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.
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.
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.