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Jul 9, 2010, 7:51:13 AM7/9/10

to Gallimaufry of Whits

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:

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.

Jul 9, 2010, 7:56:21 AM7/9/10

to Gallimaufry of Whits

> 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.

> 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.

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