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?

Take a "triangular" number of balls n(n+1)/2 of unit radius. These balls cover an area of pi*n(n+1)/2. The area of the equilateral triangle bounding these balls is √3*(n+√3-1)^2. The ratio of the area covered by the balls to the area of triangle is

{pi√3/6}*{n(n+1)/(n+√3-1)^2}. As n goes to infinity, this ratio approaches pi√3/6 ~= 0.9069.

Would all of the triangle be covered? No.

Posted by Bractals on 2004-09-21 17:39:54