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?