Solved: Hexagonal circle packing The German mathematician

Chapter 11, Problem 74

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Hexagonal circle packing The German mathematician Gauss proved that the densest way to pack circles with the same radius in the plane is to place the centers of the circles on a hexagonal grid (see figure). Some molecular structures use this packing or its three-dimensional analog. Assume all circles have a radius of 1 and let rij be the vector that extends from the center of circle i to the center of circle j, for i, j = 0, 1, c, 6. x 4 0 1 3 2 7 5 6 y a. Find r0j, for j = 1, 2, c, 6. b. Find r12, r34, and r61. c. Imagine circle 7 is added to the arrangement as shown in the figure. Find r07, r17, r47, and r75. 7