Re: Orange stacks

On Sat, 12 Jul 2025 08:41:25 -0700, Carl G. wrote:

If each orange (same size sphere) is packed so it touches 12 others,
then the packing density is the same for the stackings you mentioned.

Thanks. If I draw the layers out, I can see that.

Square base: 1, 4, 9, 16 in each layer. If I take the middle of the nine
as my reference, it touches (from the top) 0, 4, 4, 4 = 12.

Triangular base: 1, 3, 6, 10, 15 in each layer. If I take the middle of
the ten as my reference, it touches (from top) 0, 0, 3, 6, 3 = 12.

If I have this right, if I build my pyramids and glue the twelve touching
elements to the reference element, making two shapes, both of thirteen
elements each, then I finish up with two quite different shapes with quite
different symmetries. It isn't obvious that the two arrangements have the
same packing density, but I can see that that could be the case.

Interesting, thanks.

--
David Entwistle