Some Many Ways Of Flatness

[ We didn’t break 3,100 yet, and too bad that. But over the day I did get my first readers from Turkey and the second from the United Arab Emirates that I’ve noticed. Also while my many posts about trapezoids are drawing search engine results, “frazz sequins” comes up a lot. ]

I think I’ve managed, more or less, acceptance that a piecewise constant interpolation makes the simplest way to estimate the population of Charlotte, North Carolina, when all I had to work with was the population data from the 1970 and the 1980 censuses. In 1970 the city had 840,347 people; in 1980 it had 971,391, and therefore the easiest guess to the population in 1975 would be the 1970 value, of 840,347. We suppose that on the 1st of April, 1970 — that Census Day — the population was the lower value, and then sometime before the 1st of April, 1980, it leapt up at once by the 131,044-person difference. Only … how do I know the population jumped up sometime after 1975?

Here’s the thing about my piecewise constant interpolation. I want it to match the actual data I have of 840,347 people on the 1st of April, 1970. And I want it to match the actual data I have of 971,391 people on the 1st of April, 1980. I’m supposing that somewhere in-between the population leapt up by 131,044 people. After the last essay probably the easiest supposition was that the population leapt up so on the last day before the 1980 census, or maybe the last day of 1979. That’s my first thought, anyway.

But why can’t I have the population jump up the 2nd of April, 1970? I’d still have this piecewise-constant approximation, and I would still hit exactly correctly the two pieces of data I have. Why can’t I take the whole population increase all at once and enjoy it for the whole of the 1970s?

If my only considerations are that I want to exactly match the data points I have, and I want to have a piecewise constant interpolation, and I want to take that whole increase all at once, there’s not really any reason I can’t draw my interpolation that way. It may go against instinct — it goes against my instinct, anyway, and until I hear otherwise I’ll suppose you feel the same — but there’s nothing wrong with doing so.

So there’s one of the neat little complications of my nice simplest-possible-interpolation scheme. There’s actually two schemes that fit my known data of these two points. Imagine what I could do if I had there data points. (Well, I could come up with four piecewise-constant interpolations, as I make it out, although I’d say only two of them would ever actually be used. One of them is too dull to use. Another is a little ad hoc. If you’d like to spend a little time doodling, see if I’m right or if I’ve overlooked some.)

Although, really, why should I limit myself to two interpolations? If the population can leap upwards on the 2nd of April 1970, or on the 31stof March, 1980, why couldn’t it leap up on the 1st of January, 1975? Or the 31st of December, 1975? Why not the 18th of September, 1978? Or the 29th of February, 1972? For that matter, why not make the leap at 12:37 pm, the 11th of July, 1979? 12:37 and 14 seconds that same day? 12:37 and 14 and one-quarter seconds?

In my interpolation for Charlotte’s population for 1975 — at least, if I pick out one moment in 1975 — these many variations aren’t going to matter. It’ll be either the low number or the high number. But they’re unmistakably different interpolations: they don’t agree on the projected population for every moment throughout the decade.

I find it wonderful we can come up with, literally, infinitely many different interpolations from just these two data points and the simplest possible function connecting the two.

We can get more complicated yet.