# Regular Polygons in a Triangular Grid

What Is It About?

A Mathematical Droodle

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?

What if applet does not run? |

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Copyright © 1996-2018 Alexander Bogomolny

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

What if applet does not run? |

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

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Copyright © 1996-2018 Alexander Bogomolny