# Talk:Laves graph

The article mentions two networks can be interleaved:

It is possible to interleave two copies of the structure, filling one-fourth of the points of the integer lattice, while preserving the fact that the adjacent vertices are exactly the pairs of points that are √2 units apart, and all other pairs of points are farther apart. The two copies are mirror images of each other.
Is it worth mentioning this method of interleaving 8?  Googling the 8-srs network turns up a few results - should a proper paper be cited, or does the youtube video qualify as a citable source?
JasonHise (talk) 01:17, 28 June 2019 (UTC)

This review is transcluded from Talk:Laves graph/GA1. The edit link for this section can be used to add comments to the review.
Reviewer: Ovinus (talk · contribs) 19:03, 11 September 2022 (UTC)
Seeing as the EMST review is finishing up, I'll grab another one. Excepting the more abstract graph theoretic concepts, I feel like this topic can be made pretty accessible to the layperson, so I'll try make suggestions toward that. Ovinus (talk) 19:03, 11 September 2022 (UTC)

"soap film structures" Maybe link to somewhere about minimal surfaces? Dunno the best link Ovinus (talk) 19:03, 11 September 2022 (UTC)
Added before "gyroid". But in the specific "soap film structures" phrase, I really meant actual physical soap films, summarizing the later text "has also been observed experimentally in soap-water systems", and not their mathematical abstraction as minimal surfaces. —David Eppstein (talk) 21:59, 12 September 2022 (UTC)
"Its points can be given integer coordinates" Understanding this statement requires a somewhat subtle understanding that the Laves graph enjoys the same label after any isometry. How about "In its classic shape, its points have integer coordinates", something like that Ovinus (talk) 19:03, 11 September 2022 (UTC)
Isn't the claim that this shape is "its classic shape" the sort of editorial opinion that would need sourcing and attribution? Also at this point in the article the graph has only been stated to be a specific "system of points and line segments" so it can be recoordinatized but not given a different shape. The more abstract graph-theoretic view is only introduced later. —David Eppstein (talk) 21:57, 12 September 2022 (UTC)
How about "In one form"? That's not editorializing. The thing is that "points can be given" is more confusing than simply "points have". Ovinus (talk) 23:39, 12 September 2022 (UTC)
"Form" = "shape" = its geometric description in terms independent of scale and position. The integer coordinates are specific to one scale and one position. They would not be valid, for instance, for instances of this graph coming from certain crystals, unless you picked a coordinate system that was very specific to the crystal. "Arrangement", maybe? —David Eppstein (talk) 01:43, 13 September 2022 (UTC)
Sure, arrangement is precise and understandable. Ovinus (talk) 03:27, 13 September 2022 (UTC)
I'll try make an STL of a couple cells of the graph. Ovinus (talk) 19:03, 11 September 2022 (UTC)
"The two copies are mirror images of each other" specify whether it's simply a reflection, or a less intuitive rotoreflection Ovinus (talk) 19:03, 11 September 2022 (UTC)
The source doesn't appear to say, and it's not obvious just from looking at the source's illustration of these two interleaved copies. —David Eppstein (talk) 21:54, 12 September 2022 (UTC)
while preserving the fact that the adjacent vertices are exactly the pairs of points that are sqrt(2) units apart, and all other pairs of points are farther apart So, to rephrase, this graph may be constructed by taking a particular fourth of all integer-coordinate points and connecting every pair that's sqrt(2) apart? I think the phrasing might be a confusing here; it should be made clear that the points under consideration is the one-fourth subset, not the integer lattice as a whole.
Rewrote this part. —David Eppstein (talk) 21:55, 12 September 2022 (UTC)
"add even numbers to each coordinate" ah, but we've only been speaking in abstract terms so far in this section. Perhaps prepend "interpreted geometrically" Ovinus (talk) 19:03, 11 September 2022 (UTC)
"form a three-dimensional lattice" supply the basis vectors? Ovinus (talk) 19:03, 11 September 2022 (UTC)
But then I would have to pick a basis? The lattice itself is already described: "translations that add even numbers to each coordinate (additionally, the offsets of all three coordinates must be congruent modulo four)". So some subset of three out of four of (4,0,0), (0,4,0), (0,0,4), (2,2,2) (but not the first three) can be used as a basis, but I'm not sure of a source that picks out one of those bases or any other one. —David Eppstein (talk) 02:09, 12 September 2022 (UTC)
"but does not give it the same geometric layout" doesn't have the same geometric layout, or doesn't endow it with a geometric layout at all? Ovinus (talk) 19:03, 11 September 2022 (UTC)
It assigns coordinates in 
  
    
      
        
          
            Z
          
          
            d
          
        
      
    
    {\displaystyle \mathbb {Z} ^{d}}
  
 to the vertices, which one can think of as a layout that is different than the standard one for this graph. —David Eppstein (talk) 02:06, 12 September 2022 (UTC)
"the theory of homology" any reason to not just say "homology"? Ovinus (talk) 19:03, 11 September 2022 (UTC)
I guess not. Removed. —David Eppstein (talk) 02:03, 12 September 2022 (UTC)
"four-edge dipole" any place we could link this? Otherwise, maybe gloss in a footnote or smth Ovinus (talk) 19:03, 11 September 2022 (UTC)
But the same term is already linked only one line up? —David Eppstein (talk) 02:02, 12 September 2022 (UTC)
Mention the minimum-count cycle in the Laves graph in the lead, I think. It's interesting, and the lead image is a bit... dense. My first impression was that the cycle count was eight, a la cyclooctasulfur.  Ovinus (talk) 19:03, 11 September 2022 (UTC)
Ok, done. —David Eppstein (talk) 01:03, 12 September 2022 (UTC)
Any way to put a period after the OEIS parentheses? Can't seem to figure that out. Ovinus (talk) 19:03, 11 September 2022 (UTC)
Done. You just put a period after the OEIS template, like "{{OEIS|A342235}}." —David Eppstein (talk) 00:51, 12 September 2022 (UTC)
I blame the visual editor Ovinus (talk) 03:27, 12 September 2022 (UTC)
Fixing a point on the Laves graph, what is the symmetry group about that point? Ovinus (talk) 23:36, 11 September 2022 (UTC)

  
    
      
        
          S
          
            3
          
        
      
    
    {\displaystyle S_{3}}
  
. Essentially the same thing is already stated in the article, as: "More strongly, for every two vertices 
  
    
      
        u
      
    
    {\displaystyle u}
  
 and 
  
    
      
        v
      
    
    {\displaystyle v}
  
, every one-to-one correspondence between the three edges incident to 
  
    
      
        u
      
    
    {\displaystyle u}
  
 and the three edges incident to 
  
    
      
        v
      
    
    {\displaystyle v}
  
 can be realized by a symmetry." The only part missing from that sentence to be sure it's really 
  
    
      
        
          S
          
            3
          
        
      
    
    {\displaystyle S_{3}}
  
 is the statement elsewhere that the whole graph is chiral, so mirroring in the plane of the three edges doesn't work. —David Eppstein (talk) 00:48, 12 September 2022 (UTC)
That makes sense, visually as well. Any source for that? Ovinus (talk) 03:20, 12 September 2022 (UTC)
That's all for now. Will do a second pass per usual after we discuss. As an aside, I realize that the letter of WP:PAREN deprecates the usage of {{harvnb}}, which is ridiculous. Haven't seen anyone instating an inferior alternative, but maybe that guideline should be adjusted to reflect actual practice. I kind of understand deprecating references like The sun is hot (Smith 1920, Doe 1999), but harvnb in this context seems totally anodyne. Ovinus (talk) 19:03, 11 September 2022 (UTC)

WP:PAREN is, literally, about references that use parentheses, and more specifically about such references in the article text. The uses of {{harvtxt}} (not {{harvnb}}, which does not use parentheses) in this article are not intended as references. You can tell that because the same text that incorporates harvtxt usages in the actual text, also has footnotes as references, which are not even always exactly the same as the names and dates incorporated into the text using harvtxt. If I were making parenthetical references in the old deprecated style, as references rather than the text of the article, they would use {{harv}}, not {{harvtxt}}. For instance, "Coxeter (1955) named..." is a piece of text, but "The Laves graph is embedded in this skew polyhedron (Coxeter 1955)." would be the deprecated format of parenthetical referencing. —David Eppstein (talk) 20:31, 11 September 2022 (UTC)
I agree with your interpretation; I just think the guideline should be edited to make that clear. Ovinus (talk) 20:51, 11 September 2022 (UTC)


### Second pass
Seeing the recent WP:ANI case I feel obligated to do some (somewhat pro forma) spot checks.
[1]: All good.
[3,5]: Fine
[9]: All good. For the first ref I like how it talks about duality but that's too much for this article.
[10]: Can't access
[15]: Fine, also WP:EXPERTSPS
[21]: Fine

I do appreciate the spot-checking, actually. Not because it indicates a lack of trust, but because I think it should be done more generally as part of all GA reviews, and because it has a chance of turning up places where I was sloppy in sourcing or didn't myself adequately check sourcing that others put in. —David Eppstein (talk) 07:33, 14 September 2022 (UTC)
Cool, will do that going forward. I definitely don't want to be a rubber stamp, although I don't think I've been thus far in my reviews. Ovinus (talk) 08:04, 14 September 2022 (UTC)
For the fundamental domain (this is new math for me), my understanding/visualization is a "half-open" parallelepiped to represent the quotient 
  
    
      
        
          
            R
          
          
            3
          
        
        
          /
        
        Λ
      
    
    {\displaystyle \mathbb {R} ^{3}/\Lambda }
  
, topologically a 3-torus, where 
  
    
      
        Λ
      
    
    {\displaystyle \Lambda }
  
 is our lattice, and the parallelepiped has volume 
  
    
      
        V
      
    
    {\displaystyle V}
  
. Is that correct? I think the fundamental domain should be also announced as a "parallelepiped", for a slightly broader understanding—people probably wouldn't confuse it with some other random choice of parallelepiped. Also, this is why it'd be nice to supply the basis vectors, so that the reader can more quickly follow along with the 
  
    
      
        
          L
          
            3
          
        
        
          /
        
        V
      
    
    {\displaystyle L^{3}/V}
  
 calculation. Is there a verifiability/OR concern there?

This is all pretty standard, and much of it probably sourceable in sources that talk about the body centered cubic lattice (which is what we have here, scaled by a factor of four), but I think the details are off-topic for this article. Fundamental domains are not always parallelepipeds. They can be any shape that tiles space under lattice translations. Escher's animals, maybe. For something I did recently involving 2d lattices I found it more convenient to use axis-parallel non-convex hexagons (see last image in https://11011110.github.io/blog/2022/04/03/dissection-into-rectangles.html). For the Laves graph a more intrinsic choice might be the union of four of those 17-sided plesiohedron things. For the BCC lattice, ignoring the Laves graph that it is the translational symmetries of, a simpler choice is the bottom half of the unit cube, with a tiling that arranges the half-cubes into planar checkerboards and then stacks up the checkerboards into 3d, but offset by a half-unit in each direction from one layer to the next. (These are not the parallelepipeds you would get from a basis, and their tiling is not face-to-face the way the parallelepiped tiling would be.) But you don't even need to decide on a fundamental domain in order to compute the volume. Just pile together your basis vectors as the rows of a matrix and calculate its determinant. The point of the L^3/V formula is that it is unitless, so the coordinates of your starting arrangement don't affect the result. But if we use the integer coordinates, we get four vertices and 12 halves of length-
  
    
      
        
          
            2
          
        
      
    
    {\displaystyle {\sqrt {2}}}
  
 edges per fundamental domain, so total length 
  
    
      
        6
        
          
            2
          
        
      
    
    {\displaystyle 6{\sqrt {2}}}
  
, and a fundamental domain with the volume of a half-cube scaled by four units, so volume 32. 
  
    
      
        (
        6
        
          
            2
          
        
        
          )
          
            3
          
        
        
          /
        
        32
        =
        27
        
          /
        
        
          
            2
          
        
      
    
    {\displaystyle (6{\sqrt {2}})^{3}/32=27/{\sqrt {2}}}
  
. Also, before you ask, we should not link to Body-centered cubic lattice in this article, because that goes to an article that describes it in terms of crystallography rather than mathematics, and crystallographically the Laves graph is very different than the BCC lattice even though mathematically the translational symmetries of the Laves graph are a BCC lattice. (Part of the issue is those other symmetries that are not translations.) So the link would be much more likely to confuse than illuminate. —David Eppstein (talk) 04:55, 14 September 2022 (UTC)
Exciting! As a student I think the article fundamental domain could include a simple example in R^n that's not just a parallelepiped. It makes sense that the choice is arbitrary. Is it too off topic for even a footnote? The determinant calculation seems straightforward, and perhaps enlightening in lieu of a detailed diagram of the integer coordinates/simple basis (but perhaps not straightforward enough for WP:CALC to apply). If that particular calculation is unsourceable then meh. Ovinus (talk) 05:42, 14 September 2022 (UTC)
I rewrote this paragraph to try to make it more self-contained. —David Eppstein (talk) 07:15, 14 September 2022 (UTC)
Beyond that, I'm quite happy with the article. Discussion regarding improvements/location of my STL can go on the article talk. Ovinus (talk) 02:37, 14 September 2022 (UTC)

Oh also, how is Laves pronounced? [1] suggests /'lavəs/, but I don't know German. Ovinus (talk) 05:42, 14 September 2022 (UTC)
I don't either. That looks like it could be a German pronunciation except that I'm pretty sure the vowel marked as a schwa would have a distinct sound from other schwa-vowels that most native speakers of English would be unable to distinguish. LAH-vess not LAH-vass or LAH-vuss even though those would all sound almost the same to English and American ears and be pronounced so briefly as to be heard as a schwa. But that's only a guess. —David Eppstein (talk) 06:17, 14 September 2022 (UTC)
Unfortunate. Anyway, I quite enjoyed your new explanation, in particular the part that the edges should cross, not partially lie within, the fundamental domain—cleared my remaining confusion. Will probably pass tomorrow. Ovinus (talk) 08:04, 14 September 2022 (UTC)

David Eppstein: Thoughts on this STL model? I might try reduce the file size; not sure why it's so large. (Edit: silly me, the cylinders are doubled.) I can also remove extraneous points, or perhaps omit points that aren't fully connected. Ovinus (talk) 20:52, 11 September 2022 (UTC)

Having a 3d model seems like a useful addition. I like the color and animation and smoother rendering in https://www.shadertoy.com/view/wddBRX better but I don't know that it's possible to incorporate anything like that here except as an external link. —David Eppstein (talk) 21:06, 11 September 2022 (UTC)
Ah, that is nice. Appears to use WebGL-based ray marching, which is super cool but impossible for us, and as far as I know Wikipedia doesn't support 3D file formats with coloring. Conceivably the balls could have different sizes, but I think that'd only work for two colors, maybe three. Do you think the cylinders/spheres should be thinner? Ovinus (talk) 21:12, 11 September 2022 (UTC)
I've plopped the STL file in Physical examples as I don't think there's particularly special place it should be put. Ovinus (talk) 02:38, 14 September 2022 (UTC)
Could a version of this be done, where the eye location is a bit farther from the grid? Or maybe even at infinity? The issue is that when rotating it, it is hard to see the symmetries. So, from one angle, you can see hexagons, but due to the perspective, the other nearby hexagons are ruined. There are other symmetries too, but the perspective blurs them out, making them very hard to discern. Ever-so-slightly smaller spheres and cylinders might help, so they don't collide when viewed from some of the symmetric directions. 67.198.37.16 (talk) 20:44, 1 December 2023 (UTC)