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I have read a children's book where alien race of "square people" used a pair of compasses that drafted a perfect square when used. Now I wanted to explain to the child that it is not possible to have such a pair of compasses, but then I was not really sure. So is it possible to construct a mechanical tool with one fixed point and one moving point that would actually draft a square with one continuous movement?

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This isn't what you're asking, but it might be worth mentioning that there's a mechanical tool that allows one to drill (almost-)square holes. See here. – Blue Jun 13 '13 at 12:22
up vote 23 down vote accepted

It depends on how restrictive you are about what counts as a 'pair of compasses'. For instance, if you require the writing end to be a fixed (Euclidean) distance away from the fixed point at all times, then that pair of compasses will always draw a circle, since a circle is exactly the set of points at some fixed distance from a fixed point.

One way you might be able to build such a pair of compasses is by using linkages. A linkage is like a more complicated version of a pair of compasses, where you have some collection of fixed points, movable joints, and bars connecting them, with a writing stylus attached at some point. A pair of compasses, then, can be viewed as an extremely simple type of linkage. A very famous example of a linkage, dating from 1784 is Watt's linkage, which was used in the first steam engines. It is special because it draws a nearly perfect straight line:

enter image description here

James Watt once said that of all his inventions, this linkage was the one he was most proud of. It wasn't until 1864 that a linkage was discovered that could draw an exact straight line. It is called the Peaucellier-Lipkin linkage, and earned its inventors the Prix Monyon:

enter image description here

How do you get from here to a square? The following remarkable theorem was proved in 1994:

Theorem (Thurston, Kapovic-Millson): given any polynomial map, there is a linkage and a vertex of that linkage such that the vertex traces out that polynomial map.

In this context, a polynomial map is any curve on the plane that can be described by polynomial equations. Certain curves (like $y=\log(x)$) do not have polynomial maps describing them; however, any curve can be approximated to arbitrarily high accuracy by polynomial maps. We therefore have two nice corollaries:

Corollary 1: There is a linkage that signs your name!

Corollary 2: There is a linkage that draws a square (to arbitrarily high accuracy).

Bear in mind that the theorem does not guarantee that the linkages will be simple! In fact, if you wanted to build a linkage that actually did sign your name, it would probably be extremely complicated! But it would nevertheless exist. I have no idea how complicated a linkage would have to be to draw a square.

Further reading: the reason I know about all this is that I've read Joseph O'Rourke's excellent book How To Fold It, which has a website here. If this kind of thing interests you, I strongly recommend buying a copy.

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So it doesn't actually draw a square, just a very good approximation of one? You might want to rephrase corollary 2. – Potato Jun 13 '13 at 14:22
I don't understand corollary 2. Why is a square a polynomial map? – MJD Jun 13 '13 at 14:22
Or do you need one linkage for each line? Also, I'm not sure what you mean by "any line can be approximated to arbitrarily high accuracy by polynomial maps." A line is a polynomial map. – Potato Jun 13 '13 at 14:23
There is a website with live demo of this theorem. It builds and animates aforementioned linkages for (approximations of) any curves you can draw. Including your handwritten name. – ulidtko Jun 13 '13 at 15:28
@MJD: $x^n+y^n=1$ for very large $n$ is an implicit polynomial equation that approximates a square. – Raskolnikov Jun 14 '13 at 11:14

You might want to explore this link about drilling square holes. Your "central point" constraint goes - a curve of constant radius is a circle, but a curve of constant diameter need not be.

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+1. "Great minds ..." :) – Blue Jun 13 '13 at 12:23
@Blue Indeed! I was interested in this when I was young ... – Mark Bennet Jun 13 '13 at 12:25

Does the book actually describe its use as "One continuous movement"?

If not, it seems likely the "pair of compasses" in your book is not referring to any exotic contraption. "Pair of compasses" = "compass"

Definition courtesy of Wiktionary

pair of compasses (plural pairs of compasses) A tool used to draw circles, usually consisting of two arms joined at one end in such a way that the arms can be opened and closed; one of the arms is pointed at its free end while the other holds a pencil, pencil lead or some other implement for making marks.

Synonyms compass

When used correctly, (with a pencil and a straight edge) you can easily draw a perfect square.

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(draw lightly until step 12, most marks will be temporary)

  1. Draw a line with the pencil and straight edge.
  2. Choose/mark a starting point (Corner A of square).
  3. Set compass to a small radius.
  4. use compass to mare two points
  5. Centered A use the compass to mark the two points it crosses the line.
  6. Increase radius of compass marginally.
  7. Using marks from step 5, draw two arcs so they cross above and below the line.
  8. Using the straight edge and pencil draw a line through the intersection points of the arcs from step 7. This line will be perpendicular to the original and should also cross through your original starting point)
  9. Set compass radius to the desired length for the side of the square.
  10. Use compass centered at A to mark the corner points B and C on the two lines.
  11. Use compass to draw two arcs centered at B and C. The point these arcs intersect is corner point D.
  12. Use the straight edge to draw your line segments BD and CD, and darken AB and AC.
  13. Erase arcs, temporary marks, and lines

Similar instructions with pictures

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Yes, the book is a fiction. It's written in Czech. The "square people" have a compass, in Czech "Kružítko". It is even more funny because that literaly translates as "circle-drawer". Also "square people" have many others squareized tools and items, but this one was the only thing I remember since it was an interesting mathematical problem. – sm4 Jun 14 '13 at 0:18

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