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  1. Besides a square, what regular polygons can be constructed so that the points of that polygon lie on the points of a regular, planar, orthogonal grid?

  2. Besides a triangle and hexagon, what regular polygons can be constructed so that the points of that polygon lie on the points of a regular, planar, triangular grid?

  3. Are there any rules for constructing regular polygons on square or triangular planar grids?

  4. If the points of the constructed polygon are not restricted to grid-points only, but can also include points on intersections of straight lines drawn between existing points, what additional regular polygons can be created on triangular or orthogonal grids?

(No, this is not homework. I was doodling on square graph paper during a meeting and found I couldn't draw a regular octagon referencing only the grid points.)

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Take a look here: web.me.com/paulscott.info/lattice-points/5regular.html#anchor2 . You can search google for regular polygons lattice points and see the results –  Beni Bogosel Sep 6 '11 at 21:37
    
Based on the link given by @Beni, and this one on wolfram.com, it looks like regular lattice n-gons only exist for n = 3, 4, and 6. This answers questions 1, 2, and 3, but what about #4? If we are allowed to construct new points that are not on the lattice, can we create other regular polygons? –  oosterwal Sep 6 '11 at 21:59
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@oosterwal, any intersection of two lines must have rational coordinates (in the lattice basis). So for any finite set of non-lattice points constructed in this manner, you can just rescale the lattice so that they all lie on the lattice (rescale by the lcm of the denominators of the coordinates). –  Craig Sep 7 '11 at 0:05
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up vote 8 down vote accepted

We answer Part $4$ for a square grid. We may assume that this is the grid of points with integer coordinates.

Then any line between grid points has rational coefficients. Thus, if two such lines meet, they meet at a point that has rational coordinates.

Suppose that we draw a finite number of distinct lines. Let $N$ be the least common multiple of the denominators of coordinates of intersection points of these lines

Now imagine constructing a figure that uses some of these intersection points (including possibly original grid points). If we scale this figure by the factor $N$, we have scaled it to have integer coordinates. Thus, up to similarity, no more figures can be drawn using intersection points than can be drawn using original grid points. In particular, intersection points do not help in drawing a regular octagon.

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