```From:           David Eppstein <eppstein@ics.uci.edu>
To:             sci.math
Subject:        Re: Regular polygon in R^D
Date:           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/
```