Circles and Triangles

[Sean B. Palmer]

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:

http://en.wikipedia.org/wiki/Circle_packing#Packings_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.

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.

Cf. http://swhack.com/logs/2010-07-08#T22-31-32

[Sean B. Palmer]

> 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.

Oh, and six is the next multiple of three, so 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 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.