All of the lengths A(n) are irrational. Each length is the hypotenuse of a right triangle with integral legs.

Assume all the vertices of the 1994-gon lie on the lattice points. Then the legs of the triangles corresponding to each side must follow the grid lines. Substitute the legs for each side of the 1994-gon to create a 3988-gon.

The perimeter of the 3988-gon must be even. Its perimeter is calculated as 2*1994 + 1994*1995/2 = 1993003. This is odd, a contradiction. Therefore the vertices of the 1994-gon cannot all lie on lattice points.