X² = ?

[Mabubeezareel]

Back when I was a teenager many moons ago, I recognized an interesting pattern in squares, one that to this day I have never seen repeated elsewhere. Granted, I haven't looked that hard in the two decades since, and never went on to study any higher mathematics past high school level trig, either, but it's always bothered me on/off throughout the years on whether or not I came across something rather unique.

The pattern that I noticed, when spoken logically, is:

The square of a number is equal to the number less one squared, plus the number less one, plus the number.

Algebraically, that would equate to:

x^2 = (x-1)^2 + (x-1) + x

Factoring out the right side of the equation, you end up with:

x^2 = x^2

While I realize that all basically ends up boiling down to simple algebra and pattern recognition, I'm wondering if there's an actual established principle under which this would fall, or if there's any real relevance to it other than "oh, that's kind of neat, but not terribly useful".

[Aumaan_Kujata]

It's the geometric view, more or less. Picture the square number as a square gridded up into little 1x1 squares, then count how little squares you have to knock off the top and side to get the last square number. There's your extra "number and number less one".

[Blubbartron]

In general it's referred to as a sequence. More specifically a recursive sequence because you would need to know the value of the preceding place to calculate the next one. The formula would be written as:

a(n) = a(n-1) + (n - 1) + n, a(1) = 1

[TheDirtyHobo]

It's the geometric view, more or less. Picture the square number as a square gridded up into little 1x1 squares, then count how little squares you have to knock off the top and side to get the last square number. There's your extra "number and number less one".

A quick mspaint to help visualize.

[BerenTebogo]

It's a quirky thing geometric property of 2-D squares that is really easy to see and relate to because it is in 2-D. As Blubb and Hobo have pointed out, you can quickly generalize this to a square of sides "n" and have it still work (whether or not it is useful is neither here nor there).

The interesting thing, for me, that might be what got you interested in it as first as well, is that it is pretty easy to see and play around with for low numbers and is instructional in both an algebraic and geometric way for low numbers. That's why it probably stuck in your head, probably was a really easy way to think about doing some of the basic algebra and geometry work you were doing at the time.

As for any higher relevance or does it lead into other geometric or algebraic theory? I'm pretty sure no. It could be argued, and I suppose I will briefly just for your own piece of mind, that the underlying theory behind what you're looking at becomes the basis for adaptive mesh refinement methods in computational math and science. Granted, you don't AMR with equally-spaced points, but the basic idea is kinda nicely described in Hobo's pictures so yeah. I'll spoiler it so you people can read it if they want to, or ignore completely if they dont give a fuck.

Spoiler: show

The basis for almost all of computational science revolves around solving equations (differential equations) on computers. To do this task, the most fundamental concept is in taking continuous objects and actions and finding discrete representations of them (like 1, 2, 3, 4, 5 as individual points instead of a solid line from 1 to 5). Without going into any detail at all, the two main objects and actions that need to be discretized to solve these equations are the grid (x-axis, y-axis, etc, the line or plane or surface on which you are solving the problem) and the derivative operator (think of a function cos(n*cos^-1(x)) for n points x between x = 0 and x = 1, in equal steps of .1, so 0, .1, .2, .3, ... , 1 (n=10) would be your x-axis and you have a value of cos(n*cos^-1(x))at every point) which the derivative operator then acts on your function at every individual point.

At the lowest level, the way you do this is like I described above, equally-spaced points and since you are APPROXIMATING (key word) continuous objects as discrete, there is inherent error in your calculation. I have told you that there is a function cos(n*cos^-1(x)), but lets say you have cos(5*cos^-1(x)) (I'm picking this example because it was a HW problem for me like 4 years ago and I still have my plots and pdf on dropbox so it's easier to explain lmao), if you don't take enough points (higher n) it becomes impossible to distinguish from cos(25*cos^-1(x)), as you see below.


One way to get around the error is to take higher n, but that is more computationally expensive. It's essentially the same type of thing as upping your graphics quality, stuff gets smoother but you will hit a point where your PC cant keep up. Another way is AMR where you say okay I have a square grid and I'm solving a function over all of it, but maybe I don't care what happens in the lower left 8x8 set of squares, maybe shit just isn't interesting there. So you can break that 8x8 square up instead of small points like 1x1 as your smallest unit, say 4 blocks of 4x4. You get way less specific information about what is going on over there, but you don't care so who gives a shit. Maybe you REALLY need to know what is happening in the upper right 2x2 block. Like that is representing the place of highest stress on an airplane wing or something. So you want REALLY high detail, so you break it up into .001 x .001 blocks and get very accurate solutions over the area. And you do similar things for other areas of the square, so you have it blocked off not unlike Hobo's pictures above.

So how does this fit in with what you were thinking? Well at the end of the day, the theory of refining the structures is good and all, but you need an equation to write as a code before anything happens. So you look at your square and you break it up into 8 differently sized representative blocks each with their own internal block size, and write out an 8 term equation representing all of them, great, good job. But your boss comes by and says "I don't like 8, do 10" so you do it all again and do 10, but then you are on a conference call with your Chinese partners and 10 means like death or something in chinese so they want lucky numbers that look mean cat in chinese or some shit so they want 13 blocks. (This is a highly exaggerated example of shit that does actually happen). The moral is you need to go to something like what Blubb wrote, and what you picked up on initially - generalization. You could find an expression involving x-2 and x-1 instead of x-1 and x, you could find one involving x-3 and x, etc etc to your heart's desire.

[Mabubeezareel]

Thanks very much for the info/insights. I never thought to think of laying it out on a grid like that, but it makes so much sense now to visualize it in that manner.

The interesting thing, for me, that might be what got you interested in it as first as well, is that it is pretty easy to see and play around with for low numbers and is instructional in both an algebraic and geometric way for low numbers. That's why it probably stuck in your head, probably was a really easy way to think about doing some of the basic algebra and geometry work you were doing at the time.

This is pretty much exactly what was happening for me at that time - sitting in the back of the class, bored as all fuck and playing around with low numbers and noticed the sequence. From there, I proved it out with higher numbers, and then used that to back into the actual equation. Simplistic for sure, but that light-bulb moment was a fantastic feeling. I also remember following a similar pattern out for higher exponents, but really don't recall what they were now - makes me wish I had kept all of my old notebooks!

[Cadsuane]

Indeed, the interested reader may verify that the function which maps a natural number to its square is computable. By Church's thesis, this function must then be recursive. Your work merely verifies this special case.

...

A mathematician would find this post hilarious, take my word for it.

[BerenTebogo]

Indeed, the interested reader may verify that the function which maps a natural number to its square is computable. By Church's thesis, this function must then be recursive. Your work merely verifies this special case.

...

A mathematician would find this post hilarious, take my word for it.

I hate you so much and yet love you deeply.

[Cadsuane]

I've been reading "Mathematics Made Difficult" and it's been rubbing off on me.