Little did they know that the universe is four dimensional and even the orbit of planets is an illusion produced when straight motion in a four dimensional space is projected into three dimensions (or something).

immediately reminded me about projective geometry which is one of the most beautiful systems of mathematics.

So the universe is 4D? I don’t know. But if it was true then we could use the concepts of projective geometry to interpret something interesting.

So the universe is 4D! But how can we imagine a 4D space? It’s nearly impossible to get a perception of 4D without going out of the current 3D space. What would we meet when we went out of the 3D space? Intuitively, the answer is points at infinity, lines at infinity and a plane at infinity. This begs a question: what are the coordinates of these infinite stuff? Of course using 3 components like `(x, y, z)`

is insufficient. That’s why we need 4 components `(x, y, z, w)`

to represent all points/lines/planes including finite and infinite ones. Hence the name 4D.

The funny part is when you travel to a point at infinity and meet a stranger there. If you ask him about the coordinates of the current point you may get a very different answer: the point is not at infinity in his eyes! What does this mean?

The 3D world we are living in and perceiving is just one particular 3D space projected from a 4D space – the universe. Infinite points of this world are finite ones of other worlds. The only way to move from a world to other worlds is travelling to infinity of that world. The number of worlds may be infinite too.

Ironically …

Points at infinity do not exist. It is just a notion to refer to a certain subset of points in 4D space that cannot be projected into a certain 3D space and therefore are missed and non-existed in that 3D space. In 4D space every point is treated equally, nothing is called infinity.

I think there is another way to interpret 4D. Just place 3D world to time axis

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I really like your projective geometry series. Do you have any good resource that I can read on? I’d like to know how Fourier transform looks like after applying perspective transformation.

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You can find all the books I have on the topic of Projective Geometry here: https://drive.google.com/drive/folders/1_HF0g6gGURhlsfA9QIPI95dJE7uEUFrw?usp=sharing

The ones I marked with “#” are my favourites.

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