A pool rack is an equilateral triangle, filled with 15 equal-sized balls. Seen from above, we'd see a triangle, with 15 circles within.

Imagine we used smaller and smaller balls. The more the balls, more area of the triangle would be covered.

In the limit, with infinite balls, would all of the triangle be covered?

If unit-radius billiard balls fill a plane, the plane can be tesselated with triangles whose edges connect centers of adjacent balls.  Each such equilateral triangle has sides that measure 2 units, and have part of their area covered by 1/6 of each of 3 circles of unit radius.  The total area of each triangle is sqrt(3). Each circle sector has area pi/6, for a total area that's covered by circle sectors within a triangle to be pi/2 so that 90.689968...% of each triangle, and therefore the entire surface, is covered with some circle.  As the totality is shrunk, this percentage remains the same, and so is the limiting percentage as more and more circles (balls) are packed into the triangle.

The edges and corners of the containing triangle (the big one around all the balls), some of these tesselating triangles are truncated by the boundary.  All of the truncation takes place in places that would otherwise be occupied by a piece of circle, so the uncovered percentage is larger (covered percentage smaller) in the edge triangles. Therefore initially, as a larger proportion of the tesselating triangles are on the edge, the covered percentage is less than 90.689968...%.  The coverage increases, but approaches this number as the limit.

 

Posted by Charlie on 2004-09-17 13:23:25