Information networks: special closed paths, to touch all corner points in a 3 – dimensional cube. part 9

In an information network we can investigate a lot of interesting questions.

Typical ones in multidimensional cubes are:

  • what closed  path, touching all corner points, has the maximal length
  • what closed path, touching all corner points, has the minimal length
  • how many different closed paths exist, which touch all corner points

To simpliy the investigation, we agree, to start always with point P0 and end with point P7

We already constructed one path, touching all corner points, in which we keep the sequence  P0, P1, P2, …. P7

Paths3dimCubeVersion8

  • 4 connections are on the polygon, each has a length of 1
  • 2 connections are between the polygon and the first circle, each has the length 1.414
  • 1 connection is between the polygon and the second circle, it has the length 1.73

The sum is:   4 + 2*1.1414 + 1.73 = 8,558

  • if the path would be closed, the sum would be 9,558. We call such a path a closed, well ordered path of maximal order. It is not yet clear, if this path has also the maximal length.

Now we construct a path, which also touches all corner points, but we sacrifice, that all points must  touch in the same sequence and use only  connections, which are edges.

PfadEntlangKanten3dimCubeVersion1

The path length  is 7 and it is the shortest, because only edges are included.

  • If this path would be closed, then its length would be 8. We call such a path a closed path of maximal order and minimal length. It is clear, that this path must have the minimal length, because the shortest length, which can occur between 2 points is the length of an edge and this is 1.

Assumptions:

  • A closed path in an n-dimensional cube, which touches all corner points and has only edges as connections has the length 2^n
  •  ( length of  the closed, well ordered path )   /  ( length of closed path of maximal order and minimal length )   is always  less than   ( n^1.5)  /2^n
    • question:  is this quotient a constant number or does it converge to a limit with increasing dimension of the cube ?

For some time we will leave the topic of multidimensional cubes, because the calculations for multidimensional cubes of higher dimensions are too time consuming, to be done manually.

In future posts  will investigate some other applications of information networks.

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