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In one dimension, there is only one way of packing, that is keeping the balls next to each other.

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In two dimension, we can keep a line of spheres on top of another line directly or we can keep the second line in the cavities of the first line.

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In my book, for three dimensional packing, only structure is shown when keeping the square packing layers on top of each other:

enter image description here

Would it be possible to get another form of 3D packing from square close packing in two dimensions by making the spheres go into cavities? (similar to the way we kept the line in a staggered way to get hexagonal close packing)

Similarly for packing 2D hexagonal packed spheres in 3D, could we keep the spheres on top of each layer on top of each other( in axis of 3-D packing)?

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The textbook missed a beautiful opportunity to show a relationship between simple cubic and close-packed cubic structures. As students learn, mental (spatial) operations need to be explained - once shown, they are there. if not explained, these operations grow in unclarity as more and more spatial operations and pictures are piled upon the original, unclear one.

Here is an explanation: Take the pile of spheres (4 x 4 x 4) and shift them sideways as in the picture below:

enter image description here

I did a shift in only two dimensions, but you see the change in what is obviously not closest packing to a tilted pile with a lot closer packing. It's still two dimensional, but the diagram says "hexagonal packing"; to get real closest packing, you have to shift the spheres in the third dimension (into the paper, or screen). Then the spheres will lie in the lowest valleys of a packed plane. Then the densest plane of spheres is not one of the original x,y,z axes, but a tilted one. You will have converted the simple cubic lattice into a closest packed lattice.

You can play with this idea in order to make it more concrete in your mind: imagine a closest packed lattice and unfold it to a simple cubic lattice.

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  • $\begingroup$ I am not sure how this answers the question. The image you provide is already contained in the question, albeit a bit less accurate. In two dimensions the is only one way to pack the circles according to a square grid closest: square (close) packing. There is a more dense packing: hexagonal (close) packing. The point of the question is, in my opinion, what kind of structure you would get (in three dimensions) when you extend the hexagonal packing in the same way as you would go from square packing to simple cubic. Hence I believe the answer should focus on FCC and HCP structures. $\endgroup$
    – Martin - マーチン ♦
    Jul 28, 2021 at 21:58
  • $\begingroup$ @Martin: You have great vision and long experience. For those of us who take one simple step at a time, the answer above spreads out the movement of spheres into two steps, the first in 2D, and then the second in the third dimension, with one of two possibilities. Doing it all at once is a wonderful mental capability - not all of us have it. Another way to "get it" is to play with about 64 (or 125) magnetic spheres. Alas - we have here only a flat screen, typed words and some pictures. $\endgroup$
    – James Gaidis
    Jul 29, 2021 at 13:27

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