This question is regarding two algorithms for squaring/almost squaring the plane.

  1. the Henles' method of squaring the plane. pdf here

  2. my method of tiling $n^2$ squares. I worked out* a simple gap-free arrangement of squares length $1..n^2$, with $n$ odd. And realized that with $n=1,3,5,7,9\dots$ I could similarly tile any desired number $n^2$ of tiles whatever. And taking the limit as $n\to\infty$, going on forever, 'sort of' square the plane.

I mentioned this as an update in a previous question on here - there's a picture of the $n=11$ arrangement on that page.

But I'm more confused the more I think about it. I'm no mathematician so any simple explanation would be most appreciated. (I read Rudy Rucker's book on infinity once, that's the extent of my experience in such matters.)

As I quoted on that page, Jim Henle wrote to me that "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 made sense. No $n\times n$ arrangement actually has infinite area.

But then, any infinite diverging series is finite at any particular point you stop adding the terms (the partial sums); it relies on saying "repeat that step forever", and we humans pretend that that's possible. Or possible for a machine. But it's not. Any time you stop the process and look, you will have a finite sum. In the same way as method 2, the total is never infinite until you have (done the impossible and) repeated the step an infinite number of times. A block of steps in the Henle method have to be repeated an infinite number of times too. It only tiles the plane after an infinite number of repeats, at the limit. You add 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 the series of $n\times n$ squares is so different.

Henle also pointed out that his method doesn't redraw squares. It starts drawing at the origin and adds squares around those, whereas my method redraws the whole thing each time. I can't see why that makes a qualitative difference. Imagine a third method, 'just like' mine, with the same constituent squares and total area, but not needing redrawing at each step. Could that be said to tile the plane, just because it doesn't need redrawing? I can't see that it would be so different. Just the method of drawing each stage would be different.

Also, 'infinite', whether infinitely large or infinitely small, I've seen defined with the $\epsilon$ method - for any given huge/tiny value given, the infinite amount is smaller/larger. Method 2 has that 'infiniteness', if not 'actual' in the sense Henle said. Most times I come across the infinitely large or small, it's defined in the $\epsilon$ sense. And any infinite series is 'not infinite' in the same way as method 2 isn't finite - it 'never actually gets there' until an infinite number of steps. And "there is, strictly speaking, no such thing as an infinite summation or process."

I hope it's somewhat apparent what I'm confused about! Method 2 'only tiles an arbitrarily large area', but the more I think about it, so does any sum adding to infinity, like the Henle method, any divergent series etc. Sorry I don't have the mathematical lingo. At the same time, I think these issues are more..interesting than people who've commented so far realize. Ah, I realize I haven't literally asked a question. I hope it's evident. "Can you talk about these things, help me understand this, or point me to where I can read about it?' etc. Thanks!

*well, I wrote a program that came up with it.

Response to Joel's 2 comments (as I can't comment on here yet): Thank you, yes I read them. I responded to them in my answer back in 2014, as I couldn't comment. No-one responded, in the 2 years that followed. One of them totally confused the two different things I was talking about. I guess I should have asked 2 separate questions. The other comment I understand, but it didn't help me. Like I don't know how your (similar) second comment is supposed to help me. I get that. Did you read my question and think I don't understand that? ((tries to think how to explain without repeating entire question again)) In a nutshell, I can't see how the Henle method escapes that objection as well. It's adding finite things on and claiming an infinite result. (This question could be about any similar 2 series; this 'squaring the plane' thing is merely a convenient example) I thought I would try asking a more focused question, just on the infinite series aspect of that question (which was many questions in one.)

Response to Joel's 3rd comment (I still can't comment) : Maybe so, but using my method, you are guaranteed to use all the squares $1..n^2$ once each.

  • 1
    $\begingroup$ Have you read the two answers to your previous question? If you have trouble understanding them, you should comment on the answers so that the ones who posted them could explain. $\endgroup$
    – JRN
    Jul 8, 2016 at 3:43
  • $\begingroup$ I could attempt to cover the real number line by drawing a line from the origin and going towards the positive direction. I would be covering longer and longer distances, but I would never cover the whole number line. $\endgroup$
    – JRN
    Jul 8, 2016 at 3:45
  • $\begingroup$ Adding to the left and right of an existing structure, such as {0}, {0,1}, {-1,0,1}, {-2,-1,0,1}, {-2,-1,0,1,2}, etc., will cover all the integers sooner or later. But "redrawing" everything at each step, such as {0}, {1,2}, {0,1,2}, {1,2,3,4}, {0,1,2,3,4}, etc., is not guaranteed to cover all the integers. $\endgroup$
    – JRN
    Jul 12, 2016 at 3:44


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