A Little Geometric Creativity

There’s a nice old geometric pattern… …that Eric Broug presents in his book Islamic Geometric Patterns as being from the Great Mosque of Herat.

Its construction is based on a hexagon, and the pattern repeats as additional hexagons are tiled around the first one. The underlying hexagons (which are not drawn in the finished pattern) are shown in red here: This makes a fine star/wedge/triangle pattern that is quite satisfying to look at, and suitable for decorating things in your house.

The construction of one of the hexagon-based units of the pattern begins with drawing a circle, and then a hexagon within that circle. Once the hexagon is drawn, the lines of the pattern can be drawn within in it.

But in the real world, a medium on which we are drawing (or painting) always has edges, and those edges do not go along the edges of the hexagons. For example, let’s say we want to draw one of the pattern units, plus however much extra fits, on a square board or tile. Something like this: Draw in the supporting hexagons and you will see that the one complete pattern is indeed surrounded by only parts of adjacent copies of the pattern. It’s a bit of a construction conundrum. How will we fill in the portion of the pattern that lies outside the one central hexagon that we will have room to draw? How will we extend the pattern to the edges of the square when we cannot place a compass foot at the centre of any of the adjacent circles that would define the basic hexagons?

There is a way.

Let’s review how the pattern is made, and then examine how to extend it without being able to draw more circles. Follow along with your own piece of paper, pencil, compass and straightedge.

There is something magical in the fact that this pattern, with all of its exact proportions, can be constructed solely with a compass and a straightedge. You never need to measure an angle or a line.

To construct a hexagon, one begins by drawing a circle. Within the circle inscribe a hexagon using the standard method of walking the legs of your compass (still set to the same radius) around the circle. We’ll call this Hexagon A. It is oriented so it is a point-up hexagon. Draw radii through all the vertices of the hexagon. We’ll call these the ribs of hexagon A. Now we need to construct a second hexagon of the same size, and on the same centre, but rotated 30°. To do this we will connect the midpoints of the six arcs from the hexagon already made.

If we had plenty of room, we would find those midpoints by first drawing six more circles (of the same radius) centred on the six vertices of hexagon A; and then draw lines from the centre of the original circle to the outer points of intersection of adjacent secondary circles. These ribs (purple, below) would show us the midpoints of the arcs. But with the limited space we are working with, we need instead to bisect one side of the hexagon. This can be done with two compass arcs from adjacent vertices of hexagon A; we can then connect their intersection outside the circle with the circle’s centre. This line bisects one of the arcs, and we can use that as a starting point for making a second hexagon. We’ll call this Hexagon B, and it is oriented so it is a side-up hexagon.

Draw ribs for hexagon B. Note that the ribs of hexagon B (purple) go through the midpoints of the sides of hexagon A (red).

Connect the hexagon A midpoints to make a six-pointed star. We’ll call these star 1 lines (green in the next figure). Similarly, we can note that the ribs of hexagon A pass through the midpoints of the sides of hexagon B. Connect the midpoints of hexagon B to make Star 2 — being sure, however, in this case to extend the star lines beyond hexagon B as far as the first edge of hexagon A you encounter. We’ll call these star 2 lines (blue in the next figure). Finally, we ink certain portions of the Star 1 and Star 2 lines to make the final pattern. Note that the entire pattern lies within hexagon A. The pattern that is inked consists only of portions of star 1 and star 2 lines. In fact we drew hexagons A and B, and their ribs, only to create the star 1 and 2 lines. And we drew the initial circle only to create hexagons A and B.

In order to extend the pattern we need to draw more star 1 and 2 lines outside our original hexagon. On a larger medium we could easily place the centres of new circles outside hexagon A, but on this limited surface where we are working we cannot.

It may help to look at what we need to find. From our earlier construction of adjacent hexagonal cells, we know a few things about what should be happening. Star 1 lines leaving the hexagon (A in the diagram above) go a certain distance to a point Y (which we’re not really sure how to locate) and then turn through 120° to follow a line D.

Star 2 lines leaving the hexagon (B in the diagram above) go a certain distance to a point X (which we’re not really sure how to locate) where they cross a star 2 line (C) in the adjacent hexagons. At some farther point Z they also go through 120° turns.

As well, it’s valuable to make a few observations about what happens as these lines enter adjacent hexagons…

1. Hexagon A sides, when extended, become the ribs of other hexagons A, and vice versa. 2. Star 1 lines become some of the star 1 lines of adjacent hexagons. 3. Star 2 lines become some of the star 2 lines of adjacent hexagons. This means that by extending these hexagon sides, ribs and star lines, we have much (but not all) of the construction information for the pattern outside the original hexagon. What’s missing are lines C and D, and points X, Y and Z.

But now we can see that point X is the place where the extended hexagon A sides meet the extended star 2 lines. Drawing a line through them gives us line C. Similarly, Point Y is the place where the extended hexagon A sides meet the extended star 1 lines. Drawing a line through pairs of points Y gives us line D. At this point we have all the construction lines we need to ink the rest of the pattern outside the original hexagon… …and then remove the construction lines. It’s very satisfying to be able to construct this figure in a limited space, and to solve the problems associated. But now, as a bonus, it appear that the pattern presents us with a fascinating geometry problem!

As the pattern is extended, a new, larger hexagon has appeared, a hexagon that is similar to hexagon B, but is formed by star 1 lines that pass on the outside of the six small triangles. We’ll call it hexagon C. In the illustration below, hexagon B is purple, and hexagon C is blue. It’s a bit of a puzzler, but I’ll just leave the problem here for the intrigued reader to solve. In terms of the side length of hexagon B, what is the side length of hexagon C?