To form an A-B-A-B-... hexagonal close packing of spheres, the coordinate points of the lattice will be the spheres' centers. Suppose, the goal is to fill a box with spheres according to hcp. The box would be placed on the x-y-z coordinate space.
First form a row of spheres. The centers will all lie on a straight line. Their x-coordinate will vary by 2r since the distance between each center if the spheres are touching is 2r. The y-coordinate and z-coordinate will be the same. For simplicity, say that the balls are the first row and that their y- and z-coordinates are simply r, so that their surfaces rest on the zero-planes. Coordinates of the centers of the first row will look like (2r, r, r), (4r, r, r), (6r ,r, r), (8r ,r, r), ... . The sphere centered at x = 0 is immediately omitted because part of the sphere would lie outside.
Now, form the next row of spheres. Again, the centers will all lie on a straight line with x-coordinate differences of 2r, but there will be a shift of distance r in the x-direction so that the center of every sphere in this row aligns with the x-coordinate of where two spheres touch in the first row. This allows the spheres of the new row to slide in closer to the first row until all spheres in the new row are touching two spheres of the first row. Since the new spheres touch two spheres, their centers form an equilateral triangle with those two neighbors' centers. The side lengths are all 2r, so the height or y-coordinate difference between the rows is . Thus, this row will have coordinates like this:
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Two spheres that are congruent are the same size and shape. Therefore, they would have the same surface area. So this statement is always true.
It is crucial to measuring circles, spheres, etc. 3.1415926535897932384626433832795028841 (cont.)
No. An adverb is a word that describes a verb. Rowed itself is a verb form, very rarely used as an adjective (e.g. rowed boats).
No, it is not. It is the past tense and past participle of the verb to climb, and might rarely be used as an adjective (e.g. climbed mountains as opposed to unclimbed).
pi is used to work out things like the circumference (length around the outside) of a circle from the diameter (length from side to side (pi x d). It can also work out a circle's area (2pi x r(radius)), and much more with spheres and cylanders etc.