A Method of Statistical Inference on Two-dimensional Object Ratios
Hello all:
I stumbled on this site by chance. I was searching for “ROM” “random oracle model”, and had not cleared the search bar of a previous search, … patterns of squares
A link appeared with … Squares….Postby Rom … patterns of squares … Out of curiosity, I clicked on it and it brought me to your site.
The question was simple and the answer to the point. The post question was HOW MANY .745" SQUARES CAN BE CUT FROM 16"X14.25. Here is what brings me to post this.
I recently compiled a program based on several algorithms that allows for a method of statistical inference. The values in the post were ones I had not tested. There are infinite combinations of testable values. I used the posted values from the question in my program, and a solution was generated. This is a rare event as most are null except for trivial answers.
An outline with an explanation and results are below.
The algorithms and the programs that use them perform an Expansive Least Squares Multi-regression Vector analysis on two-dimensional object ratios. With this method, a statistical inference is applied to various two-dimensional geometries of equal area, by expanding them into three-dimensional spheroids. The results generate a hypothetical maximum with a residual for each of the iterations.
By subtracting the square of the inverse of the harmonic average of the residuals, from the least squares of the hypothetical maximums for each vector, the points of intersection recede to infinity. This effectively collapses the three-dimensional spheroids back to two-dimensional planes, yet leaves a residual. The residual combines every vector at every point. (Basically, the infinities of: Aleph sub2 with Aleph sub1).
Though both the Alephs are infinities, they are not the same size. As the intersections are subtracted to the predefined subset of square area, the numbers that remain are sometimes greater than the theoretical maximum. This number is referred to as an s-value. The reason for this is the intersections defining the square area are not all square. The shaped areas are all polygons (n-gons) with finite (n) values. Some (n) values are extremely large.
Though intuitively, it would appear that the s-values have to be smaller (no matter their shape) because more of them are occupying the same space. It is conjectured that the s-values retains the space of the z-dimension in a complex form that is not understood.
One theory hypothesizes the extra space exists out of phase in two-dimensional space, effectively allowing simultaneous occupancy. Another theory hypothesizes that time does not exist in two-dimensional space as it does in three-dimensional space, effectively dismissing simultaneity.
On the more tangible side: Some argue that a curve remains after collapse, implying the plane is three dimensional, so it is not a plane. Others argue, the surface is the same square area and is a two-dimensional plane, and the shapes that occupy it are all polygons, without curves, circles, or ellipsoids that would be expected in three dimensions.
Two algorithms perform the functions conveyed above. They are too complex to describe here. However, the independent theories of several mathematicians, including Befure DeCarts, Theo Horas, and Peter Guthrie Tait give a primer overview. These theories, recombinant with portions of the “Cycle Double Cover Conjecture” and, to a lesser extent, the “5-Flow Conjecture” are the basis for the algorithms.
The recombinant theories are currently considered conjecture, and the generated outputs are regarded as mathematical artifacts. The primary reason for dismissal of the conjecture is an infinity is divided by an infinity. Though the infinity sizes are different, the results are believed to be an undefined value.
What is observed is very few ratios (<.05%) generate theoretical solutions greater than the natural maximum (in the allotted polynomial time). A greater value has not yet appeared for any ratio, if the natural maximum value is a perfect fit. The smaller the natural residual the higher the probability of greater (n) values existing, and the higher probability of more than one value. (Both seem counterintuitive).
Other observations: When (n) is small more of its space is occupying the complex component of two-dimensional space. When (S) is large, more of its space is occupying the complex component of two-dimensional space, in a significantly different manner than is suggested by (n). The lengths of the (n) component in the (n-gons) are not necessarily the same in its own unit, nor from unit to unit. Basic analysis of these phenomena will require a supercomputer (or possibly a quantum computer).
Both Programs use similar functions based on theories described above. The second program uses an additional algorithm, Dimensional Quotient Demodulation (DQD) adding to its operational time. Both algorithms run in quasi-polynomial time: one Approximating 2^poly (log n), the other Approximating N^log (log n^2).
The complied program forms of the algorithms were executed on two separate, and virtually identical, distributed systems, comprised of five Lenovo 4354D3U with four NVIDIA GPUs operating as ALUs. One of the five Lenovo 4354D3U, in each group, controlled the distribution. Program operational speed is consistent with quasi-polynomial time described above, but may intentionally slow if NVIDIA GPUs temperatures exceed predetermined specifications.
The program notes and records the standard theoretical maximum then begins the expansion and contraction subroutines. For these machines: Program 1 terminates after 1440 minutes. Program 2 terminates after 2880 minutes.
The following data were generated via the two algorithms for input ratios ((16*14.25) /0.745)
As in the natural form, the integer portion of (S) is the expanded number of value spaces that may fit in a defined area.
(.745" SQUARES CAN BE CUT FROM 16"X14.25”)
Natural result: N=4: S= 410.792.
Expanded Results.
Program 1: N=89; S= 453.647 --- Time: 177 minutes
Program 2: N=9641 S= 1519.254 --- Time: 963 minutes
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You have a great forum, here. I am sure I will see some of you in the academic world of math.
Thank you
C. L. Dodgson
+6251 
