From Newsgroup: rec.puzzles

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 rCo which asks how to cram balls into a (high-dimensional) box as efficiently as possible rCo is no exception. It has enticed mathematicians
for centuries and has important applications in cryptography, long-
distance communication and more.

ItrCOs 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
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

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
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

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
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

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
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

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
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

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
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

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
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

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
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

On Sat, 12 Jul 2025 15:19:27 -0000 (UTC), 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. 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,

There have been many answers, but, having been on a break, I just
saw your question.

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.

The chapters on crystal structure in Kittel's book have excellent
diagrams of various structures. The hexagonal close packed
(hcp) and the face centred cubic (fcc) have the highest packing
density mentioned before. But, there are crystals in nature that
have a ower packing density. To add to the confusion,
non-crystalline solids abound. Nature does not always
favour the global minimum/maximum!!
--
This email has been checked for viruses by Avast antivirus software. www.avast.com
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

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
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

On Wed, 16 Jul 2025 08:05:40 -0000 (UTC), David Entwistle <qnivq.ragjvfgyr@ogvagrearg.pbz> wrote:

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.

Not sure you should *buy* one. Much of it is reall solid state stuff
at the undergraduate level. The crystal structure part is just a
couple of chapters in the beginning. Maybe you can borrow it
from a library?
--
This email has been checked for viruses by Avast antivirus software. www.avast.com
--- Synchronet 3.21a-Linux NewsLink 1.2

From Newsgroup: rec.puzzles

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
--- Synchronet 3.21a-Linux NewsLink 1.2