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 Post subject: Packing 4D spheres
PostPosted: Thu Oct 10, 2013 10:31 pm 
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Earlier today I dug up this image for another thread.

Image

However it got me thinking. These are the 3 ways to pack all of 3-space with the same size spheres. And I believe there are two primary ways to pack 2-space (a plane) with circles. They can be centered on a square grid or a triangular grid. The hexagonal grid would be degenerate with the triangular one. So we have:

2D = 2 packings
3D = 3 packings

Do 4D spheres have 4 basic packings? Does this work in general, does N-space have N packings of N dimensional spheres?

I did a very quick google search and didn't find something but I figure this question has to have been answered already.

Carl

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 Post subject: Re: Packing 4D spheres
PostPosted: Thu Oct 10, 2013 11:01 pm 
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The best synopsis I can find is http://mathworld.wolfram.com/HyperspherePacking.html

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 Post subject: Re: Packing 4D spheres
PostPosted: Fri Oct 11, 2013 1:42 am 
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Also: http://mathworld.wolfram.com/KissingNumber.html

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 Post subject: Re: Packing 4D spheres
PostPosted: Fri Oct 11, 2013 9:41 am 
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Too bad we can't model polyhyperspheres like we can polyspheres... :lol:


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 Post subject: Re: Packing 4D spheres
PostPosted: Fri Oct 11, 2013 10:37 am 
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Jared wrote:
Too bad we can't model polyhyperspheres like we can polyspheres... :lol:

We'll just have to do it one 3D crossection at a time :D

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 Post subject: Re: Packing 4D spheres
PostPosted: Fri Oct 11, 2013 7:34 pm 
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Thanks for the links guys. They allow me to clean up some of my terminology and give me faith that my question is a valid question... but I don't think either really answers the question that I asked.

In 1D we have 1 periodic packing of identical "circles". Not sure its fair to call these red and blue line segments circles but here is what it looks like.
Attachment:
1Dspheres.png
1Dspheres.png [ 1.56 KiB | Viewed 1712 times ]


In 2D have 2 periodic packings of identical circles. That is almost a strait quote from here. And here is what they look like.
Image

I was concerned that maybe the triangular packing which I show here might be considered a third but I'm happy to see that it is not. Its really just an example of the hexagonal packing with voids placed in it.
Attachment:
2Dtriangular.png
2Dtriangular.png [ 6.54 KiB | Viewed 1712 times ]


In 3D we have 3 period packings and here is what they look like.
Image

So at this point we see:
1D has 1 periodic or lattice packing. Does this lattice have a name? Surely it does... just not sure what it is.
2D has 2 periodic or lattice packing of identical circles. Those are the square and hexagonal lattices.
3D has 3 periodic or lattice packing of identical spheres. Those are the simple cubic, face centered cubic, and hexagonal lattices.

So how many periodic or lattice packing of identical hyperspheres are there in 4D? Do they have names? The links you sent talk alot about the densest lattice packings but say almost nothing about the others. I'm curious if this trend continues... does 4D have 4 basic/simple lattices? Does 5D have 5? etc?

Going through the links, statements like this one blow my mind:
"Exact values for lattice packings are known for n=1 to 9 and n=24 (Conway and Sloane 1993, Sloane and Nebe). Odlyzko and Sloane (1979) found the exact value for 24-D."

How is it possible to work something out in 24D that still isn't known for 10D through 23D? I really struggle with trying to understand 4D and there are days where I have a really hard time extrapolating from 2D images to 3D. The thread which started me along this line of thinking is a prime example.

And I can't wrap my mind around the last 2 paragraphs here. One ends with:
"However, for n=9, the central hypersphere just touches the hypercube of centers, and for n>9, the central hypersphere is partially outside the hypercube."
and the next states:
"The radius of the central sphere is therefore sqrt(n)-1. Now, the distance from the origin to the center of a facet bounding the hypercube is always 1 (one hypersphere radius), so the center hypersphere is tangent to the hypercube when sqrt(n)-1=1, or n=4, and partially outside it for n>4."

Don't these statements contradict each other? Why 9 and then 4? And doesn't this imply that in 3D the sphere of radius=1 isn't tangent to a cube of size 1? I'm lost.

Oh and I love that table. In 24D I can pack 196560 hyperspheres such that they all "kiss" one centered at (0,0,0,0,0,0,etc). Yes that is a picture I'll NEVER be able to fit into my brain. I'd love to be able to understand how someone goes about proving that but I'm sure its well over my head.

Carl

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 Post subject: Re: Packing 4D spheres
PostPosted: Fri Oct 11, 2013 8:32 pm 
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If you factorize the kissing numbers into primes, perhaps this might help you understand the series progression?

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 Post subject: Re: Packing 4D spheres
PostPosted: Fri Oct 11, 2013 11:36 pm 
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wwwmwww wrote:
Going through the links, statements like this one blow my mind:
"Exact values for lattice packings are known for n=1 to 9 and n=24 (Conway and Sloane 1993, Sloane and Nebe). Odlyzko and Sloane (1979) found the exact value for 24-D."

How is it possible to work something out in 24D that still isn't known for 10D through 23D?

I had a similar thought so I went and skimmed most of

Leech, John (1964), "Some sphere packings in higher space", Canadian Journal of Mathematics 16: 657–682, doi:10.4153/CJM-1964-065-1, ISSN 0008-414X, MR 0167901
and
Leech, John (1967), "Notes on sphere packings", Canadian Journal of Mathematics 19: 251–267, doi:10.4153/CJM-1967-017-0, ISSN 0008-414X, MR 0209983


I think the MathWorld statement about lattice packings could be written in a much less surprising way:
Sphere packings that form a lattice happen to be the densest-known packings for dimensions 1-9 and 24. For all other dimensions the best-known packing doesn't form a lattice.

Thinking about packings in N dimensions... If you don't know a dense packing for N dimensions then pick the best-known packing for N - 1 dimensions and just replicate "rows" of that packing over and over into the Nth dimension. This construction is easy to see starting an the obvious 1-dimensional packing where all spheres are in a line and the 2-dimensional one where you take lines of spheres as rows and you replicates the rows down into the 2nd dimension. Obviously you can keep extending this into any dimension.

Interestingly, this inductive step doesn't always produce a lattice. After placing two N - 1 dimensional "rows" in N dimensions, if the third "row"'s sphere centers don't lay on the lines formed be the centers in the previous two rows then the packing won't be a lattice. This is how non-lattice but dense packings are known for higher dimensions.

For the Leech Lattice, the construction is based on the parity of binary strings and the parity is extended into the idea of a k-parity of binary strings that differ in at least k digits. I don't quite follow the construction but this allows the formation of a lattice whenever the dimensions is divisible by 4. In the case of 24D the lattice happens to beat out the best known non-lattice packing.


wwwmwww wrote:
I really struggle with trying to understand 4D and there are days where I have a really hard time extrapolating from 2D images to 3D.
I have trouble with 4D even though I "understand" and have solved the N^4 Rubik's cube equivalents. 4D isn't intuitive.

wwwmwww wrote:
Oh and I love that table. In 24D I can pack 196560 hyperspheres such that they all "kiss" one centered at (0,0,0,0,0,0,etc). Yes that is a picture I'll NEVER be able to fit into my brain. I'd love to be able to understand how someone goes about proving that but I'm sure its well over my head.

I don't think mathematicians have any more intuition for > 3 dimensions than we do. I think they stop using physical interpretations of the math and just start to think of it as tables of numbers and degrees of freedom rather than physical dimensions.

For example, when I'm programming, I can declare 1-dimensional array:

int foo[N];

I think of this as a list / row of numbers.

For D = 2:

int foo[N][M];

I think of this as a matrix. For D = 3 I think of it as a cube.

For D = 4:

int foo[N][M][O][P];

I think of each foo[N] as containing a whole cube. That is, I have a list / row of cubes.

For D = 5:

int foo[N][M][O][P][Q];

I think of this as a matrix of cubes.

Obviously this extends to a cube of cubes and then a list of a cubes of cubes, etc. After a while I don't bother embedding them and just think of it as a N-dimensional array and I think of the indexes as independent variables with no real physical meaning.

I'm pretty sure mathematicians just stop thinking about the physical meaning of things and just look at the numbers. If you read through Leech's paper it's pretty clear he never really thinks of the geometry of 24D space in physical terms at all.

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Last edited by Brandon Enright on Sat Oct 12, 2013 12:03 am, edited 2 times in total.

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 Post subject: Re: Packing 4D spheres
PostPosted: Fri Oct 11, 2013 11:38 pm 
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Also check out http://members.shaw.ca/quadibloc/math/pakint.htm which takes an interesting approach to discussing sphere packing and the Leech Lattice.

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 Post subject: Re: Packing 4D spheres
PostPosted: Sat Oct 12, 2013 5:32 pm 
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bmenrigh wrote:
Interestingly, this inductive step doesn't always produce a lattice. After placing two N - 1 dimensional "rows" in N dimensions, if the third "row"'s sphere centers don't lay on the lines formed be the centers in the previous two rows then the packing won't be a lattice.
Why wouldn't the 3rd row be in line with the first 2? Was it shifted before being copied intentionally? I can see going from the lattice in 1D to the square packing of circles. And I can see going from the 2D square packing of circles to the simple cubic packing of spheres in 3D. And I assume 4D hyperspheres can be packed by just extending this to 4D producing something that I assume could be called the Simple Hypercubic packing. I don't see anything that would ever force this out of alignment unless someone is just wanting to try a new packing.

Still my main question remains unanswered as best I can tell. How many simple periodic packings or simple lattices are there in higher dimensions? The 1D one is probably called something like Unit Spacing if I had to guess. The 2D and 3D ones have been named above. Surely there aren't too many more available in 4D so I would have expected all of them to have been named and explored just a bit. Per Mathworld/Wolfram we have, "The analog of face-centered cubic packing is the densest lattice packing in four and five dimensions. In eight dimensions, the densest lattice packing is made up of two copies of face-centered cubic. In six and seven dimensions, the densest lattice packings are cross sections of the eight-dimensional case." If the densest packing is known in all these cases surely the density of the other packings have been checked to verify they are less dense. So what are they?

Here is a link that I found with an interesting table:
http://www.hermetic.ch/compsci/lattgeom.htm
Attachment:
Lattice.PNG
Lattice.PNG [ 17.18 KiB | Viewed 1655 times ]
Though that article starts off with "A lattice (in the sense used in computational physics)..." so I'm not sure we are using the same definition. I.e. a simple periodic packings. Mathworld/Wolfram says there are 2 in 2D and I'm more inclined to trust them. So maybe the question I should be asking is what is the definition of a simple periodic packing or simple lattice that I should be using here? I know if one goes off in the direction of crystals one can introduce many different lattices in 3D as seen here. But I believe I'd still argue there are 3 basic or simple ones. Am I drawing lines where no one else is? Maybe the water is muddy enough on the definition that my main question doesn't make sense. I'm just not sure.

Here this sounds interesting. Wish I could follow the logic.
http://neilsloane.com/doc/Me59.pdf

Oh and in 3D it appears that what I've been calling the Hexagonal Packing or Lattice is also called the Body Centered Cubic Lattice. So if the names haven't been pinned down in 3D I guess its easy for me to see why the 4D ones haven't been named. Yet we haves names for 24D lattices!? Oh well.
http://chemistry.umeche.maine.edu/~amar/spring2012/bur25542_1216.jpg

Carl

EDIT: That last image is too big to post as an image and I had temporally posted a smaller copy till I noticed it said "Permission required for reproduction or display". So I figure its just best to post as a link.

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 Post subject: Re: Packing 4D spheres
PostPosted: Sun Oct 13, 2013 5:41 pm 
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Let's step back to just 3D packing of spheres again. We have these 3 basic/regular packings... correct?

(1) Simple Cubic
(2) Hexagonal (I believe this is the same as Body Centered Cubic... PLEASE correct me if I'm wrong)
(3) Face Centered Cubic

Something I remember from dealing the SiC Polytyes and sphere packing is illustrated here:
Attachment:
800px-Close_packing.svg.png
800px-Close_packing.svg.png [ 129.01 KiB | Viewed 1619 times ]

If you place spheres using the 2D hexagonal packing method you get the A layer in the above image. You can now place a second layer of spheres packed via the 2D hexagonal packing method on top of this one. If you place it as close as possible to the first layer you end up placing the spheres in the divots between the joining of 3 spheres in the layer below. Let's call this second layer B. However when you go to place the 3rd layer using this same method you are faced with a choice. You can place the spheres in the divots directly above those spheres in the A layer and thus copy the A layer and simply translate it up as is seen here on the left, or you can pick the other location and create a "C" layer that isn't directly above either the A or B layer, as seen on the right.

ABA as I recall is the Body Centered Cubic or Hexagonal lattice.
ABC is the Face Centered Cubic latice

Now for my question. The Face Centered Cubic lattice is supposed to produce the densest packing in 3-space. Why? Shouldn't these 2 methods produce lattices of exactly the same density? Each 2D layer is identical and all the divots certainly look to be the same depth to me so how does the choice of where the 3rd layer is placed affect the density? Again I'm certain that I'm missing something.

Carl

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 Post subject: Re: Packing 4D spheres
PostPosted: Mon Oct 14, 2013 7:13 am 
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There are 4 distinct "pure" types of 3D sphere packing:

Simple cubic
Body-centered cubic (BCC)
Face-centered cubic (FCC)
Hexagonal close packing (HCP)

FCC and HCP are equally the most dense, but differ in packing sequence:

http://www.ndt-ed.org/EducationResource ... cc_hcp.htm
http://en.wikipedia.org/wiki/Close-pack ... al_spheres

But remember you can also have intermediate forms, which are not stable. For example in 2D, just imagine skewing the layers of square packing to give many different patterns of rhombic packing, before you get the triangular/hexagonal packing. Also, you can get mixed mosaics of different packing forms within the same crystal, as if you just shove a bunch of balls quickly in a box. For example, different layers may be packed as square, rhombus, triangle, or anything in between, and the layers themselves may be offset to varying different degrees in different regions. You could even have alternating bands of 100 layers FCC and 100 layers HCP, which would even have macroscopic symmetry!

So sphere packing is a lot more complex than the pure and simple symmetric crystalline forms you are looking at, and it gets exponentially more complex with more dimensions.

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 Post subject: Re: Packing 4D spheres
PostPosted: Mon Oct 14, 2013 4:32 pm 
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KelvinS wrote:
There are 4 distinct "pure" types of 3D sphere packing:

Simple cubic
Body-centered cubic (BCC)
Face-centered cubic (FCC)
Hexagonal close packing (HCP)
Interesting... I was working myself to that conclusion too. What I don't get is why there are papers which seem to ignore BCC as seen here. And I guess it makes sense that the articles on "Cubic Latices" would ignore HCP. I guess one could argue that BCC isn't a "pure/simple" lattice as it could be considered two interwoven simple cubic lattices. Not sure if that is a valid argument or not though. I'm not sure if there is a definition of pure or simple lattice out there.
KelvinS wrote:
FCC and HCP are equally the most dense, but differ in packing sequence:
But it seems the FCC packing is always singled out as the most dense. Why isn't this tie mentioned more often?
KelvinS wrote:
But remember you can also have intermediate forms, which are not stable. For example in 2D, just imagine skewing the layers of square packing to give many different patterns of rhombic packing, before you get the triangular/hexagonal packing. Also, you can get mixed mosaics of different packing forms within the same crystal, as if you just shove a bunch of balls quickly in a box. For example, different layers may be packed as square, rhombus, triangle, or anything in between, and the layers themselves may be offset to varying different degrees in different regions. You could even have alternating bands of 100 layers FCC and 100 layers HCP, which would even have macroscopic symmetry!
Agreed. I'm an epi process engineer and have grown several of the polytypes of SiC. I'm aware things can get very complex very fast which is why I was just trying to look at the "simple" or "pure" lattices (if there is such a thing) for the purposes of (1) seeing how the trend progressed into higher dimmensions and (2) seeing if there was anything here that could be used/exploited for an interesting new puzzle.
KelvinS wrote:
So sphere packing is a lot more complex than the pure and simple symmetric crystalline forms you are looking at, and it gets exponentially more complex with more dimensions.
Yes... that is why I was explecitly trying to stay with the "pure and simple symmetric crystalline forms".

And now I have another question. Think of a set of points which belong to each of these lattices:

Simple cubic
Body-centered cubic (BCC)
Face-centered cubic (FCC)
Hexagonal close packing (HCP)

If one takes all of 3D space and assigned each point to the closest lattice point one can see the following:

Simple Cubic cuts all of 3D space up into cubes.
BCC cuts all of 3D space up into truncated octahedrons.
FCC cuts all of 3D space up into rhombic dodecahedrons.

What is the shape produced if this is applied to HCP?

In another thread Jared made this statement:
Jared wrote:
There aren't many polyhedra which can tile space alone, and of the ones that can there are three which are highly symmetric. The first of course is the cube, which provides us with polycubes. The second is the rhombic dodecahedron, which makes polyrhons. The third is the truncated octahedron.
So shouldn't there be a 4th highly symmetric polyhedra which can tile space alone? What is it?

I suspect it could be used to make interesting polyforms.

Carl

P.S. It just hit me... maybe the HCP packing produces 2 shapes which are the mirror images of each other. That's just a guess at this point.

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 Post subject: Re: Packing 4D spheres
PostPosted: Mon Oct 14, 2013 7:07 pm 
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It would seem that the HCP produces this lattice: http://en.wikipedia.org/wiki/Trapezo-rh ... _honeycomb
There are some other highly symmetric polyhedra that fill space, such as the triangular prism and the hexagonal prism.

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 Post subject: Re: Packing 4D spheres
PostPosted: Mon Oct 14, 2013 7:37 pm 
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contrabass wrote:
It would seem that the HCP produces this lattice: http://en.wikipedia.org/wiki/Trapezo-rh ... _honeycomb
Thanks... yes that must be it. I wonder if anyone has made polyforms with these. They look interesting. Imagine combing a set of Tetrarhons and a set of Tetrapezorhons (a name I just made up). In principle these sets could be made to fit together to make structures which related to ABCBABCBA... type structures. Then imagine adding to that set the pieces which mix rhombic dodecahedrons and trapezo-rhombic dodecahedron in the same piece. Sounds complicated... but it might be fun if it produced any interesting shapes.
contrabass wrote:
There are some other highly symmetric polyhedra that fill space, such as the triangular prism and the hexagonal prism.
Agreed... but these make more interesting 2D polyforms then they do 3D ones. In principle if one tried to make 3D polyforms with these they could be free to arbitrarily rotate and shift the pieces between layers. Though I guess you could do that with cubes too and no one does so maybe its a non-issue. And to some extend the trapezo-rhombic dodecahedron shape would force a layered structure. If one had a set of Tetrapezorhons it would be clear that each piece had a plane which must align with all the others. In some sense you would be fitting hexagonal prisms which forced a certain alignment between the layers so it would feel more like a 2D puzzle then the other symmetric shapes which are free to rotate along more then one axis. Still I think they could be interesting as they could be made to fit with the Tetrarhons.

Carl

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 Post subject: Re: Packing 4D spheres
PostPosted: Mon Oct 14, 2013 9:08 pm 
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KelvinS wrote:
There are 4 distinct "pure" types of 3D sphere packing:

Simple cubic
Body-centered cubic (BCC)
Face-centered cubic (FCC)
Hexagonal close packing (HCP)
I was thinking about this while driving home from work. Is there any reason to exclude the simple hexagonal packing of spheres simply translated over and over in the 3rd direction? Let's call it the Simple Hexagonal Packing (SHP) but I assume its got an official name already. Just not sure what it is.

In the sense that:

FCC relates to the pattern ABCABCABC... ect.
and
HCP relates to the pattern ABABABABA... etc.

Then

SHP would relate to the pattern AAAAAAAAA... etc.

So that should be 5 distinct "pure" types of 3D sphere packing... shouldn't it? I'm now wondering if we are missing others. Without a definition of "pure" this could be a slippery slope.

Here is a nice stereo image:
Image

I just pulled that image from here, which is a very nice link. Looks like the official name is Simple Hexagonal (SH).
http://www.cartage.org.lb/en/themes/sciences/physics/solidstatephysics/atomicbonding/ReciprocalLattice/CrystalLattice/sh_stereo.gif

Though the bold lines I think give me a reason why it could be excluded. This could be viewed as simply a squished cubic packing.

Carl

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 Post subject: Re: Packing 4D spheres
PostPosted: Tue Oct 15, 2013 7:44 am 
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And what about face or body-centered hexagonal packing? Or tetrahedral packing? Or octahedral packing? Or face or body-centered tetrahedral or octahedral packing? How pure and simple is "pure and simple"? What is the minimum number of contacts per sphere? The mystery deepens...

Hint: Before you can answer any question, you first need to make sure you are asking the right question. So, what is the objective, and what are the "rules" or constraints? :wink:

For example, one constraint may be that every sphere must contact at least 4 other spheres in 3 dimensions (out of plane); and/or every sphere must be equivalent, with the same coordination geometry (which must be symmetric) in the same (local or global?) environment.

I think you need to go out and buy/play with a bunch of marbles! :lol:

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 Post subject: Re: Packing 4D spheres
PostPosted: Tue Oct 15, 2013 1:23 pm 
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KelvinS wrote:
How pure and simple is "pure and simple"? What is the minimum number of contacts per sphere? The mystery deepens...

Hint: Before you can answer any question, you first need to make sure you are asking the right question. So, what is the objective, and what are the "rules" or constraints? :wink:
My initial objective was to see if the perceived trend that I noticed for 1D, 2D, and 3D continued to higher dimensions. I've since been convinced that what I had percieved as a trend is based on just partial information. I still contend that is 1D there is one simple lattice. In 2D I think it can already get more complicated. The hexagonal packing can be viewed as a distored square packing if one allows something other then a 90 degree alignment between the layers. If the lattices with 90 degree alignment and 60 degree alignment are considered unique then I think one could argue for an infinite number of lattices for all the angles inbetween. It all depends on how one sets up the problem and I admit that I've done a very poor job at that. So the objective has pretty much changed into an effort to educate myself a bit more on this topic.
KelvinS wrote:
For example, one constraint may be that every sphere must contact at least 4 other spheres in 3 dimensions (out of plane); and/or every sphere must be equivalent, with the same coordination geometry (which must be symmetric) in the same (local or global?) environment.
I've so far resisted trying to define "simple" lattice as I assumed such a definition must have already been made and if I defined it differently then I'd just be adding to the confusion. Now I'm not sure sure... Here Wolfram Mathworld states "In two dimensions, there are two periodic circle packings for identical circles..." as their opening sentence. Yet now that I think about it there are also rhombus packings that are periodic? Why is Wolfram Mathworld comfortable totally ignoring those? I had assumed there were some understood unwritten rules that everyone in the field was aware of. I now think the rule are being changed to fit the needs of each article or link that I'm finding. It certainly doesn't seem to be as clear cut as I was hoping it would be.

Carl

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 Post subject: Re: Packing 4D spheres
PostPosted: Tue Oct 15, 2013 2:11 pm 
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wwwmwww wrote:
I had assumed there were some understood unwritten rules that everyone in the field was aware of. I now think the rule are being changed to fit the needs of each article or link that I'm finding. It certainly doesn't seem to be as clear cut as I was hoping it would be.

I agree there must be some rules for what's simple. Otherwise I think the answer, even in 2D, would be that there are an infinite number of periodic packings.

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 Post subject: Re: Packing 4D spheres
PostPosted: Tue Oct 15, 2013 2:14 pm 
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So you may as well just go ahead and choose your own lattice constraints/criteria (or at least identify the established ones, if there are any), then you will be able to start working through the potential solutions/configurations...

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