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 r ij be the vector that extends from the center of circle i to the center of circle j , for i , j = 0, 1, …, 6. a. Find r 0 j , for j = 1, 2, …, 6. b. Find r 12 , r 34 , and r 61 . c. Imagine circle 7 is added to the arrangement as shown in the figure. Find r 07 , r 17 , r 47 , and r 75 .
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 r ij be the vector that extends from the center of circle i to the center of circle j , for i , j = 0, 1, …, 6. a. Find r 0 j , for j = 1, 2, …, 6. b. Find r 12 , r 34 , and r 61 . c. Imagine circle 7 is added to the arrangement as shown in the figure. Find r 07 , r 17 , r 47 , and r 75 .
Solution Summary: The author explains the calculation of r_0j, which is as follows: There are six circles encircling one circle.
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, …, 6.
a. Find r0j, for j = 1, 2, …, 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.
Quantities that have magnitude and direction but not position. Some examples of vectors are velocity, displacement, acceleration, and force. They are sometimes called Euclidean or spatial vectors.
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