Time has its own dimensions

mathematics: More than length, width, height


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Is our world three-dimensional? Not really. Even a digital vacation photo on the computer screen can be viewed as at least a five-dimensional object: Each pixel has two coordinates that indicate its position, as well as a color that is made up of three values, the red, green and blue components. Seen in this way, the flat vacation photo suddenly becomes a point cloud in a five-dimensional space. And because of length, width, height: the dimensions are also a very colorful matter here.

In fact, when they are supposed to visualize four- or higher-dimensional spaces, mathematicians sometimes use color as a "substitute dimension" in order to make the unimaginable become imaginable. Because one thing is certain despite all the aids: it is not so easy to imagine more than three dimensions. The mathematician Vašek Chvátal even warns the readers of his textbook on linear programming with a wink: "Never try to imagine n-dimensional objects for n greater than or equal to 4. That is not only doomed to fail in the first place - it could also be intellectual Harm health. "

Nevertheless, it would not occur to a modern mathematician to only research in three dimensions. Of course, mathematics today takes place in many dimensions. Example: If the digital vacation photos are transmitted via WLAN, transmission errors occur that have to be ironed out again. To do this, the image is broken down into packages - and each of these packages is understood as a point in a space that has a few dozen dimensions.



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A hundred years ago, many mathematicians would not have been able to deal with it in a relaxed manner. It was not until 1913 that the Dutch mathematician Luitzen Egbertus Jan Brouwer defined what a dimension is and thus laid the foundation for a systematic theory of dimensions.

Previously, dimension was only intuitively understood as length, width and height. Four dimensions had already been discussed, but the fourth quickly disappeared into the drawer. "A shrewd acquaintance of mine believes that a period of time can be viewed as the fourth dimension; this idea may be criticized, but I believe that it has some value, even if it is new," wrote D'Alembert in 1754 Diderot's more famous encyclopedia under the keyword "dimension", long before Einstein's four-dimensional space-time.

But mathematics in everyday business remained almost exclusively three-dimensional until the end of the 19th century. Motto: It cannot be what you cannot think. In 1827, for example, August Ferdinand Möbius, professor in Leipzig and discoverer of the famous winding "infinite" ribbon, discovered that two-dimensional bodies cannot be transformed into a mirror image by rotating in a plane, but by rotating them in space - which everyone knows the ever an overhead slide upside down a placed on top of the projector. Möbius recognized that this also works in the same way with three-dimensional bodies. If you skilfully rotate it in four dimensions, you get a three-dimensional reflection. But then the thinker gave up: "But since such a space cannot be thought, coincidence in this case is also impossible." Basta and ad acta.

The dimension can also have fractional values

It was not until the end of the 19th century that people began to break the ban on thinking. The fourth dimension had been struck again. The mathematician Georg Cantor had discovered a figure that shows a line of length 1 on cubes of any dimension with edge length 1, in such a way that each point in the cube corresponds exactly to one image in the segment. Ergo, a one-dimensional segment had to contain the same number of points as a two-dimensional square, a three-dimensional cube, a four-dimensional hypercube and so on. Obviously, the dimension with the verb "measure" - that is the original meaning of the word stem - only had to do something to a limited extent. Cantor wrote to his friend Richard Dedekind in disbelief: "As long as you have not agreed with me, I can only say: Je le vois, mais je ne le crois pas. " - "I see it, but I don't believe it."