Six Circles / by sbp

When you put circles around another circle, how come there are six of them? This question is related to the problem known as circle packing in the plane.

But this doesn't really answer the question of why a hexagon should form out of circles. The circle seems such a smooth and symmetrical figure, and six a rather strangely arbitrary figure for the packing thereof.

This morning I think I managed to come up with a partial explanation. I'd been thinking of the basic unit of this arrangement as being the circle — the circle in the middle with the other six circles surrounding it. But it would be more sensible to think of the basic unit as being the triangle, a unit made of three circles arranged triangularly. The hexagon can then be seen to be made from six triangles.

This makes sense when you think about the triangles because an equilateral triangle must have internal angles of 60 degrees in order to add up to 180, which means that you're going to get six of them around a 360 degree whole rotation. As six is the next multiple of three, it makes sense that you can nicely form a hexagon out of triangles, and that when you do so using circles you could get this nice regular single-circle-sized space in the middle.

So the question of the circle packing then becomes one more of why we use triangles rather than, say, squares. Squares don't pack properly with all the edges touching one another, whereas a triangle formed out of circles is somehow entirely neat. Why do three circles form a really neat equilateral triangle?

To put it another way, when you radiate outwards from the vertices of an equilateral triangle, how come all the radiations meet simultaneously, as opposed to when you do the same thing from a square they meet unevenly?

Perhaps the way to think about that is to realise that the circle is just enlarging the points of the vertices, it's not really changing their natures. So what we're talking about is the relationships of the vertices to one another. And of course in a triangle the vertices are equidistant. In the square, they are not equidistant.

In conclusion, the fact that circles are symmetrical and smooth means that they are excellent things to use in the construction of equilateral triangles (and therefore hexagons), which are themselves very regular shapes since their vertices are equidistant.

9 July 2010