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  • Orange stacks

    From David Entwistle@21:1/5 to All on Sat Jul 12 15:19:27 2025
    Possibly going off-topic, but I hope you don't object too much. I have a question. Please don't spend a lot of time on it, but if you happen to
    know the answer, that would be appreciated. The excellent Quanta magazine
    have an article about sphere-packing:

    https://www.quantamagazine.org/new-sphere-packing-record-stems-from-an- unexpected-source-20250707

    It starts:

    "In math, the search for optimal patterns never ends. The sphere-packing problem — which asks how to cram balls into a (high-dimensional) box as efficiently as possible — is no exception. It has enticed mathematicians
    for centuries and has important applications in cryptography, long-
    distance communication and more.

    It’s deceptively difficult. In the early 17th century, the physicist
    Johannes Kepler showed that by stacking three-dimensional spheres the way
    you would oranges in a grocery store, you can fill about 74% of space. He conjectured that this was the best possible arrangement. But it would take mathematicians nearly 400 years to prove it".

    If I had a grocery store, I think I would stack oranges in a square-based pyramid, but I assume that a triangular-based pyramid would lead to more efficient packing. To what does the "74% of space" figure refer, square-
    based, or triangular-based? I can't see that they would be the same thing,
    but I could be wrong.

    Thanks,
    --
    David Entwistle

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  • From Carl G.@21:1/5 to David Entwistle on Sat Jul 12 08:41:25 2025
    On 7/12/2025 8:19 AM, David Entwistle wrote:
    ...
    If I had a grocery store, I think I would stack oranges in a square-based pyramid, but I assume that a triangular-based pyramid would lead to more efficient packing. To what does the "74% of space" figure refer, square- based, or triangular-based? I can't see that they would be the same thing, but I could be wrong.

    Thanks,

    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. A
    cubic close packing and hexagonal close packing both have a density of pi/(3*sqrt(2)) (0.74048048...).

    https://en.wikipedia.org/wiki/Sphere_packing
    --
    Carl G.


    --
    This email has been checked for viruses by AVG antivirus software.
    www.avg.com

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  • From Richard Tobin@21:1/5 to qnivq.ragjvfgyr@ogvagrearg.pbz on Sat Jul 12 19:49:24 2025
    In article <104tudv$274e7$1@dont-email.me>,
    David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote:

    If I had a grocery store, I think I would stack oranges in a square-based >pyramid, but I assume that a triangular-based pyramid would lead to more >efficient packing.

    Contemplate the sloping faces of this square-based pyramid:

    https://en.wikipedia.org/wiki/Sphere_packing#/media/File:Rye_Castle,_Rye,_East_Sussex,_England-6April2011_(1)_(cropped).jpg

    -- Richard

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  • From David Entwistle@21:1/5 to Richard Tobin on Mon Jul 14 02:26:10 2025
    On Sat, 12 Jul 2025 19:49:24 -0000 (UTC), Richard Tobin wrote:

    Contemplate the sloping faces of this square-based pyramid:

    https://en.wikipedia.org/wiki/Sphere_packing#/media/
    File:Rye_Castle,_Rye,_East_Sussex,_England-6April2011_(1)_(cropped).jpg

    Thanks - I've been contemplating...

    I've ordered a copy of "The Six-Cornered Snowflake".

    --
    David Entwistle

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  • From David Entwistle@21:1/5 to Carl G. on Mon Jul 14 06:26:13 2025
    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

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  • From Richard Tobin@21:1/5 to qnivq.ragjvfgyr@ogvagrearg.pbz on Mon Jul 14 11:54:06 2025
    In article <10527u5$391km$1@dont-email.me>,
    David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> 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.

    It's not merely that the density is the same.

    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,

    Yes.

    then I finish up with two quite different shapes with quite
    different symmetries.

    But - hard though it may seem to believe - no!

    -- Richard

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  • From David Entwistle@21:1/5 to Richard Tobin on Mon Jul 14 13:11:16 2025
    On Mon, 14 Jul 2025 11:54:06 -0000 (UTC), Richard Tobin wrote:

    But - hard though it may seem to believe - no!

    Oh!... I do find that hard to believe. Here's what I see (think I see).

    If I use the centres as defining points:

    Then one shape (from the square pyramid) has a square top and bottom. It
    looks to me like a truncated octahedron.

    The second shape (from the triangular pyramid) is based exclusively on equilateral triangles and is entirely regular - is is what I think of when
    I think of a diamond crystal lattice.

    In my mind, to be the same, the rows in the base of the square-based
    pyramids base would need to be offset left and right, such that the
    distance between any three adjacent spheres is minimized. It is no longer square-based, but is a triangular-based pyramid standing in a square box.

    I can see I'm going to have to buy some golf balls, or table tennis balls.

    --
    David Entwistle

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  • From Richard Tobin@21:1/5 to qnivq.ragjvfgyr@ogvagrearg.pbz on Mon Jul 14 14:00:51 2025
    In article <1052vlk$3dn0r$1@dont-email.me>,
    David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote:

    Then one shape (from the square pyramid) has a square top and bottom. It >looks to me like a truncated octahedron.

    Top layer:

    . . .

    a b

    . . .

    c d

    . . .


    Middle layer:

    e f g

    . .

    h i j

    . .

    k l m


    Bottom layer:

    . . .

    n o

    . . .

    p q

    . . .

    If you consider the octahedron based on the square f, h, j, and l,
    then the balls on the top and bottom layers do not fall on its faces -
    in fact they are directly above and below the edges of the square: a
    is directly above the midpoint of hf and n below it, forming a square
    face.

    There are in fact 6 square faces: abcd, nopq, ahnf, bfoj, djql, and
    clph. And 8 triangular faces: abf, bdj, dcl, cah, nof, oqj, qpl, and
    pnh. This is a cuboctahedron.

    The second shape (from the triangular pyramid) is based exclusively on >equilateral triangles and is entirely regular [...]

    I can see I'm going to have to buy some golf balls, or table tennis balls.

    You will find that the second shape is also not what you think it is.

    -- Richard

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  • From David Entwistle@21:1/5 to Richard Tobin on Tue Jul 15 06:32:04 2025
    On Mon, 14 Jul 2025 11:54:06 -0000 (UTC), Richard Tobin wrote:

    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,

    Yes.

    then I finish up with two quite different shapes with quite different >>symmetries.

    But - hard though it may seem to believe - no!

    I'm still struggling with the idea the structures of the square-based and triangular-based pyramids are the same.

    I think we can agree that there are 90 degree angles between the centres
    of the cannonballs in the layers of the square based pyramid.

    In my mind the triangular based pyramid is entirely regular and based on
    the tetrahedral structure with a tetrahedral angle of (approximately)
    109.5 degrees between all centres. There are no right angles.

    https://en.wikipedia.org/wiki/Tetrahedral_molecular_geometry

    In my mind the square based and triangular based pyramids are not the same packing structure.

    --
    David Entwistle

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  • From Richard Tobin@21:1/5 to qnivq.ragjvfgyr@ogvagrearg.pbz on Tue Jul 15 11:28:23 2025
    In article <1054sl4$3u0lv$1@dont-email.me>,
    David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote:

    I'm still struggling with the idea the structures of the square-based and >triangular-based pyramids are the same.

    They are, just at different orientations.

    This wikipedia page makes it explicit:

    https://en.wikipedia.org/wiki/Close-packing_of_equal_spheres

    Both are face-centred cubic (FCC). The diagrams labelled FCC on the
    wikipedia page show the two orientations.

    -- Richard

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  • From David Entwistle@21:1/5 to Charlie Roberts on Wed Jul 16 08:05:40 2025
    On Tue, 15 Jul 2025 11:13:34 -0400, Charlie Roberts wrote:

    One other source I suggest you look up is any elementary/introductory
    solid state text book. "Introduction to Solid State Physics" by Charles Kittlel is a canonical one in the US. Equivalents no doubt exist across
    the world.

    Thanks for the recommendation. I've got a second-hand hardback copy on
    order for for a few pounds. It is the John Wiley & Sons Inc published
    version.

    --
    David Entwistle

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  • From David Entwistle@21:1/5 to Richard Tobin on Mon Jul 21 07:46:50 2025
    On Tue, 15 Jul 2025 11:28:23 -0000 (UTC), Richard Tobin wrote:

    This wikipedia page makes it explicit:

    https://en.wikipedia.org/wiki/Close-packing_of_equal_spheres

    Thanks. Also of particular interest is:

    https://en.wikipedia.org/wiki/Kissing_number

    Regarding the packing of twelve equally sized spheres in three dimensional space, it includes the following comment: "In fact, there is so much extra space that any two of the 12 outer spheres can exchange places through a continuous movement without any of the outer spheres losing contact with
    the center one." That is something I had failed to appreciate and my go
    some way to explain my confusion. My vision of a similarity to the
    tetrahedral carbon lattice, you may have seen at school during physics and chemistry lessons, is wrong. Or, at least, incomplete.

    I now have this Conway, Sloane book on order:

    <https://books.google.co.uk/books/about/ Sphere_Packings_Lattices_and_Groups.html?id=hoTjBwAAQBAJ&redir_esc=y>

    --
    David Entwistle

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