All calculations in part 1 and part 2 were done with the Algebra Softwareprogram Mathematica  10  from Wolfram Research.

Here is the program code:

Structure of the transcendental number Pi

Converting Pi into a list of Integers

• At first the number n of digits after the dot is chosen
• Then the real number Pi is converted into a list of natural  numbers.
  • The last member of the list defines the number of figures before the dot and is therefore dropped.
    • The command RealDigits counts all figures in a real number, but we want to know only, how many figures are after the dot.

The figures of Pi are plotted

• Thereafter the number of each figure  is calculated  ( with Tally )
• Thereafter it is calculated, how many times a sequence of figures occurs ( with PatternSequence )

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• comment:  n    defines, how many figures of the number Pi are taken after the dot.

n = 10

• Comment:   The numer Pi is converted into a list, which contains all figures of Pi, including the 3 before the dot. The last number in the list is dropped, because it gives the number of figures before the dot, in which we are not interested.

Drop[Flatten[RealDigits[N[Pi, n]]], {n + 1}]

• Comment:    The distribution of the figures is plotted

ListLinePlot[Drop[Flatten[RealDigits[N[Pi, n]]], {n + 1}]]

• Comment:    in the following,  lists are created, which contain only the same  figure at several different positions

Position[Drop[Flatten[RealDigits[N[Pi, n]]], {n + 1}], 0]
Position[Drop[Flatten[RealDigits[N[Pi, n]]], {n + 1}], 1]
Position[Drop[Flatten[RealDigits[N[Pi, n]]], {n + 1}], 2]
Position[Drop[Flatten[RealDigits[N[Pi, n]]], {n + 1}], 3]
Position[Drop[Flatten[RealDigits[N[Pi, n]]], {n + 1}], 4]
Position[Drop[Flatten[RealDigits[N[Pi, n]]], {n + 1}], 5]
Position[Drop[Flatten[RealDigits[N[Pi, n]]], {n + 1}], 6]
Position[Drop[Flatten[RealDigits[N[Pi, n]]], {n + 1}], 7]
Position[Drop[Flatten[RealDigits[N[Pi, n]]], {n + 1}], 8]
Position[Drop[Flatten[RealDigits[N[Pi, n]]], {n + 1}], 9]

• comment:  The last figure in the list is dropped, because it gives the number of figures before the dot and in this we are not interested.

Tally[Drop[Flatten[RealDigits[N[Pi, n]]], {n + 1}]]

• the following command counts, how often a pattern occurs. That was not yet used up to now and will be used in part 4.

Drop[Flatten[RealDigits[N[Pi, n]]], {n + 1}] /. {x__,PatternSequence[1, 4, 1], y__} -> {x,Style[Q, FontColor -> Red, FontSize -> 36]}

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If one takes more and more positions, then one should add  a semicolon   ;  after the commands, which show the results on the screen, because otherwise too many figures are shown on the screen.

If you use the program yourself, do not paste the comments into your Wolfram program code.

some more remarks:

I use a desktop computer with 16 Gigabytes Ram.

• For n = 10 000 000  it takes quite some time to do the calculations and therefore that is the limit, up to which I investigate the structure of Pi at the moment.
  • But this number is large enough, to find out, which questions could be of interest.
  • At present I  have the feeling, that it should be possible, to develop a mathematical equation, with which one can calculate the position of each figure, but probably, nobody can calculate it manually.

Probably a very advanced mathematical algorithm will be needed.

• this algorithm must create equations in a heuristic way, using the accumulated knowledge of mathematics, but deviating from it in an evolutionary way.
• this program should also integrate the thinking methods of mathematics, similar like chess programs have integrated all heuristic methods, which were used by chess  masters  over the last 100 years.

But if one should be successful, then it would open quite a new time for the advancement of mathematics and because physics and other sciences  heavily depend on mathematics, they would also profit a lot from it.

Therefore it is worthwhile, to go on, to investigate the structure of Pi and other transcendental numbers.