Q1. I was fiddling around with squaring-the-square type algebraic maths, and came up with a family of arrangements of $n^2$ squares, with sides $1, 2, 3\ldots n^2$ ($n$ odd). Which seems like it would work with ANY odd $n$. It's so simple, surely it's well-known, but I haven't seen it in my (brief) web travels. I have a page with pics here of the $n=7,9,11$ versions, and description of how to construct them; $n=11$ is:
It seems the same method could square the plane, well, fill greater than any specified area, no matter how huge, at least. Which is what 'infinite' means, practically, isn't it? Anyway, it seems, if somehow not known (which it must be, surely - if anyone has links etc to where it's discussed I would be very grateful) then it's another way of squaring the plane.
Q2. Each of these arrangements can be extended, but the ones I've tried (5 or 6) have a little gap to the south-east, (i.e. where the squares don't fit neatly together) but otherwise can be extended forever. Is there an $n$ for which there is no gap? There are more than 1 of each sized square in this arrangement, but still, it would be a nice tessellation with integer squares. Here's a picture of the 7x7 version extended, and detail of the centre.
Thanks for any answers or help.
P.s. This was too long for the comments section:
I wrote to Jim Henle the other day asking about this method, he hadn't seen it, thought it was nice, but not plane-filling in the way his method is. He wrote in part:
You are sort of "squaring the plane," but not in the sense that we did it. You are squaring larger and larger areas of the plane, but you don't square the whole thing. ... There are many meanings to "infinite". Aristotle distinguished between the "potential infinite" (more and more, without bound) and the "actual infinite" (all the numbers, all at once). Your procedure is the first sort, and ours is the second sort.
Which is what I had thought.
But the more I think about it.. the less clear the difference seems. Well, e.g. 'there are an infinite number of primes' means: there is no highest one; any number you can say, there's a higher prime. That's how that is defined, spoken about, to my (non-mathematician's) understanding. Similarly, there are an infinite number of these $n\times n$ square groups, there's no biggest one, any area you name, there's a bigger one. The Henle's method consists in adding more squares, ideally forever, but practically, you stop at some point and say 'and so on forever'. The procedure requires an infinite number of steps. I can't quite see how this series of $n\times n$ squares is so different. You have to start drawing again with each new $n$, sure, but I can't see that matters so much - there are an infinite number of arrangements, and there's the same 'and so on forever'.. i.e. "there is, strictly speaking, no such thing as an infinite summation or process."
Nov 2015. [I can't add comments to questions below, not sure why.] Ross M, sorry about the delay! It seems you might be confused between the 2 parts, my fault for combining them in one question. (I still haven't heard anything much about either 2 from anyone.)
The first part is the basic $n^2$ arrangements of squares. (The second part takes just one of these and tries to extend it outwards, wonders about the possibility and mathematics of there being no gaps, and has many more than 1 of each sized square) I still don't quite see why 'having to rearrange each step' makes a huge difference to anything. Imagine I had a method of going from $n^2$ to $(n+1)^2$ squares by adding more around the edges. Then, according to what people seem to be saying, I 'could tile the whole thing'? The way I've done it has exactly the same area as that would be, just it has to be redrawn. I don't see how that affects whether it 'tiles the plane' or not, or anything else. If someone could explain that to me, I would be very grateful.