How does the second dimension exist?

Some of the images and texts come from the "Mathematics and Art" exhibition at the University of Greifswald. PerMATH has received official permission from the initiator of the exhibition to take over the pictures.
Author: Brigitte Wessenberg,  PerMath-Team.

The transition from the 2nd to the 3rd dimension

A Möbius strip, named after the mathematician August Ferdinand Möbius (1790-1868), is a three-dimensional Object without back! That is the special thing about this structure, because normally you have all items front and back, left and right. They show the typical features of the room.

The "Colossus of Frankfurt" (about 5 meters high) by
Max Bill (1908-1994) was established in 1986 and is
an artistic examination of the Möbius strip.

Anyone can build such a tape themselves, all they have to do is glue a strip together at its ends turned 180 °. If you run your finger along the tape, it seems to us that the figure only has one side. You can paint through with a pen in one! So there is no "back" here.

Ants - planar beings, so to speak, that do not perceive the 3rd dimension - crawl along the ribbon forever, they are on a seemingly endless surface.

Representation by M.C. Escher (1898-1972), who for his
mathematically and technically fascinating graphics is famous.

Similar to how the surface beings of the ant type can only experience a 2-dimensional world, humans can only see 3 dimensions. He can no longer imagine the 4th dimension. But, if we succeed in comparing the transition from the 2nd to the 3rd dimension on a level higher, then we might get an idea of ​​how the 4-dimensional space could be.

  The transition from the 3rd to the 4th dimension

We imagine planar beings that live on a plane. Its plane should now move and hit a ball.

The sphere does not belong in the world of planar beings. But when she enters their world, the surface dwellers first experience the first point of contact between the plane and the sphere. A little later, an ever-growing circle emerges for the area residents. At a certain point in time, the circle develops backwards, becomes smaller and finally ends again in a point.

We 3-dimensionally gifted beings recognize the situation from the outside at a glance, the surface beings have to experience this gradually in a temporal sequence.

Why shouldn't we be able to imagine that everything we humans experience in the course of time would not be instantly manageable by a 4-dimensional mind? Perhaps it is not by chance that Hermann Minkowski (1864-1909) presented Einstein's theory of relativity in 4 dimensions and added the time t as a new dimension to the 3 spatial directions x.y.z?

With the help of a few models we try to track down the fourth dimension.

The two-dimensional equivalent of a cube is a square, and its one-dimensional equivalent is a line. If you move this line perpendicular to it by a distance that is equal to its length, you create the square.

If you move the square perpendicular to its plane by a distance that is equal to its side, you sweep over a cube.

So if you still had a direction perpendicular to the three-dimensional space available, you could move the cube in this direction by a distance equal to its edge length and you would get one four dimensional cube. A physically meaningful interpretation of this fourth dimension is, as already mentioned, time. If a line exists in a time interval, the length of which we set equal to its side length after selecting the appropriate unit of measurement, we can represent this as a square in a coordinate system of the plane, analogously to the square existing in a time interval as a cube.

The cube can, however, be mapped into the plane through a central projection, whereby the front cube surface appears larger than the rear one, although both are the same size.

Model of the 4-dim. Cube

Similarly, the four-dimensional cube can be mapped into space through a central projection, whereby its initial state (the outer cube of the model) is greater than its final state (the inner cube of the model). The 8 further edges realized as red threads represent the Lanes of the cube corners in the time direction.

The picture "Corpus hypercubicus" by Salvador Dalí has ​​to do with these considerations, note the title of the work and also the key to it that Dalí gave on the floor in the form of the development of the three-dimensional cube. One can be a little amazed at how deeply an artist has penetrated into the abstract thoughts of mathematics.

Salvador Dalì: Corpus hypercubius

The mathematician Felix Klein (1849-1925) found a peculiarity of the 4th dimension.
A knot that is in a closed string can only be loosened by cutting the string.

This no longer applies in 4-dimensional space. There the knot could be loosened by simply twisting it!

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