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Replies: 2   Last Post: Jul 13, 2014 11:20 PM

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 Einstein Posts: 59 Registered: 11/22/07
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
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/4cuber-inc.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.

Date Subject Author
7/6/14 Einstein
7/8/14 Einstein
7/13/14 IV