Summarize the following article:

home
publications
hobbies
blog
Counterintuitive Properties of High Dimensional Space
Our geometric intuition developed in our three-dimensional world often fails us in higher dimensions. Many properties of even simple objects, such as higher dimensional analogs of cubes and spheres, are very counterintuitive. Below we discuss just a few of these properties in an attempt to convey some of the weirdness of high dimensional space.

You may be used to using the word “circle” in two dimensions and “sphere” in three dimensions. However, in higher dimensions we generally just use the word sphere, or \(d\)-sphere when the dimension of the sphere is not clear from context. With this terminology, a circle is also called a 1-sphere, for a 1-dimensional sphere. A standard sphere in three dimensions is called a 2-sphere, and so on. This sometimes causes confusion, because a \(d\)-sphere is usually thought of as existing in \((d+1)\)-dimensional space. When we say \(d\)-sphere, the value of \(d\) refers to the dimension of the sphere locally on the object, not the dimension in which it lives. Similarly we’ll often use the word cube for a square, a standard cube, and its higher dimensional analogues.

Escaping Spheres
Consider a square with side length 1. At each corner of the square place a circle of radius 1/2, so that the circles cover the edges of the square. Then consider the circle centered at the center of the square that is just large enough to touch the circles at the corners of the square. In two dimensions it’s clear that the inner circle is entirely contained in the square.


Figure 1: At each corner of the square we place a circle of radius 1/2. The inner circle is just large enough to touch the circles at the corners.
We can do the same thing in three dimensions. At each corner of the unit cube place a sphere of radius 1/2, again covering the edges of the cube. The sphere centered at the center of the cube and tangent to spheres at the corners of the cube is shown in red in Figure 2. Again we see that, in three dimensions, the inner sphere is entirely contained in the cube.


Figure 2: In three dimensions we place a sphere at the each of the eight corners of a cube.
To understand what happens in higher dimensions we need to compute the radius of the inner sphere in terms of the dimension. The radius of the inner sphere is equal to the length of the diagonal of the cube minus the radius of the spheres at the corners. See Figure 3. The latter value is always 1/2, regardless of the dim Your answer should have exactly 10 paragraphs. Make sure that the last paragraph starts with the word "recover" and that each paragraph is separated by two new lines.
user