# Squaring the plane, with consecutive integer squares. And a related arrangement

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.

• Quite interesting. But I don't think this belongs on this web site. The AG tag is certainly wrong. Combinatorial geometry maybe. I think you could try asking this on MSE. For what it's worth, I have a paper with some similar pictures: math.uvic.ca/faculty/aquas/papers/paper43.pdf May 14, 2014 at 3:43
• Your pictures show that you can square an arbitrarily large area. For the infinite version of squaring the whole plane, you'd have to produce a single arrangement that covers all of it, where each positive integer side length appears exactly once.
– Matt F.
May 14, 2014 at 3:54
• Thanks Anthony, I will try there. I'm not sure why it doesn't belong here, but I don't know the guidelines etc. I hope to understand your paper on rectangles one day! Looks very interesting. May 14, 2014 at 3:58
• You may find this guy's paper of interest: math.smith.edu/~jhenle/stp It was also written up in the January 2008 issue of the American Mathematical Monthly.
– user940
May 14, 2014 at 17:29
• I don't see how having a single arrangement or not makes any difference. If the Henles' method needed redrawing with each step - a different arrangement, but equal squares and area as currently - it would work the same, wouldn't it? In both cases, the infinite area is only reached as a 'limit' of a series of steps each adding finite area. (Sure, the last in the series of nxn arrangements is hard to imagine :-) (harder than a spiralish pattern extended forever like the Henles' method) .. so is anything extended to infinity. ) May 15, 2014 at 6:14

There is a big difference between "arbitrarily large" and "infinite". This example shows the difference clearly:

For any positive integer $n$, there exists a strictly decreasing sequence of positive integers of length $n$.

True, obviously. But this is false:

There exists a strictly decreasing sequence of positive integers of infinite length.

If you like, you can call these "potential infinite" and "actual infinite".

It looks like you can tile an arbitrarily large section of the plane, but not the whole thing. As you say, it looks like you cannot extend it toward the southeast. If I challenge you to cover a $1,000,000 \times 1,000,000$ square you can do it, but you need to plan ahead by putting the correct number of small squares starting with $1$ on the first diagonal. You couldn't take a pattern that started out covering $1000 \times 1000$ and extend it to cover $1,000,000 \times 1,000,000$. This doesn't say there is anything wrong with what you have done-it is quite neat. You just need to be careful what you claim to be able to cover.

(OP here)

Update: Well, I answered Q2 "Yes" after a couple of years, and found the rules governing when an arrangement "has no gaps" or not. It seems every arrangement actually tiles the plane in 2 layers, but only for some the layers of squares coincide. I wrote a paper a while back about exactly when that is, and lots more about them. This page has some more pics of the plane tilings, plus a program to draw the $1\dots n^2$-type square arrangements of Q1, which were christened Ponting square packings by Ed Pegg, who wrote a Mathworld demonstration about them. It seems they were new.