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# Combinatorial Fractalization

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The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3938 Posts
Offline
 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)}
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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 4 5 6 7 8

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Order is a Subset of Chaos
Knowledge is Beyond Belief
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Douglas Paul Smallish
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The Quantum Master
West Concord, MN
United States
Member #21
December 7, 2001
3938 Posts
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 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 4 5 6 7 8

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