# Regular Polygons in a Triangular GridWhat 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?

### This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

 What if applet does not run?

Explanation 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)?

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.

### This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at https://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

 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 n = 3, 4, 6. • A problem with equilateral triangles: an interactive illustration
• Six Incircles in Equilateral Triangle
• Equilateral Triangle, Straight Line and Tangent Circles
• Equilateral Triangles on Diagonals of Antiequilic Quadrilateral
• Equilateral Triangles On Sides of a Parallelogram
• Equilateral Triangles On Sides of a Parallelogram II
• Equilateral Triangles on Sides of a Parallelogram III
• Triangle Classification
• Equilateral Triangles and Incircles in a Square
• Equal Areas in Equilateral Triangle
• A Circle Rolling in an Equilateral Triangle
• Viviani's Theorem
• Common Centroids Lead to Equilateral Triangle
• 