Talk:Heawood graph

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Geometric and topological properties paragraph

Hi @David Eppstein: I was hoping to talk a bit about the first paragraph of the section on geometric and topological properties. In my view, I find the current text to be pretty confusing. It discusses a geometric realization and a topological realization as if they were different instances of the same kind of thing, which they are not. The Szilassi polyhedron is a geometric realization of the map / abstract polytope. I attempted to change it to clarify that relationship, but you have reverted it. I don't quite understand what you are saying in your revert summary, so I thought I'd elaborate my perspective and ask you to elaborate yours here. AquitaneHungerForce (talk) 22:17, 2 February 2024 (UTC)

You changed it to conflate two very different geometries, calling them the same: one being a tiling of a flat torus by geometrically regular hexagons and the other being the faces of the Szilassi polyhedron. They have the same topology but your version gives only the Szilassi geometry; it removes the geometric description of the flat torus. Additionally, you instead described the abstract regular map as being made of regular hexagons, confusing different levels of description; the abstract map is a topological structure without a geometry. —David Eppstein (talk) 23:19, 2 February 2024 (UTC)

Ok. It was unclear to me that the second description is geometric. I can see now that that is the intention. I think we should make it clearer what the geometry actually is. It's also a pretty informal description, I think it could benefit from being more precise. I also still think this is conflating two pretty distinct notions. The Szilassi polyhedron is a faithful realization in Euclidean space while the flat torus is a faithful and symmetric realization (symmetric on the map not the graph). They are notable for entirely different reasons. AquitaneHungerForce (talk) 23:33, 2 February 2024 (UTC)

The geometry of a flat torus is exactly a quotient of the Euclidean plane by a lattice, or the result of gluing a Euclidean parallelogram's opposite edge pairs. It is described in non-technical terms in this article, for reasons of WP:TECHNICAL, but there is nothing actually imprecise in the description. Flat tori can also be embedded in Euclidean space: for instance, the subset of $\mathbb {R} ^{4}$ with $x^{2}+y^{2}=z^{2}+w^{2}=1$ is a flat torus (a square one, not the hexagonal one used here). —David Eppstein (talk) 23:55, 2 February 2024 (UTC)

This coversation is beginning to fork a bit so I will separate two points.

1. Yes we don't want to run foul of WP:TECHNICAL .I think there are two things here that can be made more precise without any issue there. I think it should be clear what is meant by "folding", maybe wikilinking to a page on quotients would be sufficient. I think it should also be precise in the kind structure it is building. On my initial reading I thought this was entirely topological. It does call it a realization, but it requires the reader to fill in most the gaps to get to a realization. At least in the sense of McMullen & Schulte, maybe realization has a different definition that this article is using. I think this bit could be clearer with the meanings of a couple of its terms. Let me know what you think.
2. The conflation is not between Euclidean and the flat torus realizations. As you point out the flat torus does have a natural Euclidean analog, but it's not full rank. The distinction is between full rank and faithful versus faithful symmetric.

AquitaneHungerForce (talk) 00:27, 3 February 2024 (UTC)

To step back for a second, I think, based on our conversation, there are two key facts this section is trying to convey:

1. The Heawood graph is the skeleton of the Szilassi polyhedron.
2. The Heawood graph is the skeleton of a toroidal map with a high degree of symmetry (chiral).

Is this a correct assessment or is there something more? AquitaneHungerForce (talk) 00:35, 3 February 2024 (UTC)

It's the skeleton of a topological toroidal map with a high degree of symmetry, and all the same symmetry can be realized as a geometric structure. The flat torus is geometric, no more and no less than Euclidean 3-space is geometric. —David Eppstein (talk) 03:02, 3 February 2024 (UTC)

Yes, naturally. The geometry comes for free. I'll make an edit tomorrow. Thank you for your help. AquitaneHungerForce (talk) 03:55, 3 February 2024 (UTC)

I tried a couple drafts but I've had some issues. I've decided to find some sources so I can see how authors with a different perspective discuss it and to establish which details are notable. It will probably take a little longer then. AquitaneHungerForce (talk) 02:48, 4 February 2024 (UTC)

I rewrote the section based on the sources I could find. I really doubt the geometric structure on the torus is notable, since every regular map has a space form and this one doesn't seem to have an special properties. However if there is a source discussing this, I'd love to see it, and I would support restoring this information (and hopefully further) to the article. AquitaneHungerForce (talk) 17:55, 9 February 2024 (UTC)

One case for its significance is that this geometric structure, unrolled to its universal cover on the plane, gives the best-known upper bound for the Hadwiger–Nelson problem. —David Eppstein (talk) 00:12, 10 February 2024 (UTC)

Ah interesting. That gives me a new term to use in my literature search. I'll have another look and see what I can add. Thanks. AquitaneHungerForce (talk) 03:30, 10 February 2024 (UTC)

It's probably going to be easy to find sources about the resulting 7-coloring of a hexagonal tiling of the plane [1] [2], and hard to find sources that say that the skeleton of its quotient is the Heawood graph. —David Eppstein (talk) 06:29, 10 February 2024 (UTC)

The geometric realization by regular hexagons on a hexagonal flat torus is discussed in [3] which calls it "probably the most famous regular tiling of the torus". —David Eppstein (talk) 07:11, 10 February 2024 (UTC)

Also Coxeter 1957 gives a very geometric rather than topological description of regular torus maps (for instance referring to 120° angles) that includes an explicit reference to Heawood and his seven-color example (but described as its dual triangular tiling of a torus) on pp. 106–107 of [4]. —David Eppstein (talk) 07:19, 10 February 2024 (UTC)

Ceballos and Doolittle [5] write "Our favorite representation of Heawood’s graph is Leech’s highly symmetric representation using regular hexagons". Here the Leech reference is to Fig.2 of "John Leech. Seven region maps on a torus. Math. Gaz., 39:102–105, 1955" [6]. Leech begins his paper by discussing exactly the construction you removed: cutting out a parallelogram from the plane and gluing opposite edges. —David Eppstein (talk) 07:25, 10 February 2024 (UTC)

Nice. At this point I would suggest we break the Heawood map off to its own section. I'll try to incorporate the information from these references into that unless you would like to do so first. It will likely take me a while to obtain all these references, and even then it would take a bit to synthesyse that into prose. AquitaneHungerForce (talk) 06:10, 11 February 2024 (UTC)