0

It is well-known, that it's impossible to square a circle. There are good papers on the matter and good videos, like from Mathologer or Numberphile

So, if rules of the game are that we have a

  • Ruler (NOT a measured one) for straight lines
  • Compass for circles

it's proven to be an impossibility. I'm not a professional mathematician but I'm able to understand the material (I think) well.

My question is:

What is the "minimal" change in the rules of the game so that we can square a circle?

I put "minimal" into quotation marks because, while this is bugging me, I fail to come up with a proper definition of "minimal". I.e. I have no clear way of comparing one change of rules of the game with another to clearly answer which one does "fewer changes" to the rules of the game. But I still am curious about what could be possible answers to the question bearing in mind this "handwave-y" definition.

EDIT: (Bonus) if the construction process with the changed rules is the one we can reproduce in real life. For example, adding a new tool to the tool set and/or allowing other operations with tools.

Alma Do
  • 499
  • 2
    Consider the field of numbers that can be generated by a straight-edge and compass, and a unit interval. What you are asking for is a simple extension to this field that adds $\pi$ to the field, since that is what is needed for squaring the circle to be possible. The issue is, since $\pi$ is transcendental, there likely is no clean way to create such an extension, without doing something like drawing a line segment of length $\pi$ for use in the construction. I can't think of anything pretty to answer this. – Rushabh Mehta Aug 28 '19 at 16:30
  • @DonThousand The exact thing which led to this question is this statement about impossibilities by Burkard Polster. He explicitly said that "if you change the rules of our game just a tiny, tiny little bit" it will be possible. So I wonder what would that be. – Alma Do Aug 28 '19 at 16:42

2 Answers2

1

As I noted in my comment, there are no really pretty ways to make squaring the circle possible. However, the following is as close as I imagine one can get to a loosening of rules to allow squaring the circle.

The new rule is to allow for infinite sums. For each $n\in\mathbb N$, we construct a segment of length $n^2$, and then one of length $\frac1{n^2}$ via this construction. Then, we line up all of these segments to construct a new segment of length $x$. Finally, we construct a segment with length $6x$, and then construct a segment of length $\sqrt[4]{6x}$ by repeating this construction. Finally, construct $\frac1{\sqrt[4]{6x}}$ by applying this construction.

Note that $\frac1{\sqrt[4]{6x}}=\frac1{\sqrt\pi}$, so a circle with radius $\frac1{\sqrt[4]{6x}}$ has area $1$, the same as that of the unit square.

Rushabh Mehta
  • 13,663
  • It sounds mathematically plausible, but well.. we will never be able to do this thing in reality – Alma Do Aug 28 '19 at 16:58
  • @AlmaDo Isn't that the point? You can't square a circle. – Rushabh Mehta Aug 28 '19 at 16:58
  • Well, ancient Greeks had a way of "doing it" in reality for some stuff and just couldn't do this with a ruler and a compass. But maybe if we extend our tool set a bit or allow some other operations with the tools - then we will be able to complete the constructing of the desired square in reality. Don't get me wrong, your answer is a valid one, just not the one we can do in reality. – Alma Do Aug 28 '19 at 17:00
  • @AlmaDo Can you source the claim that Greeks had a way of doing this? – Rushabh Mehta Aug 28 '19 at 18:05
  • They didn't. They could do "some stuff" like I wrote above (like pentagons or angle bisectors) but they couldn't do it for squaring a circle - because.. well, they only had ruler and compass. – Alma Do Aug 28 '19 at 18:06
  • What does the following sentence mean -- Well, ancient Greeks had a way of "doing it" in reality for some stuff and just couldn't do this with a ruler and a compass. – Rushabh Mehta Aug 28 '19 at 18:07
  • "It" meant "some stuff" (those objects which they could construct). I probably should've chosen better phrasing – Alma Do Aug 28 '19 at 18:09
  • So why do you expect that there is a solution that can be done in reality? – Rushabh Mehta Aug 28 '19 at 18:12
  • Because we can change the rules? Add other tools (marked ruler? Marked circles) ? Allow some other operations? – Alma Do Aug 28 '19 at 18:13
  • @AlmaDo But, what kind of operations do you even consider "doable in real life"? Am I allowed to have a ruler with interval markings of length $\pi$? – Rushabh Mehta Aug 28 '19 at 18:19
  • I can't tell which are "doable" for sure, but I can probably tell which aren't. Surely it's Impossible to finish an infinite number of steps – Alma Do Aug 28 '19 at 18:20
  • @AlmaDo It's pretty obvious to me that this isn't suited for this site, given the lack of criteria presented. I will have to vote to close. – Rushabh Mehta Aug 28 '19 at 18:21
  • That is your right. Sad it is to see that a wish for insights and curiosity isn't welcomed here – Alma Do Aug 28 '19 at 18:22
  • @AlmaDo I hope you are not serious. I'd suggest we take this to chat, but the reason for me voting to close is most certainly not your wish for insights. I wouldn't have answered otherwise. – Rushabh Mehta Aug 28 '19 at 18:23
  • I doubt that the ancient Greeks used only compass & straight-edge when making buildings & boats & bridges. But some of them were in love with the "ideal" figures (circles & lines), perhaps because every part of one looks like every other part. BTW Archimedes invented an incredibly simple device (his "linkage") for trisecting angles. – DanielWainfleet Aug 29 '19 at 05:13
  • 1
    @DanielWainfleet Oh I definitely agree. I just think that this question is too broad. – Rushabh Mehta Aug 29 '19 at 12:18
1

This is a comment.

It has been shown that the unmarked straight-edge is redundant. Starting with 2 points in a plane, any point that can be constructed with drawing compass & straight-edge can be found with the compass alone.

Also if we are given a single circle and its center and one other point, we can drop the compass and use only the straight-edge and still construct every point that can be reached by edge+compass. Note that we do NOT assume all points on that given circle are constructed. The only constructible points on it are the intersections of it with lines through pairs of (previously) constructed points.

I am sorry I cannot recall references for these. I believe they are from the 1700's.

BTW in an earlier era, what we call a compass was called "compasses".

DanielWainfleet
  • 57,985