Are the vertices in triangular tiling, or in hexagonal tiling, considered to be a mathematical lattice?
e.g. excuse the different colours of the vertices but pretend they're the same colour.
An example of hexagonal tiling
Or of triangular tiling (ignore the different colours and pretend the vertices aren't joined up)
I did once read that to be a lattice in the mathematical sense, it has to meet the criteria of being "a group" https://en-academic.com/dic.nsf/enwiki/11776 so every point should be able to be determined by an algebraic formula. So i'm wondering if maybe those don't meet the criteria.
I did read that the vertices in trihexagonal tiling doesn't meet the group criteria of the definition of what would be a mathematical lattice. I want to check if my suspicion is correct that the vertices in triangular and hexagonal tiling don't meet it either. So I suspect that the answer to my question of "Are the vertices in triangular tiling, or hexagonal tiling, considered to be a mathematical lattice?" Is No, but I want to check.
(I see a bunch of different lattice tags and am not sure which one to choose but I chose a popular one that hopefully is general enough for my question) (added- I don't see a tag as general as lattice in group theory..
added
I see there are two definitions of lattice, there's lattice in order theory https://en.wikipedia.org/wiki/Lattice_(order) And there's lattice in group theory https://en.wikipedia.org/wiki/Lattice_%28group%29
And from what I understand, the examples I give don't meet the definition of a lattice in order theory. I wonder though if they all meet the definition of a lattice in group theory? Or if some do and some don't.
amrsa's comment has led me to what might be an answer.. So if he wants to post an answer i'll accept that answer..or if anybody else posts an answer..
I'm no mathematician, and my answer(which may be wrong somewhere), is that there's two types of lattices. A lattice in order theory(known as an ordered lattice), and a lattice in group theory(which I think is known in physics as a lattice model). There is also a term "lattice graph"(which I think without edges would be a lattice in group theory).
I think that of the examples I show or mention...
A trihexagonal one (which I didn't picture), is, when including lines, known as a Kagome Lattice https://en.wikipedia.org/wiki/Trihexagonal_tiling#Kagome_lattice And when not including the lines, so, when looking just at the crossing points, wikipedia says, they "do not form a mathematical lattice" (and when wikipedia says lattice, it refers to the lattice in group theory. So, based on that plus the comment from amrsa, I conclude that it doesn't fit either the criteria for a group theory lattice, or the criteria for an ordered lattice(which seems to be a stricter criteria. (It looks to me like maybe the group theory lattice is a superset of the ordered theory lattice). And I think the reason why the crossing points of the Kagome lattice / crossing points of the trihexagonal tiling, don't meet the definition of group lattice, is to do with it being composed of two different shapes. That's a big difference between that and the other examples.
As for the other ones. Based on amrsa's comment that they don't make for an ordered lattice, i'll take his word for it that they don't. But I think they do fit the definition of group theory lattice. Since looking at the wikipedia page for group theory lattice I see a picture of
And while one could look at that as a triangle and an upside down triangle.. that could be seen as one shape, a diamond shape, that is repeated. But also even the idea of triangles oriented one way and oriented another way, seems to fit for example the wikipedia "lattice(group)" page includes this picture which seems to me to have the same dots as the one above, and shows that one can tile it with those triangles
And it looks to me like it could also be tiled with hexagons.
And that's one polygon repeated so i'd think it'd meet the group theory definition of lattice. I did hear that it'd meet the definition of lattice graph (but not lattice). Though now I understand there's two definitions of lattice, and it makes sense that there's a group theory definition of lattice that correlates to the concept of a lattice graph. And when many mathematicians think of lattice, they think of ordered lattice and it wouldn't meet the definition of that.