From:David Eppstein <eppstein@ics.uci.edu>To:sci.mathSubject:Re: Regular polygon in R^DDate:Fri, 05 Jan 2001 10:26:03 -0800

In article <3A5609D5.560394E@studNOSPAM.hia.no>, Jan Kristian Haugland <jkhaug00@studNOSPAM.hia.no> wrote: > tim_brooks@my-deja.com wrote: > > > If n>=3 it is always possible to find d such that there exists a > > regular polygon with n sides in R^d such that all of its vertices have > > integer coordinates (lattice polygon) with respect to the standard > > lattice of R^d ? > > I don't think this is possible for n = 5, since the > ratio between the distances between non-neighbours > and between neighbours is (1+sqrt(5))/2, while the > ratio between two distances in the lattice has the > form sqrt(r) with r rational. Nice argument. But it looks like n=5 is impossible even for non-integer lattices in R^d: the intersection of the lattice with the plane containing the pentagon would have to itself be a planar lattice, but no planar lattice can contain the vertices of a regular pentagon, for if one had vertices abcde (say, in clockwise order) then (a+d-e) (b+e-a) (c+a-b) (d+b-c) (e+c-d) would be the vertices of a smaller pentagon, ad infinitum. -- David Eppstein UC Irvine Dept. of Information & Computer Science eppstein@ics.uci.edu http://www.ics.uci.edu/~eppstein/