Information networks: paths in multidimensional cubes part 4

The following shall demonstrate, how powerful the concept of information networks is.

We apply this concept, to calculate the paths in multidimensional cubes.

People, who have not studied mathematics or physics, in general have no idea, what a space with 4 dimensions or even many more dimensions could be and how it is possible, that one can tell anything about them.

One astonishing fact we can tell directly:

  • After having studied information networks already in some detail, we know, that we can calculate, how many different paths of any order  exist in any multidimensional cube.

How is that possible, if we not yet have any idea, how such a multidimensional cube looks like ?

Simple !  We just ignore the dimensions.  They are irrelevant, if we are interested only  in the number of  paths.

  • The only number, which we have to know is, how many corners a multidimensional cube has and that we will calculate in some of the coming posts. And that is enough, to calculate  the number of  all paths from order 3 up to order n, which is the dimension  of the cube.
    • Thereafter we apply, what we have found out about paths in information networks.
  • if we also want, to construct the paths, then we have to do some more work.
    • we must find a clever way, how to label all corners of a cube, which is easy for a 3-dim cube, but not intuitively clear, how to do it with multidimensional cubes, because we cannot visualize them. And that is, because during the evolution of our brain, men never lived  in multidimensional spaces and therefore no brainstructures do deal with them, has been developed.

But we can overcome that problem, by using mathematical reasoning. Of course, even with that, we cannot visualize multidimensional cubes. It is similar to optical illusions; even if we know, that it is an illusion, we cannot avoid to get misled by it, because we have the wrong structure in  the system of our senses, nerves and in the brain.

Example:

A  cube in 3 dimensions has 8 corners, which is easy for us to visualize.

We directly can answer, how many different paths are possible, which include each of these corners, that means, we ask, how many paths of order 8 exist:

number of paths =  8! / ( 8-8)! = 8!/1  =  8!  =  1*2*3*4*5*6*7*8  = 40 320

Of course, probably  nobody really would be able  to construct all these paths  manually, but it is no problem, to do it, using a  mathematical  software program like Mathematica and we will show it for paths of lower order, like for paths of order 3, for which there exist

8!/(8-3)! = 8!/5! = 6*7*8 = 386  different paths.

We will develop the concept of information networks step by step further  in the next blog posts and also use multidimensional cubes  to demonstrate that concept.

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