This may be more of a recreational mathematics question than a research question, but I have wondered about it for a while. I hope it is not inappropriate for MO.

Consider the standard assumptions for ruler and compass constructions: We have an infinitely large sheet of paper, which we associate with the complex plane, that is initially blank aside from the points 0 and 1 being marked. In addition we have an infinite ruler and a compass that can be stretched to an arbitrary length.

Let us define a *move* to be one of the two actions normally associated with a ruler and compass:

- Use the ruler to draw the line defined by any two distinct points already marked on the paper.
- Stretch the compass from any one marked point to another and draw the resulting circle.

Assume that all intersection points among lines and circles drawn by these operations are automatically marked on the paper.

Now define $D(n)$ to be the maximum distance between any two marked points that can be constructed in this way with $n$ moves.

Questions:

- Is anyone aware of results about the function $D(n)$ or something equivalent?
- It is not difficult to prove $D(n) > 2^{2^{cn}}$ for some positive constant $c$ for sufficiently large $n$. Can one do better? If so, can one prove an upper bound on $D(n)$?

segmentbetween the two points. – Mark Meckes Jul 23 '10 at 3:26