Octagons with integral coordinates?

[quickfur]

In 2D, it is impossible to represent the coordinates of an equilateral triangle using only integer coordinates. However, in 3D this is possible, as a face of the alternated cube: (1,1,1), (-1,-1,1), (1,-1,-1), for example.

Similarly, a regular octagon cannot have integer coordinates in 2D; but can it have integer coordinates in 3D or higher? What about regular pentagons?

[wendy]

A regular octagon can not have integer coordinates in any dimension. This is because an chord parallel to an edge would be of length 1+sqrt(2), and the implication here is that sqrt(2) can be expressed in integers.

On the other hand, one can get really close with the sets of points (17,0), and (12,12), with all permutations, change of sign. The area of this figure differs from the octagon by a measure of 1/169 of the unit-square.