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 $1,2\ldots n^2$ (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.

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.

  • $\begingroup$ This AMM article might interest you. $\endgroup$
    – JRN
    May 14, 2014 at 4:24
  • $\begingroup$ Thanks, yes I saw the Henle's paper. 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. $\endgroup$ May 14, 2014 at 8:36
  • $\begingroup$ 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." $\endgroup$ May 14, 2014 at 8:37
  • $\begingroup$ 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 - 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 nxn 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'. $\endgroup$ May 14, 2014 at 8:37
  • $\begingroup$ I can't quite see how this series of nxn 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, and there's the same 'and so on forever'.. $\endgroup$ May 14, 2014 at 8:38


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