JCSBimp's Hexagonal Orthogonal

Early Tuesday morning, I popped awake, but no mathematical insight was waiting to compel me to stay awake. However, i did walk quietly downstairs to work on my mathematical study, as I have been doing every early morning since the insights started coming – about a week and change now, I think.

I took a two-hour nap later in the day, but that did not seem to be enough, and I was sleepy at a fairly early bedtime hour. I checked in with my wife who is out of state, texted her messages full of love, and went to bed.

Sometime between 11 p.m. Tuesday night and midnight, the dogs woke me up for their DentaStix treat, and, like magic, there came the insights, jumping into my head as soon as I got the dogs back upstairs and in bed with me. I thought about them, let one drop from my mind so that I needed to work to pick it back up again later, and then decided I had them sorted out enough to work on them at whatever time my brain, my mind, Azarya’s Angels, or The Queen of the Sciences instructed me to do so.

Then my cell phone dinged with a Messenger notification!

Like Mee-Maw in “The Heart, She Holler” once cackled, This Was Starting To Get Interesting! (No, don’t look up the scene. It’s unjustifiably gruesome.)

The insight began simply as a multi-category sort, according to mostly simple criteria, of the positive integers:

• Even or Odd
• Prime or Composite
• Perfect Square or Not
• Expressible as the Difference Between the Ceiling Square and a Smaller Perfect Square (and thus, if composite, having an easily obtained non-trivial factorization)

The Messenger ding, by the way, was a friend from my college days who shares his insights and art on Facebook. I love his art, but hate some of his political insights. Such is our fraught friendship.

Once I had responded adequately to my friend’s message to me, I came downstairs, looked at and commented on both his most recent political post and his art postings of the past few months. I commented on what I saw in both categories.

Then I produced the following triangular arrangement of integers, colored according to the categories above, as explained in the text box of the screenshot below.

If arranging the non-negative integers (I included zero for completeness) and coloring them this way does not help me and my brain/Azarya’s Angels/The Queen of the Sciences to produce more amazing insights, it will not be my fault for not trying!

See y’all again after dawn’s early light. Here’s hoping it’s A GOOD BIT AFTER. 😀