The applet below depicts a family of equilateral triangles erected inwardly on the sides of regular polygons. Because of the built-in symmetry their third vertices form another regular polygon = smaller than the original one. This simple fact is a key to a more profound assertion. What is it?
The applet below provides an illustration to David Radcliffe's solution of a problem posed on twitter.com by James Tanton:
Is it possible to draw a square on an equilateral triangle lattice of points (each vertex of the square on a lattice point)?
David gave a short answer:
If a triangle lattice contains a square ABCD, then it contains a smaller square EFGH and so on.
He supported the claim with an illustration generalized in the applet. He then also added a more general assertion: The same construction shows that the equilateral triangle lattice contains no regular n-gon unless n = 3 or 6. The applet serves to illustrate the latter as well.
David's solution is based on a fundamental property of a triangular grid, viz., a segment joining any two nodes of the grid serves as a base of exactly two equilateral triangles with the third vertices the grid points. This follows from the fact that the grid is invariant under rotations through 60° around any of its nodes.
David also offered another solution. If the vertices of a regular n-gon are a, b, c, d, ..., then points (a + c - b), (b + d - c), ... form a smaller regular n-gon, unless n = 3, 4, 6.
