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Combinatorial Fractalization

Topic closed. 1 reply. Last post 11 years ago by JADELottery.

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Thread Starter
The Quantum Master
West Concord, MN
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December 7, 2001
3938 Posts
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Posted: November 10, 2006, 1:04 pm - IP Logged

Combinatorial Fractalization

    C(n, r) =  [from z = 1 to z = n - r + 1]  å C(n - z, r - 1)

    C(n, r) = C(n - 1, r - 1) + C(n - 2, r - 1) + C(n - 3, r - 1) + ... + C(r + 3, r - 1) + C(r + 2, r - 1) + C(r + 1, r - 1) + C(r, r - 1) + C(r - 1, r - 1)

    Fractals of C(n, r) -  {C(n - 1, r - 1), C(n - 2, r - 1), C(n - 3, r - 1), ... , C(r + 3, r - 1), C(r + 2, r - 1), C(r + 1, r - 1), C(r, r - 1), C(r - 1, r - 1)}
    Fractals of C(n, r) - [from z = 1 to z = n - r + 1] y {C(n - z, r - 1)}

    Iteration of C(n, r) Fractals
      Fractals of C(n - 1, r - 1) - {C(n - 2, r - 2), C(n - 3, r - 2), C(n - 4, r - 2), ... , C(r + 2, r - 2), C(r + 1, r - 2), C(r, r - 2), C(r - 1, r - 2), C(r - 2, r - 2)}
      Fractals of C(n - 1, r - 1) - [from z = 1 to z = n - r + 1] y {C(n - z - 1, r - 2)}


      Fractals of C(n - 2, r - 1) - {C(n - 3, r - 2), C(n - 4, r - 2), C(n - 5, r - 2), ... , C(r + 2, r - 2), C(r + 1, r - 2), C(r, r - 2), C(r - 1, r - 2), C(r - 2, r - 2)}
      Fractals of C(n - 2, r - 1) - [from z = 1 to z = n - r] y {C(n - z - 2, r - 2)}


      Fractals of C(n - 3, r - 1) - {C(n - 4, r - 2), C(n - 5, r - 2), C(n - 6, r - 2), ... , C(r + 2, r - 2), C(r + 1, r - 2), C(r, r - 2), C(r - 1, r - 2), C(r - 2, r - 2)}
      Fractals of C(n - 3, r - 1) - [from z = 1 to z = n - r -1] y {C(n - z - 3, r - 2)}
      .
      .
      .

      Fractals of C(r + 1, r - 1) - {C(r, r - 2), C(r - 1, r - 2), C(r - 2, r - 2)}
      Fractals of C(r + 1, r - 1) - [from z = 1 to z = 3] y {C(r - z + 1, r - 2)}


      Fractals of C(r, r - 1) - {C(r - 1, r - 2), C(r - 2, r - 2)}
      Fractals of C(r, r - 1) - [from z = 1 to z = 2] y {C(r - z, r - 2)}


      Fractals of C(r - 1, r - 1) - {C(r - 2, r - 2)}
      Fractals of C(r - 1, r - 1) - [from z = 1 to z = 1] y {C(r - z - 1, r - 2)}

Within every combinatorial set there are subsets of combinations that are similar in characteristics to the whole combination. Example, below is a sample combinatorial set of 8 numbers taken 6 at a time. Along side the combinatorial set are a few subsets of combinations. Table 1 shows only one leg of the fractal path. The fractals are in blue and transformed to the right. The symbol Y is the superset fractal and extends to infinity.

                                      Table 1
    Fractal Path is Y ® y(8,6) ® y(7,5) ® y(6,4) ® y(5,3) ® y(4,2) ® y(3,1)

1

2

3

4

5

6

 

1

2

3

4

5

 

1

2

3

4

 

1

2

3

 

1

2

 

1

1

2

3

4

5

7

 

1

2

3

4

6

 

1

2

3

5

 

1

2

4

 

1

3

®

2

1

2

3

4

5

8

 

1

2

3

4

7

 

1

2

3

6

 

1

2

5

®

1

4

 

3

1

2

3

4

6

7

 

1

2

3

5

6

 

1

2

4

5

®

1

3

4

 

2

3

 

 

1

2

3

4

6

8

 

1

2

3

5

7

 

1

2

4

6

 

1

3

5

 

2

4

 

 

1

2

3

4

7

8

 

1

2

3

6

7

 

1

2

5

6

 

1

4

5

 

3

4

 

 

1

2

3

5

6

7

 

1

2

4

5

6

®

1

3

4

5

 

2

3

4

 

 

 

 

 

1

2

3

5

6

8

 

1

2

4

5

7

 

1

3

4

6

 

2

3

5

 

 

 

 

 

1

2

3

5

7

8

 

1

2

4

6

7

 

1

3

5

6

 

2

4

5

 

 

 

 

 

1

2

3

6

7

8

 

1

2

5

6

7

 

1

4

5

6

 

3

4

5

 

 

 

 

 

1

2

4

5

6

7

 

1

3

4

5

6

 

2

3

4

5

 

 

 

 

 

 

 

 

 

1

2

4

5

6

8

®

1

3

4

5

7

 

2

3

4

6

 

 

 

 

 

 

 

 

 

1

2

4

5

7

8

 

1

3

4

6

7

 

2

3

5

6

 

 

 

 

 

 

 

 

 

1

2

4

6

7

8

 

1

3

5

6

7

 

2

4

5

6

 

 

 

 

 

 

 

 

 

1

2

5

6

7

8

 

1

4

5

6

7

 

3

4

5

6

 

 

 

 

 

 

 

 

 

1

3

4

5

6

7

 

2

3

4

5

6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

3

4

5

6

8

 

2

3

4

5

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

3

4

5

7

8

 

2

3

4

6

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

3

4

6

7

8

 

2

3

5

6

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

3

5

6

7

8

 

2

4

5

6

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1

4

5

6

7

8

 

3

4

5

6

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

3

4

5

6

7

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

3

4

5

6

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

3

4

5

7

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

3

4

6

7

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

3

5

6

7

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2

4

5

6

7

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

3

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7

8

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Presented 'AS IS' and for Entertainment Purposes Only.
Any gain or loss is your responsibility.
Use at your own risk.

Order is a Subset of Chaos
Knowledge is Beyond Belief
Wisdom is Not Censored
Douglas Paul Smallish
Jehocifer

    JADELottery's avatar - AMeOn 02.png
    Thread Starter
    The Quantum Master
    West Concord, MN
    United States
    Member #21
    December 7, 2001
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    Offline
    Posted: November 10, 2006, 1:54 pm - IP Logged

    Here's another fractal path shown in Table 2.

                                          Table 2
        Fractal Path is Y ® y(8,6) ® y(7,5) ® y(5,4) ® y(3,3) ® y(2,2) ® y(1,1)

    1

    2

    3

    4

    5

    6

     

    1

    2

    3

    4

    5

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    1

    2

    3

    4

    5

    7

     

    1

    2

    3

    4

    6

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    1

    2

    3

    4

    5

    8

     

    1

    2

    3

    4

    7

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    1

    2

    3

    4

    6

    7

     

    1

    2

    3

    5

    6

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    1

    2

    3

    4

    6

    8

     

    1

    2

    3

    5

    7

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    1

    2

    3

    4

    7

    8

     

    1

    2

    3

    6

    7

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    1

    2

    3

    5

    6

    7

     

    1

    2

    4

    5

    6

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    1

    2

    3

    5

    6

    8

     

    1

    2

    4

    5

    7

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    1

    2

    3

    5

    7

    8

     

    1

    2

    4

    6

    7

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    1

    2

    3

    6

    7

    8

     

    1

    2

    5

    6

    7

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    1

    2

    4

    5

    6

    7

     

    1

    3

    4

    5

    6

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    1

    2

    4

    5

    6

    8

    ®

    1

    3

    4

    5

    7

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    1

    2

    4

    5

    7

    8

     

    1

    3

    4

    6

    7

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    1

    2

    4

    6

    7

    8

     

    1

    3

    5

    6

    7

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    1

    2

    5

    6

    7

    8

     

    1

    4

    5

    6

    7

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    1

    3

    4

    5

    6

    7

     

    2

    3

    4

    5

    6

     

    1

    2

    3

    4

     

     

     

     

     

     

     

     

     

    1

    3

    4

    5

    6

    8

     

    2

    3

    4

    5

    7

     

    1

    2

    3

    5

     

     

     

     

     

     

     

     

     

    1

    3

    4

    5

    7

    8

     

    2

    3

    4

    6

    7

    ®

    1

    2

    4

    5

     

     

     

     

     

     

     

     

     

    1

    3

    4

    6

    7

    8

     

    2

    3

    5

    6

    7

     

    1

    3

    4

    5

     

     

     

     

     

     

     

     

     

    1

    3

    5

    6

    7

    8

     

    2

    4

    5

    6

    7

     

    2

    3

    4

    5

    ®

    1

    2

    3

    ®

    1

    2

    ®

    1

    1

    4

    5

    6

    7

    8

     

    3

    4

    5

    6

    7

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    2

    3

    4

    5

    6

    7

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    2

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    8

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

     

    2

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    2

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    3

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    Presented 'AS IS' and for Entertainment Purposes Only.
    Any gain or loss is your responsibility.
    Use at your own risk.

    Order is a Subset of Chaos
    Knowledge is Beyond Belief
    Wisdom is Not Censored
    Douglas Paul Smallish
    Jehocifer