JCSBimp's Hexagonal Orthogonal

Or: The Asymmetric Behavior of the Ceiling Square Remainder

My Inspiration Process has been more active, more related to working on thought processes, than it has been something spontaneously occurring, which is especially noticeable during my wife’s current extended absence from our residence (to help care for grandchildren), compared to how the Process seemed to behave during her previous similar extended absence. Rather than waking me up with an immediate desire to think about some aspect of odd integers, their ceiling squares, or some way to visualize them, this morning’s inspiration process kicked in only after some thought. However, much of that thought took place while I was lying awake in bed with the dogs, and as with the Inspiration Process kicking in during the past months, this one did not let me sleep in. I like that. It helps me keep feeling that I’m possibly on the track of something worth while, and that Inner Fire is still somehow at work, albeit in a strangely less direct fashion.

To the mathematical business at hand: My notion this morning was to look at the arrangement of odd integers on the grid/pegboard corresponding to the positive integer points in quadrant I of the function in x and y given by f(x,y) = (x+y)²-x-3y+1. This is a function I described a few days ago in an earlier blog entry: The “pegboard” of integer points with x ranging from 1 to infinity and y ranging from 1 to infinity corresponds to a grid I produced of (theoretically) all the positive odd integers positioned in sequence along the lower-left-to-upper-right diagonals of that grid. The values of integers along any single row or column are consequentially values of a quadratic polynomial for positive x or y, accordingly.

I had noticed that the odd numbers which differed from perfect squares by a value itself equal to a perfect square arranged themselves in upper-left-to-lower-right diagonals, parallel to the main diagonal with the odd perfect squares, but only in the area of the grid below the main diagonal. I did some calculation and visualization on the fly, and it took me some minutes to realize that yes, the values above the main diagonal where their differences with their Ceiling Squares (the least perfect squares greater than their values) were themselves perfect squares also occurred in straight rows, but not parallel to the main diagonal.

Though that made sense to me, characterizing it was a challenge for a few minutes. (Notice that I had to enter some values into the chart that were not perfect squares, before I got the hang/understood the structure of it.) Then I realized:

The Linear Behavior Above the Main Diagonal Changes Because the Ceiling Squares for the Numbers Above the Main Diagonal Are Not the Same as Those for the Points Below It.

Were one to extend the chart, and therefore the “pegboard” of values and the x, y graph, outward, one would see that every lower-left-to-upper-right diagonal would contain all of the odd integers, and therefore the perfect squares would appear above and below the main diagonal on which they appear in this chart. The next such line above the main diagonal actually corresponds to the Ceiling Squares of the odd integers above that diagonal. That line has the slope to which the upper-portion lines are parallel.

This has an influence on how the bold red numbers in a previous chart I shared – and have now finished coloring for f(x,y) with x and y from 1 to 32 – appear to favor rows and diagonals, but chiefly in the area of that chart below the main diagonal.

The bold red numbers, as previously explained, correspond to discrete semiprimes which differ from their ceiling squares by perfect square values. They appear in the area below the main diagonal on the straight lines which have almost no prime numbers on them which correspond to s²-1, s²-4, s²-9, and so on. They and the other odd numbers along those diagonals factor trivially by Fermat’s factorization method, i.e, taking advantage of the fact that s²-t²=(s+t)(s-t). The corresponding rows above the main diagonal have a different slope and are more difficult visually for someone to perceive. Here it also at least appears that such values are more rare in the Quadrant I xy points on that side of the main diagonal. This is perhaps worth further study.

And all of this math leads, finally, to a somewhat personal anecdote: I filled out the chart’s entire square of values with x and y both ranging from 1 to 32. Since the Quadrant I arrangement of all the odd integers builds up in a triangular fashion, diagonal line by diagonal line, any square chart holding triangularly-arranged numbers will have gaps in the consecutive numbers that appear. Of personal interest to me is the gap starting with 1957 – my birth year – and 2007 – the year Wendy and I married. It is like the only part of my life range this arrangement of numbers chooses to show is the current phase of of my personal life, maritally speaking: a rather clever coincidental reminder for me to live in the present.)