How to design the infinite loop staircase?

This is still a very interesting proposition in mathematics. The winding staircase that Arthur and the dream designer Ariadne (played by Ellen Page) walked through in the movie is impressive. And incredible. Ariadne walked up four flights of stairs and kept going up, only to end up back where she started. This is actually the infinite staircase in the famous painting "Ascending and Descending" by Escher, the Dutch master of optical illusion graphics. The exquisite mathematical geometric figures in the painting create a visual illusion, but the "impossible world" in reality becomes possible in dreams. . %D%A Another impressive detail is that in order to test Ariadne’s ability to design buildings, Cobb asked her to design a maze in two minutes that could not be solved in one minute. The first two times Ariadne designed a rectangular maze, Cobb easily solved it; the third time she designed a circular maze, this time she stumped Cobb. And this maze is actually the famous Ring Snake Maze. This is also an impossible figure, a real maze that can never be exited. %D%A Arthur’s stairs and the maze drawn by Ariadne are not complicated, but they do not exist in the real world. In mathematical language, the real world is a Euclidean space (Euclidean space), while the maze in the dream is built in a non-Euclidean space (non-Euclidean space). %D%A Then when Cobb taught Ariadne, he folded the world into a box-like structure. The earth becomes the inner surface of the box, the sky is at the center of the box, and the world becomes topsy-turvy like a kaleidoscope, also a non-Euclidean space. %D%A If we set a coordinate system for each space, the coordinate system of the Euclidean space is a straight line, while the coordinate system of the non-Euclidean space will be curved into a circle. In one dimension, Euclidean space is a straight line, and non-Euclidean space can be a circle. In two dimensions, Euclidean space is a plane, while non-Euclidean space can have many kinds. %D%A The box world shown by Cobb is actually a spherical non-Euclidean space. If we were to construct an Escher staircase that Ariadne walked on, it would have to be curved into a circle in the height direction of that space. In this way, the highest point and the lowest point of the stairs are at the same height, so they can be connected. In this space, there are still upward and downward directions, but their meanings are different. Up and down do not represent an increase or decrease in height, but rather refer to drawing circles from two different directions. %D%A Just like looking in one direction, going up is clockwise and going down is counterclockwise. So as you go up and down, it's just constant repetition. There are no such stairs in life, but the way many things, such as clocks, work, have this property. %D%A In the maze design of the movie, if the dreamer wants to trap a person, he must give him the illusion of infinity. Think of the deceived person as a small insect. In a two-dimensional world, if it is a European space, it is a plane. You can only design a very large circle, but the insect will still run out one day. But if this is a non-Euclidean space, such as a sphere, the bug cannot escape. In this way, the dreamer can trap the enemy in the dream of his own design forever. %D%A The core idea of ??the maze designed by Cobb is to trap the enemy in a circle. But the story is more complex than that. Ariadne showed a maze structure different from that envisioned by Cobb, which produced infinite numbers of figures in the mirror. %D%A Ariadne took Cobb to a place, closed the door, and made two mirrors. Countless figures appeared in the two mirrors. Because a mirror can produce an image in a mirror, there is a mirror within a mirror within a mirror within a mirror? As the number of mirror layers increases, the image in the mirror will become smaller and smaller. But even if it is a very small image, after magnification, there is still a mirror within a mirror within a mirror within a mirror? This is a structure called a fractal in geometry. We can think of the mirror within the mirror as a metaphor for the story structure of Inception. Because there can be a mirror within a mirror, so there is a mirror within a mirror within a mirror within a mirror? Similarly, because dreams occur within dreams, there are dreams within dreams within dreams within dreams%D%A. The fractal structure corresponds to the logic of infinite recursion. The point of view in physics is that "elementary particles can be subdivided". This is exactly the case. Molecules are decomposed into atoms, and atoms are decomposed into electrons, protons and neutrons. Modern physics is further decomposing electrons, protons and neutrons. But every time a basic particle is obtained, it must be decomposed into more basic particles. This logic determines that the most basic particles in the world cannot be found. %D%A Dreams within dreams also have such a fractal-like logic, and at the beginning and end of the movie, Cobb and Saito appear in the same scene, forming a cycle. At this point, "Inception" has already Opened the box of agnosticism.

%D%AIt’s actually almost done%D%A