A lot of arithmetic geometry has to do and took off with constructions or proofs of non-constructibility of specific figures only with ruler and compass. See e.g. the non-constructibility of the regular heptagon and the constructibility of the regular heptadecagon by Gauss.

I wonder what it means that there is a simple device that can be constructed with the help of a ruler and a compass that together with ruler and compass allows to construct arbitrary regular polygons very easily.

The device is nothing but a solid cone of finite height and a flexible string of length smaller than the circumference of the cone.

Now arrange $n$ dots on the string at equal distances and close the string such that also the first and the last dot have the same distance. I.e. you arrange $n$ dots equally on a closed string. This can be achieved with a compass alone.

After that you place the string on the cone, keeping all dots at same height until they all touch the cone. Projecting them vertically on the plane gives the corners of a regular $n$-gon.

Note that the cone is not a magic device like the angle trisector which may help to construct a rectangular heptagon, but adds nothing genuinely new to ruler and compass (because it can be constructed with the help of these). On the other hand, a flexible string is enough to create a ruler (as a physical device), and you need two rulers to make a compass. So a string might be enough for everything, in a specific sense? Note further, that angle trisectors - and more generally angle $k$-sectors - can easily be constructed for any angle $\alpha = 2\pi / n$: just place $kn$ dots on the string.

My question is:

Is there something wrong with my device? Might it be not constructible with the help of ruler and compass alone? Do other parts of the described procedure add something else beyond ruler and compass (the cone itself is supposed not to)?

I guess, it's leaving the plane that "disqualifies" this kind of construction. Might this have to do with leaving the real number line and entering the complex plane when solving polynomial equations? ("Suddenly all polynomial equations can be solved!") On the other hand: The compass is a genuinely three-dimensional device, too!

[Counter-historical note: If geometers would not have restricted themselves (deliberately?) to ruler and compass, there would have been no necessity to study the constructibilty of regular $n$-gons - and lots of arithmetic might not have been discovered (or only considerably later). On the other side: Wouldn't complex numbers have been discovered earlier, when the cone device had been considered more seriously?]

  • $\begingroup$ How do you project vertically with compass and ruler? Is plane construction through a point perpendicular to another plane a known thing? $\endgroup$ – dEmigOd Aug 21 '18 at 10:27
  • $\begingroup$ That's the kind of objections I was asking for. One could use the light of the sun (at least the ancient Greeks could have come upon this). $\endgroup$ – Hans-Peter Stricker Aug 21 '18 at 10:32
  • $\begingroup$ If you are talking about light of the sun, then should not beams be actually not parallel, nor gravitational pull, as it supposed to be directed to the center of the earth. $\endgroup$ – dEmigOd Aug 21 '18 at 10:42
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    $\begingroup$ You can also do things with a marked ruler that compass and straightedge are not able to. -- Even though you can certainly draw a picture of the marked ruler with a straightedge, that does not count as a "compass and straightedge" construction. $\endgroup$ – Henning Makholm Aug 21 '18 at 10:46
  • $\begingroup$ @dEmigOd: The greeks - when they were to think about it - would most surely have assumed, that the light of the sun comes in parallel. $\endgroup$ – Hans-Peter Stricker Aug 21 '18 at 10:54

The basic misunderstanding here is that "ruler and compass" is not really about which physical tools you're allowed to use.

When this phrase appears in mathematics it is a shorthand reference to the idea of basing constructions on the three first postulates of Book I of Euclid's Elements:

  1. To draw a straight line from any point to any point.

  2. To produce a finite straight line continuously in a straight line.

  3. To describe a circle with any center and radius.

(The fourth and fifth postulates are not basic constructions but are claims about what happens when certain other combinations of constructions are performed).

These three postulates are the definition of "ruler and compass". Often in English the alternative phrase "compass and straightedge" is preferred, to underscore the fact that there is none of the basic allowed constructions that depend depend on having measuring marks on your ruler.

In fact even without leaving the two-dimensional paper there are things you can imagine doing with a ruler that has measuring marks on it -- so-called neusis constructions. These are nevertheless not part of the basic operations. They can, for example, be used to trisect angles.

Mathematics is not about physical tools. Saying "ruler and compass" is merely supposed to remind you what the fundamental three constructions in Euclid are. You may, if you wish, object that it is a confusing or misleading shorthand -- that is ultimately a subjective judgment -- but this does not change the fact that what we really care about is not the tools named by the shorthand, but the three postulates it attempts to point to.

  • $\begingroup$ What does "continuously" in the second postulate mean, BTW. What would happen if we dropped "continuously"? $\endgroup$ – Hans-Peter Stricker Aug 21 '18 at 11:12
  • $\begingroup$ Let me repeat my question: Might it be the case that we need a marked ruler to create a cone? (Even though you might be saying that this doesn't matter.) $\endgroup$ – Hans-Peter Stricker Aug 21 '18 at 11:13
  • $\begingroup$ @HansStricker: I'm not sure whether "continuously" has any real content here, except perhaps to point out that the original finite straight line together with its extension is supposed to form a single straight line rather than a curve that has a bend where you started extending it. The quotation is from an English translation of the Greek original text, and Euclid was not quite up to today's standard of precision particularly in his basic definitions. $\endgroup$ – Henning Makholm Aug 21 '18 at 11:21
  • $\begingroup$ To be sure: "straightedge and compass" doesn't allow to draw equidistant marks on the straightedge with the compass (resulting in a marked ruler)? I.e. we may draw a two-dimensional circle in the plane but not a one-dimensional circle on the ruler? We are allowed to do the more complicated but not the more simple? Or do I miss the point? $\endgroup$ – Hans-Peter Stricker Aug 21 '18 at 11:48
  • $\begingroup$ To sum it up: The straight edge has to be thought of as a beam of light (in which we cannot make marks). We must not think of it as a straight physical body. But how may we think of the compass? Doesn't it have to be a physical body with a definite length? (Nevertheless, Euclidean geometry is about drawing.) $\endgroup$ – Hans-Peter Stricker Aug 21 '18 at 12:07

That's actually very clever! It shows that the ruler and compass model may actually be narrow-minded, since flexible strings are available, as is paper.

However, I believe that if flexible things are not allowed, then we still have unconstructability proofs. For instance, suppose everything we cut out is solid. Then we are only allowed to place it on the plane, and everything we get will have been constructed in the plane before cutting out the shape. So only if we are allowed to bend and stretch things, the third dimension gives additional possibilities.

  • $\begingroup$ What could you do without flexibel strings (or let's say flexible rulers)? How would you measure the length of the circumference of a circle? (This is what you must do: 1. Draw a circle on a piece of wood. 2. Cut it out perfectly. 3. Lay the flexibel ruler around it. 4. Create a straight ruler of equal length.) $\endgroup$ – Hans-Peter Stricker Aug 21 '18 at 10:57

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