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

DSC_2284

DSC_2318

DSC_2319

DSC_2320

DSC_2321

Advertisements

Magformers Pythagoras set

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).

dsc_2040388899318.jpg

Chinese square Proof:

dsc_20481571486190.jpg

dsc_2051927828638.jpg

Area of Parallelograms:

dsc_20541926246679.jpg

dsc_20521360054958.jpg

Applying Pythagorean theorem to other shapes:

dsc_2043900843144.jpg

Extended theorem by the Greek mathematician Pappus:

dsc_20502120717258.jpg

dsc_20491455294093.jpg

Areas of Similar Triangles:

dsc_20561750730228.jpg

dsc_2055530778092.jpg

More of Pythagorean theorem with various shapes:

dsc_2044717909933.jpg

dsc_2046825305280.jpg

dsc_2047946430086.jpg

dsc_2042172543324.jpg

dsc_2045353646649.jpg

Puzzles:

dsc_2062565567867.jpg

dsc_2064278481395.jpg

dsc_20691197824981.jpg

 

The Magformers saga continued

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!”

And there is more! Magformers the company has actually contacted Simon on his YouTube channel, saying they loved his Magformers Table program he made in JavaScript and wanted Simon to send them an e-mail and to talk to him about it! Simon put his code on GitHub and shared it, you can view his Magformers Table online here:

https://simon-tiger.github.io/magtables/magtable/

He hasn’t made it interactive yet though. This was the original plan but he got stuck.

DSC_1823

Automatic Magformers Table

Magformers (magnetic building sets involving maths) used to be Simon’s greatest passion when he was six (just a year ago!) and this week he has been travelling in time to revisit this old love, after his little sis received a new Magformers set as a present. What Simon did next was to combine Magformers and programming: he created an automatic table listing various Magformers models (in HTML/ JavaScript). The sets that can be used to build those models were to get filled in automatically, depending on the number of specific shapes needed for every model and the number of such shapes available in every set. As you might imagine, this involved many lines of code and a whole lot of computational thinking. At one point, when Simon was nearly done, he realized that the column listing the sets wouldn’t get filled in properly. He had a bug in his program that he couldn’t find, so he turned to his older friends in Slack for help. It’s such a pleasure to see him communicate with these experienced programmers on a regular basis now and unbelievable how eager and resourceful they are. One of Simon’s friends from Slack even created his own version of Simon’s program in CodePen! What makes it even more wonderful is that Simon no longer hesitates to apply the good advice he gets. The next day he wrote some “helper functions” and the table worked!

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’s Rhombi Numberline

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.

  1. 4 squares became a big square.
  2. The cubes are different sizes because 4 cubes became a big cube