如何想象第十維空間

【原始音頻解說詞的相關文字如下】：

0. A point (no dimension)

We start with a point. Like the 「point」 we know from geometry, it has no size, no dimension. It』s just an imaginary idea that indicates a position in a system.

1. The first dimension – a line

A second point, then, can be used to indicate a different position, but it, too, is of indeterminate size. To create the first dimension, all we need is a line joining any two points. A first dimensional object has length only, no width or depth.

2. The Second Dimension – A Split

If we now take our first dimensional line and draw a second line crossing the first, we』ve entered the second dimension. The object we』re representing now has a length and a width, but no depth. To help us with imagining the higher dimensions, we』re going to represent our second dimensional object as being created using a second line which branches off from the first.
Now, let』s imagine a race of two-dimensional creatures called 「Flatlanders」. What would it be like to be a Flatlander living in their two-dimensional world? A two-dimensional creature would have only length and width, as if they were the royalty on an impossibly flat playing card. Picture this: a Flatlander couldn』t possibly have a digestive tract, because the pipe from their mouth to their bottom would divide them into two pieces! And a Flatlander trying to view our three-dimensional world would only be able to perceive shapes in two-dimensional cross-sections. A balloon passing through the Flatlander』s world, for instance, would start as a tiny dot, become a hollow circle which inexplicably grows to a certain size, then shrinks back to a dot before popping out of existence. And we three-dimensional human beings would seem very strange indeed to a Flatlander.

3. The Third Dimension – A Fold

Imagining the third dimension is the easiest for us because every moment of our lives that is what we』re in. A three dimensional object has length, width, and height. But here』s another way to describe the third dimension: if we imagine an ant walking across a newspaper which is lying on a table, we can pretend that the ant is a Flatlander, walking along on a flat two-dimensional newspaper world. If that paper is now folded in the middle, we create a way for our Flatlander Ant to 「magically」 disappear from one position in his two-dimensional world and be instantly transported to another. We can imagine that we did this by taking a two-dimensional object and folding it through the dimension above, which is our third dimension. Once again, it』ll be more convenient for us as we imagine the higher dimensions if we can think of the third dimension in this way: the third dimension is what you 「fold through」 to jump from one point to another in the dimension below.

4. The Fourth Dimension – A Line

Okay. The first three dimensions can be described with these words: 「length, width, and depth」. What word can we assign to the fourth dimension? One answer would be, 「duration」. If we think of ourselves as we were one minute ago, and then imagine ourselves as we are at this moment, the line we could draw from the 「one-minute-ago version」 to the 「right now」 version would be a line in the fourth dimension. If you were to see your body in the fourth dimension, you would be like a long undulating snake, with your embryonic self at one end and your deceased self at the other. But because we live from moment to moment in the third dimension, we are like our second dimensional Flatlanders. Just like that Flatlander who could only see two-dimensional cross-sections of objects from the dimension above, we as three-dimensional creatures can only see three-dimensional cross-sections of our fourth-dimensional self.

5. The Fifth Dimension – A Split

One of the most intriguing aspects of there being one dimension stacked on another is that down here in the dimensions below we can be unaware of our motion in the dimensions above. Here』s a simple example: if we make a M