Here it is:

Method for constructing a line of length π:

  1. Construct a circle labeled A, with a radius of 1.
  2. Bisect circle A. Each of the resulting arcs is now length π. Label one arc B.
  3. Align one end of your straightedge with one end of arc B, and form with it a tangent line of arc B.
  4. Rotate the straightedge to form a tangent with the other endpoint of arc B while constantly keeping it tangential to the arc throughout the rotation.
  5. Draw a tangent line labeled P from the second end of arc B to the the end of the straightedge that was tangential to the first end of arc B.
  6. Line P is length π.

Method for squaring the circle:

  1. Use the above method to construct a line P of length π.
  2. Bisect line P.
  3. Take the length of the bisection of line P and construct an equilateral right triangle with legs equal to this length.
  4. The length of the hypotenuse of this triangle is ${\sqrt π}$.
  5. Construct a square labeled C with sides of length ${\sqrt π}$.
  6. Square C has the same area as circle A.

Now I suspect that my operation of using the partially completed construction as a guide for the movement of a strait edge while simultaneously using the strait edge as a point reference is simply not allowed in Greek geometry, but I'm not sure if it's actually something that breaks the rules or is just sufficiently far removed from the normal ways that the Greeks used the tools that define the limits of their geometry that no one thought of it before now.

So my question is: "Is this valid, if not why not, and has anyone come up with this method before I did?"

Also, I didn't realize it was the day before π day when I came up with this. so happy π day.

  • 1
    $\begingroup$ You can't do the tangential operation that you're describing. $\endgroup$ – Michael Burr Mar 14 '16 at 4:02
  • $\begingroup$ You need to understand exactly what is meant by "squaring the circle". In particular, the straight edge can be used for one thing only: to draw a straight line between two points that you have already constructed. $\endgroup$ – David Mar 14 '16 at 4:03
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    $\begingroup$ This is called a Neusis construction. Step 4 is not allowed under normal construction rules. $\endgroup$ – Laars Helenius Mar 14 '16 at 4:04
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    $\begingroup$ @LaarsHelenius That looks like something that would be more compelling as an answer than a comment. (If you have time to write such a thing) $\endgroup$ – Milo Brandt Mar 14 '16 at 4:09
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    $\begingroup$ @MiloBrandt will attempt a complete answer later. Right now all I have is my phone and it is quite tedious to enter anything substantive. $\endgroup$ – Laars Helenius Mar 14 '16 at 4:11

Just a quick answer for now.

A compass and straightedge construction is a finite sequence of steps each of which is one of the following:

  1. Construct a point that is the intersection of two lines/circles.
  2. Construct a line between any two (previously constructed) points.
  3. Construct a circle centred at a point that passes through another point.

When talking about constructing lengths, we are originally given two points that are a unit distance apart, and the goal is to construct two points that are the required length apart. These lengths are called constructible numbers.

$π$ is not constructible, nor is $\sqrt[3]{2}$. In general the only numbers that are constructible are those that can be expressed using only integers, plus, minus, times, divide and square-root. This needs some Galois theory.


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