Part 1: The structure of the transcendental number Pi

Properties of Pi

The number Pi  is quite important in mathematics.  It shows up in many  geometric equations, which describe circles or spheres, but also in trigonometry.

Pi is a  transcendental  number, because  there exist infinite many figures after the dot

3.14……..

, which do not repeat with the same sequence of figures again and again.

But there exist finite sequences of figures, which repeat many times and it is quite interesting to find out, which sequences that are and how often they repeat and at what positions they start to repeat.

We want to answer following questions:

  • do all figures   0, 1, 2, …. 9  occur
  • do patterns like  11, 111, 111, 1111  etc occur
  • what other patterns occur

At first let us have a look at a longer presentation of  pi:

The first 101 figures of  Pi are:

{3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, \
4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, 4, 1, 9, 7, 1, 6, 9, 3, 9, 9, \
3, 7, 5, 1, 0, 5, 8, 2, 0, 9, 7, 4, 9, 4, 4, 5, 9, 2, 3, 0, 7, 8, 1, \
6, 4, 0, 6, 2, 8, 6, 2, 0, 8, 9, 9, 8, 6, 2, 8, 0, 3, 4, 8, 2, 5, 3, \
4, 2, 1, 1, 7, 0, 6, 8}

Just by inspection we see, that all figures from 0, 1,2,3, …. 9  show up.

Positions of the figures 0, 1,2,3, …. 9

0    is at  positions  {{33}, {51}, {55}, {66}, {72}, {78}, {86}, {98}}

1    at        {{2}, {4}, {38}, {41}, {50}, {69}, {95}, {96}}

2   at          {{7}, {17}, {22}, {29}, {34}, {54}, {64}, {74}, {77}, {84}, {90}, \
{94}}

3  at           {{1}, {10}, {16}, {18}, {25}, {26}, {28}, {44}, {47}, {65}, {87}, \
{92}}

4   at          {3}, {20}, {24}, {37}, {58}, {60}, {61}, {71}, {88}, {93}}

5   at          {{5}, {9}, {11}, {32}, {49}, {52}, {62}, {91}}

6  at          {{8}, {21}, {23}, {42}, {70}, {73}, {76}, {83}, {99}}

7  at          {{14}, {30}, {40}, {48}, {57}, {67}, {97}}

8   at         {{12}, {19}, {27}, {35}, {36}, {53}, {68}, {75}, {79}, {82}, {85}, \
{89}, {100}}

9   at        {{6}, {13}, {15}, {31}, {39}, {43}, {45}, {46}, {56}, {59}, {63}, \
{80}, {81}}

 

Does there exist an equation, to calculate the positions of the figures in the number Pi ?

Can we express  the sequence of the positions of the zeros and the positions of the other figures by a mathematical equation, as it is possible for many sequences of natural numbers ?

To  answer that question, the simplest method is, to look up the sequences in:

The On-Line Encyclopedia of Integer Sequences

which was founded by N.J.A.Sloane and which at present has equations for about 250 000 different sequences of natural numbers.

This website is therefore always the first choice to look at, if one investigates sequences of natural numbers.

Now let  us start with the zeros  in the number Pi :

  • if we put in only 33, 51, 55  into the search window, then it directly finds out, that this sequence are the zeros in the number Pi.  The other sequences, which are shown below it, have also many other figures in their sequences and are therefore no valid answer
  • for the sequence of the other figures, we have to give in eventually more than only 3 figures, so that the search engine finds out, that they are a sequence  in the number Pi

This demonstrates, that probably up to now no mathematician could find an equation, with which the sequence of the positions of the figures in the number Pi can be calculated.

It also demonstrates, that it can be quite hard or even impossible,  to find simpler patterns, with which one can construct a more complex pattern, even if these simpler patterns exist.

Here is  the plot, how those figures are distributed:

  • the verical axis is for the figures 0, 1, 2, 3, … 9
  • the horizontal shows the  position of these figures in the number  Pi.

 

Pi

It looks like an oscillation.

 

How often do the figures 0, 1,2,3, .. 9  occur ?

Answer:      {{3, 12}, {1, 8}, {4, 10}, {5, 8}, {9, 13}, {2, 12}, {6, 9}, {8, 13}, {7, 7}, {0, 8}}

3 occurs  12 times

1 occurs  8 times

4 occurs 10 times

etc.

In the next blog post, we will investigate, what patterns show up, when we investigate more figures of  Pi.

  • how often occur the figures 0,1,2,3..9, if we take more and more positions of Pi ?
  • how often occur patterns  like
    • 00, 000, 0000, 00000  etc.
    • or any combinations of a finite sequence of figures

Mental attitude

This approach is that of a physicist, who sees a pattern in nature and tries to find out, if he can construct it, using simpler patterns.

This mental attitude is the opposite of  that, which  Stephen  Wolfram uses in his book:

A new kind of Science

in which he starts with very  simple  rules,  to construct patterns, some of which get more and more complex.

The expectation is, that both approaches will converge in the far future and help to understand nature   much better.

 

 

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