
Advanced Combinations
Posted:
Jul 6, 2014 8:20 AM


The idea of Advanced Combinations is simple, where I have a laymans description and the math definition.
The laymans argument is one of cubes (though the pending patent allows for any method, this is the easier challenge). In a given dimension space a certain number of single cubes, or of larger shapes made up of cubes joined together, can fit, as well as represented empty spaces.
This creates a condition where it would seem possible, in a new manner, to encode more data via the act of creating Combinations, than the binary would normally allow. Note I said seem, were we have not yet advanced far enough to test if this is true.
 definition: legal object 
Let S be the set of solid unit cubes in R^3 with all vertices at lattice points.
A legal object is a subset X of R^3 such that
(1) X is a nonempty, finite union of elements of S.
(2) The interior of X is connected.
Congruent objects are considered the same.
_______________________
Empty spaces are allowed, which will increase the total possible outcomes. _______________________
Now we were able to do a variety of lower level efforts, solving the exact count for a 2x2x2 container, and maximum shapes, a 3x3x3 container for maximum shapes, and a 3x3x4 container for maximum shapes. The exact results are below
A 2x2x2 container has 9472 total outcomes from 14 possible shapes. A 3x3x3 container has 1,585,580 total shapes with an estimated quadrillion plus possible outcomes. A 3x3x4 container has 1,828,003,418 possible shapes, and defies an estimation of possible outcomes (most certainly exceeds the quadrillion by a factor or two)
2x2x2 = 9472 total combinations For a 2x2x2 container, the number of distinct legal objects is 14.
Exact Shapes count for a 3x3x3 array Units Shapes 0: 0 1: 1 2: 1 3: 2 4: 7 5: 25 6: 111 7: 485 8: 1,844 9: 6,134 10: 17,322 11: 41,998 12: 86,803 13: 152,959 14: 226,410 15: 277,767 16: 277,390 17: 223,802 18: 145,803 19: 77,251 20: 33,413 21: 11,772 22: 3,356 23: 769 24: 138 25: 22 26: 4 27: 1
Exact Shapes Count for a 4x3x3 Array. #Bits #Shapes 1: 1 2: 1 3: 2 4: 10 5: 34 6: 181 7: 959 8: 5,383 9: 27,582 10: 124,741 11: 486,690 12: 1,655,982 13: 4,934,805 14: 12,969,641 15: 30,109,904 16: 61,740,472 17: 111,353,404 18: 175,542,991 19: 239,819,535 20: 281,599,067 21: 282,326,036 22: 241,011,786 23: 175,295,801 24: 109,002,113 25: 58,119,504 26: 26,620,643 27: 10,455,818 28: 3,509,584 29: 997,606 30: 238,069 31: 46,638 32: 7,435 33: 901 34: 92 35: 6 36: 1 Total: 1,828,003,418
So the question is, can anyone here make better software for this? Or perhaps someone here has access to a cluster and can run the computations faster than our systems?
Here is links to the existing software
http://pat7.com/js/3shapes/1cellcount.c http://pat7.com/js/3shapes/4cuber.c http://pat7.com/js/3shapes/4cuberinc.c http://pat7.com/js/3shapes/3cuber.py http://pat7.com/js/3shapes/rotate24x8.py
It is my observation that the combinations rapidly spiral out of control, where 100x100x100 would be computationally impossible without a reasonable system to shortcut from checking all possibilities and instead build a full binary tree from the results.
One issue we are having is poor software and poor computational capabilities. I would like to see all possible outcomes of a 3x3x3 array enumerated, but our efforts cannot do this. We also are unable to count the shapes in a 3x4x4 sized array. I suspect that a 5x5x5 array could make feasible a Pending Patent of mine using this math theory.

