How to understand higher dimenstion structures(i.e hypercube)?

Question

I am reading a book on ML there is a chapter on Dimensionality Reduction I found a very subtle paragraph related to hypercube paragraph is given blow

It turns out that many things behave very differently in high-dimensional space. For example, if you pick a random point in a unit square (a 1 × 1 square), it will have only about a 0.4% chance of being located less than 0.001 from a border (in other words, it is very unlikely that a random point will be “extreme” along any dimension). But in a 10,000-dimensional unit hypercube (a 1 × 1 × ⋯ × 1 cube, with ten thousand 1s), this probability is greater than 99.999999%. Most points in a high-dimensional hypercube are very close to the border.3 square, the distance between these two points will be, on average, roughly 0.52. If you pick two random points in a unit 3D cube, the average distance will be roughly 0.66. But what about two points picked randomly in a 1,000,000-dimensional hypercube? Well, the average distance, believe it or not, will be about 408.25 (roughly 1, 000, 000/6)! This is quite counter intuitive: how can two points be so far apart when they both lie within the same unit hypercube? This fact implies that high dimensional datasets are at risk of being very sparse:

Answer 1

These are some tricks I use to "get a feel" for higher dimension:

-understanding how to create a teseract (4d cube) from eight 3d cubes

-imagining a 4d object with advancing time - with the time you go along an axis and the 5d or n-d objects with arrays of such (in the first go-through, you go throug x5=0, in the second x5 = 1 and so on)

-computing how things like rotating work in higher dimension(In 4d you can rotate around a plane!)

-and especially in your case: look how the euclidian distance changes in higher dimension

Answer 2

Space seems to expand in higher dimensions simply because there are more degrees of freedom of movement. Although this is mincing words a little bit, while spacial dimensions can go as high as we wish, the typical way that we compute distance remains a one dimensional thing - the one dimensional distance between two points in 100 dimensions is far less significant than the one dimensional distance between two points in three dimsensions. I suggest you have a look at this video:

https://www.youtube.com/watch?v=zwAD6dRSVyI

Answer 3

The first phenomenon comes from the probability that all coordinates are non-extremal falling exponentially with the dimension $n$. The second comes from the longest diagonal being $\sqrt{n}$.