂
- Equilateral triangle
- Regular pentagon
- Regular hexagon
Make the following angles using the shapes above:
- 60 degrees
- 90 degrees
- 108 degrees
- 120 degrees
- 180 degrees: tetrahedron
- 216 degrees
- 240 degrees: octahedron
- 270 degrees: cube
- 300 degrees: icosahedron
- 324 degrees: dodecahedron
- 360 degrees
You can watch this video for see this.
Simon has made a system that explains how to construct bigger 3D-objects using smaller ones. For example, in the first video he constructs Cuboctahedron using:
- cube (6 squares)
- octahedron (8 triangles)
and a Rhombicubeoctahedron by adding 12 squares.
In the next video things are getting more complicated as Simon shows how to construct a Rhombicosidodecahedron from:
- cube (6 squares)
- cube (24 squares)
- dodecahedron (12 pentagons)
- icosahedron (20 triangles)
but why is 1 cube with 6 squares and 1 with 24 squares?
Because all cubes are different sizes.
- 4 squares became a big square.
- The cubes are different sizes because 4 cubes became a big cube