The human brain can simulate a complex model of three-dimensional space, but it is very difficult to simulate four-dimensional space. The main reason is that the four-dimensional space contains too much information, which is difficult for the human brain to process, and it is difficult to show all the information in the high-dimensional space in the low-dimensional space.

For four-dimensional space, the way we can understand is analogy, and we can infer the properties of high-dimensional space by low-dimensional analogy. In order to express the law in four-dimensional space, we need to reduce its dimension step by step.

Mathematics is a very good tool, which can help us to deal with all dimensions. One of the methods of dimensionality reduction in mathematics is "projection". Projection is essentially a function transformation, which shows some information of high-dimensional objects in low dimensions.

One-dimensional projection

The zero dimension is a point. When the point is in the n-dimensional space, there are n variables to describe the position of the point. One dimension is a line. In mathematics, a straight line is a set of continuous point coordinates. If a one-dimensional straight line is projected into a zero-dimensional space, it is a point.

Two-dimensional projection

Two dimensions are a surface. In mathematics, two-dimension is a surface composed of infinite straight lines, and the projection of the surface in one dimension is a straight line.

Three-dimensional projection

Three-dimensional is a body, such as a cube in three-dimensional, and the projection of the cube in two-dimensional plane is more complicated. Different angles of projection will get different shapes, which can be rectangular or other polygons.

As shown above, no matter at which angle, the projection on the two-dimensional plane can only be a plane figure, and the figure obtained by each projection only contains part of the information of the cube; With the transformation of various angles, all the information of the three-dimensional cube will be displayed.

Four-dimensional projection

Cubes correspond to hypercubes and spheres correspond to hyperspheres, but we can't imagine things in four-dimensional space; However, by analogy with the above projection, we can infer that the projection of hypercube in three-dimensional space has the following properties:

(1) In the three-dimensional space, the projection of the hypercube is a three-dimensional figure;

(2) With the change of projection angle, 3D projection will have different shapes;

(3) The simplest stereoscopic projection is a cube;

It is difficult to imagine a hypercube according to the above properties. The above picture shows the three-dimensional shape of a hypercube under different projection angles. Hypercube contains more information than three-dimensional cube.

High dimensional projection

Humans can't imagine high-dimensional things, but mathematics can help us understand the essence of high-dimensional things. For example, the famous Calabi-Qiu Chengtong space is a six-dimensional space, and the projection of this six-dimensional space in three dimensions can be simulated by computer, as shown in the following figure.