Algebraic Proof That Higher Dimensions Should Exist

This is a Clifford torus. In geometric topology, it is the simplest and most symmetric flat embedding of the Cartesian product of two circles \(\color{#6ab825}{S^1_a}\) and \(\color{#6ab825}{S^1_b}\) (in the same sense that the surface of a cylinder is "flat"). It is named after William Kingdon Clifford. A normal torus is basically a a tube shape that looks like a doughnut or an inner tube, created by revolving a circle in the \(\color{#6ab825}{3rd}\) dimension around a circle, which is why a torus is a "surface of revolution.

Here🔗 is a link to a Desmos page that you can play with it. When you change the 3D properties (\(\color{#6ab825}{X}\), \(\color{#6ab825}{Y}\) and \(\color{#6ab825}{Z}\)), you can just see the doughnut rotating.

But when you play with the \(\color{#6ab825}{4th}\) dimension (\(\color{#6ab825}{W}\)), your doughnut seems to lose its 3D form, meaning, you can't 'imagine' what's going on physically. However,

Before playing with the Clifford Torus, I suggest reading the rest of this post. And before that, I strongly suggest watching Neil deGrasse Tyson explaining a 2D civilization here:

Now you can bla bla

In our universe, it is clear that mathematical rules are present. Below, you can find an example of a ball falling down, for the young philosophers which may say;

Math isn't real. Humanity invented it.

A bold, young philosopher

Let me just show it with a simple example and a question at the end.

When a stone falls down towards Earth, as it falls, meaning its height changes, accelerates based on the simple product of the gravitational acceleration on Earth times the square of time, divided by two. More clearly, the distance \(\color{#6ab825}{d}\) travelled by an object falling for time \(\color{#FF8B8B}{t}\):
\[\color{#6ab825}{d} = \frac{g\color{#FF8B8B}{t}^2}{2}\]

Air friction or another force of attraction can be 'included' in the calculation, but these additions complicate the formula and increase its precision, without invalidating the basic calculation. Below, you can find a kind of 20 fps recording of a ball falling.

The numbers represent positions. An initially-stationary object (the basketball) which is allowed to fall freely under gravity, drops a distance which is proportional to the square of the elapsed time. This image, spanning half a second, was captured with a stroboscopic flash at \(\color{#6ab825}{20}\) flashes per second.

During the first \(\color{#6ab825}{1/20th}\) of a second the ball drops one unit of distance (here, a unit is about 12 mm); by \(\color{#6ab825}{2/20th}\) it has dropped at total of \(\color{#6ab825}{4}\) units; by \(\color{#6ab825}{3/20th}\)s, \(\color{#6ab825}{9}\) units and so on.

Even though this image wasn't recorded in a vacuum, the observation matches with the formula because the timeframe is just half a second. If it was longer, the result would start being quite inaccurate after only 5 seconds of fall (at which time an object's velocity will be a little less than the vacuum value of \(9.8 \mathrm{m/s^{2}} × 5 \mathrm{s} = 49 \mathrm{m/s} \) due to air resistance).

Air resistance induces a drag force on any body that falls through any atmosphere other than a perfect vacuum, and this drag force increases with velocity until it equals the gravitational force, leaving the object to fall at a constant terminal velocity.

(Image Source; Wikimedia Commons, Image Credit: MichaelMaggs)

In computer graphics,