Packing in two dimensions

One simple way of filling space in two dimensions in a regular and indefinitely extending manner is shown in Figure 1a.

In Figure 1b a grid is drawn connecting the circle centres. You should check the following:

  1. that the complete pattern can be generated from one circle by repeating it at all other grid intersections;
  2. that the coordination number of each circle is four;
  3. that each circle (unit radius) is equally shared between four unit cells so there is an average of one circle (4 x 1/4) in each unit cell area, and the % space filling = 100pi/4 = 78.5%;
  4. that the radius r of the largest circle that could fit at the centre of a square of four circles in contact (an interstitial site, so called) is just r = SQRT(2)-1 = 0.414

Figure 2, shows a second arrangement with the circles still in contact but with the coordination number increased to 6.

Sketch this arrangement in your practical book. (Use blue for the circles; then draw in a grid, in red, connecting the circle centres. Finally, heavily indent a unit cell.)

Now calculate

  1. the % space filling;
  2. the radius of the interstitial site.

This arrangement is the closest packing possible in two dimensions. (Try others if you wish.)