I am laughing at the sheer trickiness, almost by design, of the mathematical relationships I am uncovering in my number study, and the most recent one in particular.
I shared yesterday the VERY SMOOT GRAPH, suggesting to me a parabola, charting out the plot of odd integer m vs. the difference between (m+1)/2 and its ceiling square. Soon after I posted that, I picked three or four equally spaced points on that graph, down on the low end, verified that they did indeed follow a quadratic functional relationship, and then, this morning, came up with coefficients for the polynomial I thought the smooth curve satisfied.
Good news and bad news, which adds up to funny news! The four points I picked had values that EXACTLY satisfied the polynomial whose coefficients I calculated. But the vast majority of the rest of the points plotted are JUST A LITTLE OFF. In fact, if one zooms in to look really closely at how the points on the curve fall, one sees the zig-zaggy pattern that suggests what I’m seeing: different quadratics depending on different conditions, which I’ve yet to figure out.
Hey, this is all part of the fun. Reality is more strange than I thought! I need a better theory!