11 Jun
2003
11 Jun
'03
2:40 p.m.
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. John conway