-----Original Message----- From: John Conway [mailto:conway@Math.Princeton.EDU] Sent: Wednesday, June 11, 2003 4:40 PM To: math-fun Subject: Re: [math-fun] What's your proof?
On Wed, 11 Jun 2003, Fred W. Helenius wrote:
At 09:26 AM 6/11/03, John McCarthy wrote:
A polygon can be scaled to have vertices at lattice points if and only if all its angles have rational tangents.
I don't think so. That would mean there was a lattice-point regular octagon, but there isn't, as JHC has just confirmed. For another example, consider a parallelogram with angles that are alternately arctan(2) and arctan(-2), and with sides alternately 1 and pi. The tangents are rational, but lattice-point distances can't be in the ratio 1:pi.
But the angle of a regular octagon DOESN'T have a rational tangent.
The "rational tangents" condition is exactly right.
I think words are being used in two different ways. If ABCDEFGH is a regular octagon, the angle ABC is 3pi/4, which has tangent -1, which is rational. But the angle ABD, for example, does not have a rational tangent, which shows that triangle ABD, and therefore the octagon, cannot be scaled to lattice points. I think the intended criterion is that a set of points can be scaled to be lattice points just if the angle formed by any 3 of them has a rational tangent. Fred's example fails to meet this criterion because the angle formed by a side and a diagonal of the parallelogram has an irrational tangent. Andy Latto andy.latto@pobox.com