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.
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
Simon made this blog entry:
What is this? That’s a castle. It doesn’t look like it, because that’s the area. You can watch this video to see if that’s a castle or not.
A scoop of what Simon builds in just a couple of hours, now in a set of short videos because the photos don’t suffice anymore.
Simon said he wanted to write a post. What he came up with makes a perfect blueprint of how his brain works.
He adds: but what are these triangles and that square there? See pictures 1 and 2 for see what they are.
Two days later Simon decided to illustrate his entry with a video:
Cubeoctahedron: 8 🔺 6 ⬜, Rhombicubeoctahedron: 8 🔺 18 ⬜ and icosidodecahedron: 12 5-angle 2d-shapes 20 🔺
Simon, Neva and the monster truck Simon has built all having a great time.