Embarrassing Question about "Squaring the Circle"

Question

I was reading about this famous problem called "Squaring the Circle" (https://en.wikipedia.org/wiki/Squaring_the_circle).

The goal of this problem is to make a square and a circle with the same surface area. However, it was shown that this is not possible - in particular, it is impossible to make a square with an area of "pi".

I am having a bit of difficulty understanding why it is not possible to make a square with an area of "pi" - for instance, a square of area "pi" would have each side measuring "square root of pi". If approximations are "good enough", it shouldn't be too hard to do this?

 ggplot() + 
  geom_rect(aes(xmin = 1, 
                xmax = sqrt(pi), 
                ymin = 1, 
                ymax = sqrt(pi))) + 
  coord_qual()

I don't think I am correctly understanding this problem - for instance, I can understand that it is impossible to draw a square with each side measuring "square root of pi", seeing as "pi" is an unending and irrational number. For instance, I could not cut a piece of wood to exactly measure "pi" meters: I could curt a piece of wood to measure 3.14159 meters, because eventually at some decimal position, no cut of wood would exactly measure "pi" meters.

But wouldn't this concept apply to almost any problem in which measurements are unending decimal numbers? For instance, if I want to build a bridge to sustain some amount of load - I might do some engineering calculations, and it might tell me that that the bridge needs to 42.1256789900031323... meters long. But of course, I would make a cut at 42.125 meters and carry on with the construction.

Thus: Is the "Squaring the Circle" problem impossible because it is impossible to draw a square having an unending length (i.e. I would spend eternity making sure that the measurement of each side was faithful to "pi" over infinite decimal places) - or is there some other reason that the "Squaring the Circle" problem is impossible?

Thanks!

Answer 1

If you have a marked ruler, it is trivial to draw a square of area $\pi.$ The problem is this: given a line segment of length 1, and a straightedge and compass, what other length line segments can you produce? The important part is you have no ruler; your straightedge is unmarked.

Answer 2

There is absolutely no problem with the existence of a square whose side length is $\sqrt \pi$, and you are right that such a square has the same area as the circle.

The problem of squaring the circle is much more limited. Here, our goal is to construct using a straightedge and compass the square with the same area as the circle.

To compare: suppose you are building a bridge and want to put a support at the midpoint of the bridge. Of course, it is impossible to perfectly measure and nail down the location of the midpoint; any actual physical measurement you took would have an error. However, the midpoint of a segment $AB$ can still be constructed by the following procedure:

• Draw a circle $O_1$ with center $A$ and radius $AB$; draw another circle $O_2$ with center $B$ and radius $AB$.
• Let $C,D$ be the intersection points of $O_1$ and $O_2$. Draw line $CD$.
• Let $M$ be the intersection point of $AB$ and $CD$. This is the midpoint.

Whenever you do this in real life, there will be some error due to the limits of your tools. However, you can achieve any precision you like with the same procedure if you can find tools that are precise enough.

The same is not true for constructing a segment whose side length is $\sqrt \pi$. Any finite procedure you could write down with straight-edge and compass (and many attempts have been made at such a procedure!) would be flawed! I am not just saying that with imperfect tools, the result would be an approximation. I am saying that the result would be an approximation of the wrong number: an approximation of an approximation of $\sqrt \pi$. If you increased the precision of your tools, eventually you'd get to the point where the precision was no longer helping.

This is specific to compass and straightedge: the tools the Ancient Greeks had at their disposal. Computers are much better at this sort of thing. A computer is also an imperfect tool - it has limited memory, and limited pixels on the screen. However, we can write a computer program which will draw a square whose side length is as close to $\sqrt\pi$ as you want - subject to acquiring enough memory and enough pixels.

Thus, we say that although $\sqrt\pi$ is not a constructible number (no compass and straightedge construction generates $\sqrt \pi$ to arbitrary precision) it is a computable number (a computer program does exist which generates $\sqrt \pi$ to arbitrary precision).

Answer 3

As Michael and Ted said, the problem is to draw a square of area $\pi$ with a straightedge and a compass only, starting from two points at distance one.

To help you understand why the problem is difficult, I advise you to find ways of constructing, with such tools only, starting from a segment of length $1$ and another of length $x$, a segment of length $\frac{1}{x}$ and another one of length $\sqrt{x}$.

Your remarks about $\pi$’s decimals are irrelevant to this problem: the diagonal of a square with all sides of length $1$ has length $\sqrt{2}$, which also has an infinite number of nonzero decimals.

Answer 4

A practical approach - that won't be very exact due to tolerances: Make a circle from wood/metal with inner diameter/radius of 1 x. Fill it to a height y with sand.

Make a square and start to fill it from one edge with sand in the height y. Adjust the square until the sand has height y everywhere.

Done. :-)