Simon and Neva make a 3D projection of a Hypertetrahedron – one of the regular solids in 4D – using straws. Simon looks up the formula for the center of the tetrahedron (radius of its circumscribed sphere) to measure the sides of the inside straws. To cut the exact length of the inside straws, he constructs a segment with the length of square root of six, divides it by 4 and multiplies the result by the original length of the straws.
Please also see our next and even cooler project – a 3D projection of a Hyperoctahedron:
The Hyperoctahedron came out to look very nice and four-dimensional. “It lands on the floor very nicely”, Simon says throwing it around – it is a very stable shape, made up of 16 tetrahedrons. Simon had to work out the centre of the triangle for this projection, which is easy to do for equilateral triangles.
The making of the Hyperoctahedron:
Measuring the center of the equilateral triangle:
Cutting the straws so that their length equals the distance between the vertex and the centre of the triangle:
The Hyperoctahedron is ready:
“I’m holding a four-dimensional shape in my hands!”
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 comparing possibilities to create 4D noise in different computer languages:
Many efforts to make a leap forward in understanding Matrix Math last week.
He was looking stuff up on Wikipedia, formulas on Matrix Math and turning matrices. Was extremely upset when he realized he wasn’t able to fully grasp the material and apply it yet, at this stage.
Simon talking to himself in English, lying on the floor under his desk, his chair upside down next to him: “Am I also 4D? Probably, because I live through time…”
Originally in Russian, very excited: “Mom, did you know it’s a geometrical progression? The A in the first-lined octave is 440Hz, in the second-lined octave it’s 880 Hz and in the third-lined octave – I don’t remember exactly, but how much is 880 times 2? Yes! 1760 Hz!”