A Geometry Puzzle: Alternating Hexagons and Squares In a Ring

Background

This intriguing pattern is an alternating sequence of six hexagons and six squares. Both basic shapes have the same edge length, and they pack perfectly around a centre.

Furthermore, this fairly complex construction can be done with only a ruler and straightedge. Typically, you start with a circle, within which you construct what will be the dodecagonal centre of the figure above. The alternating hexagons and squares then sit on the sides of this dodecagon.

Because we will have to create other hexagons along the way, I'll call the hexagons in the final pattern around the outside ring hexagons.

The construction goes like this. (I will assume you know how to construct a hexagon within a circle, and how to bisect a line segment.)

1. Draw a circle.

2. Construct a hexagon within it.

3. Bisect one of the sides of the hexagon and draw a ray from the circle's centre through it. Construct another hexagon beginning with a vertex that falls where this ray intersects the circle.

4. Connect the successive vertices of the two hexagons to make a dodecagon, a 12-sided regular polygon.

5. Using the side length of the dodecagon as radius, draw circles around each vertex of the dodecagon.

6. Using every other intersection of these circles as a centre, draw six more circles of the same radius.

7. Construct hexagons within these last six circles. One side of each will coincide with a side of the dodecagon.

8. Connect hexagons to form squares.

9. The final figure without the construction lines. Rotating it 15° counterclockwise will make it look like the one at the very top of the post.

The Puzzle

OK, now for the puzzle. If we connect the centres of the six hexagons in the ring, we get another, larger hexagon.

This hexagon, which I'll call the master hexagon, can be used as the repeating frame for tiling a larger area with the pattern.

But if you were to do this, you would draw the pattern of master hexagons first, and would then construct the ring pattern based on it.

So here's the puzzle: how do you construct the ring pattern of hexagons and squares, given only the master hexagon?

Solution

The basic problem is to locate the dodecagon that forms the inside of the ring. Once we have that, we can construct  the ring, as above. But how do we get from the master hexagon to the dodecagon? 

10. Begin with the master hexagon.

11. Locate its centre by connecting vertices, and construct the circumscribing circle.

12. Bisect one side of the hexagon, and construct a second hexagon, much as you did in step 3 above, starting from the point where this bisector meets the circle.

13. Connect vertices of one hexagon to make a six pointed star.

14. Connect the vertices of the other hexagon in a similar fashion.

15. Connect the intersections of those two six-pointed stars, to make a dodecagon. This is the dodecagon that will form the inside of the ring.

16. Using steps 5, 6 and 7 above, construct ring hexagons from the sides of the dodecagon.

17. And, as in step 8 above, connect the ring hexagons to form squares.

It's interesting to compare the construction lines one uses when beginning with a circle to those drawn when beginning with the master hexagon.

18. "Forward" construction lines (that is, those beginning from the circle that circumscribes the dodecagon) are black. "Reverse" construction lines (beginning with the master hexagon) are blue.

Commentary

Why does using this method to construct the dodecagon within the master hexagon work?

Well, we know the master hexagon has a concentric dodecagon within it somewhere. But which dodecagon?

Because each dodecagon has a different edge length, each implies a different size of hexagons arrayed around it. We want the dodecagon where the hexagon's centre will fall at a vertex of the master hexagon (the red one, below, in this case).

The "right" dodecagon will have vertices that are 60° apart when viewed from a vertex of the master hexagon. Necessarily then, these dodecagon vertices will fall somewhere on the sides of equilateral triangles drawn within the master hexagon.

Drawing the other equilateral triangle within this hexagon gives us a general idea of where these dodecagon vertices will fall, but nothing precise. As well, this pattern so far only has 6-fold rotational symmetry.

If we add another master hexagon, rotated 15°, and its inner triangles, we get a pattern with the necessary 12-fold rotational symmetry.

The set of four equilateral triangles has 3 sets of common intersections, all of which will make dodecagons (red, orange and yellow, below). But only the outermost set (red) creates dodecagon sides that subtend a 60° angle when viewed from the vertices of the master hexagon.