Clarifying the meaning of the tiling/tessellation notations?

Question

I would like to come up with a final list of "tilings", but am having hard determining what the name or even a standard representation of the tiling is. Sidenote, it appears that the terms "tiling" and "tessellation" can be interchanged, where some are arbitrary repeated patterns (like M.C. Escher's works), and others have much higher symmetry. Is there a clear separation between these two groups somehow?

I have a few scattered questions revolving around this that I am hoping to clarify.

First we have this:

It shows the:

1. Wythoff symbol (i.e. q | p 2)
2. Coxeter diagram (i.e. )
3. Vertex figure (i.e. $p^q$ or q.2p.2p)

So the Wythoff symbol is up above each image, the Vertex figure down below.

A vertex configuration/figure is given as a sequence of numbers representing the number of sides of the faces going around the vertex. The notation "a.b.c" describes a vertex that has 3 faces around it, faces with a, b, and c sides.

For example, "3.5.3.5" indicates a vertex belonging to 4 faces, alternating triangles and pentagons.... 3.5.3.5 is sometimes written as $(3.5)^2$... The notation can also be considered an expansive form of the simple Schläfli symbol for regular polyhedra.

Then:

The Schläfli notation {p,q} means q p-gons around each vertex. So {p,q} can be written as p.p.p... (q times) or pq. For example, an icosahedron is {3,5} = 3.3.3.3.3 or $3^5$.

Then:

The Wythoff construction begins by choosing a generator point on a fundamental triangle... The three numbers in Wythoff's symbol, p, q, and r, represent the corners of the Schwarz triangle used in the construction.

• p | q r indicates that the generator lies on the corner p,
• p q | r indicates that the generator lies on the edge between p and q,
• p q r | indicates that the generator lies in the interior of the triangle.
• | p q r is designated for the case where all mirrors are active, but odd-numbered reflected images are ignored.

But then they write:

The sequence of triangles (p 3 2) change from spherical (p = 3, 4, 5), to Euclidean (p = 6), to hyperbolic (p ≥ 7).

Where did the (p 3 2) and (p = 3, 4, 5) come from, what does that mean? Also, in the Vertex figure, why do they use p and q and do q.2p.2p, instead of like the example of 3.5.3.5? Finally for this, why does the Wythoff symbol use a number 2 in there?

Then we have this on the Euclidean uniform tilings page:

It shows a Coxeter diagram, Wythoff symbol, Vertex figure, and then p4m, [4,4], (*442), where does that come from, what is all that? I think that may be a group theory thing? More of those here. For example on that last like (k uniform tilings), it has stuff like [$3^34^2$; $3^26^2$; (3464)2; 3446].

Sometimes they will list several Wythoff constructions, like here, what does that mean?:

Basically in summary, we have 4 (or 5?) notations:

1. Wythoff constructions
2. Vertex figures
3. Coxeter diagrams
4. Schläfli symbols
5. Group theory symbols? [$3^34^2$; $3^26^2$; (3464)2; 3446]

They are all focusing on different pieces of the puzzle. Wythoff tells us something about the triangle generator. Vertex figures with their numbers like q.2p.2p and their p's and q's tell us about the polygons around a vertex somehow. The Schläfli symbol makes sense. And I'm not sure what the "group theory symbols" mean. Would you mind clarifying how I can grab ahold better conceptually of Wythoff, Vertex figures, and the "group theory symbols"? Is the coxeter diagram stuff important? I don't understand that one yet, but i think knowing this much should get me further along.

Answer 1

I would like to come up with a final list of "tilings"

An ambitious goal. Grünbaum and Shephard, the authors of the book Patterns and Tilings, at first intended to make it just the first chapter of a book on geometry, but ended up with a 700-pages book just on this subject.

it appears that the terms "tiling" and "tessellation" can be interchanged

Yes, Grünbaum and Shephard say in p. 16 that the word tesselation is used "synonymously or with similar meaning" to "tiling"; English Wikipedia agrees.

Is there a clear separation between these two groups somehow?

One could draw the line in a variety of ways and degrees of generality. In particular, one could require the tiling to be invariant under some discrete group of isometries. In addition, one could require the tiles to be compact. In addition, one could require them to be polygons.

The sequence of triangles (p 3 2) change from spherical (p = 3, 4, 5), to Euclidean (p = 6), to hyperbolic (p ≥ 7).

Where did the (p 3 2) and (p = 3, 4, 5) come from, what does that mean? Also, in the Vertex figure, why do they use p and q and do q.2p.2p, instead of like the example of 3.5.3.5? Finally for this, why does the Wythoff symbol use a number 2 in there?

Wythoff construction is always based on a tiling by copies of a triangle, the Schwartz triangle; it is characterized by a (sorted) tuple of three positive integer numbers. That section of the Wikipedia article is an example, as follows from its title, they consider only tilings by right triangles, hence one of the numbers is always 2.

They mix two degrees of generality, assuming two arbitrary parameters $p$ and $q$ in the top three rows of the table but setting $q=3$ in the rest of the table and in the introductory paragraph. There is no good reason for that. However, the choice $q=3$ is motivated: $(6~3~2)$ is one of the few Schwartz triangle possible in the Euclidean plane. So if we fix $q=3$, then to list tilings by Schwartz triangles, we should consider $p=3,4,5,6,7\ldots$. So Schwartz triangles $(3~3~2)$, $(4~3~2)$, $(5~3~2)$ tile the sphere, $(6~3~2)$ tiles the plane, $(7~3~2)$ etc. tile the hyperbolic plane. These tilings are shown in the leftmost column. Each produces many different tilings depending on the particular Wythoff construction.

It shows a Coxeter diagram, Wythoff symbol, Vertex figure, and then p4m, [4,4], (*442), where does that come from, what is all that? For example on that last like (k uniform tilings), it has stuff like $[3^34^2; 3^26^2; (3464)2; 3446]$. Sometimes they will list several Wythoff constructions, like here, what does that mean?

These are different ways to describe the tilings.

• Wythoff symbol fully defines a tiling as described in the Wikipedia article devoted to it. Different Wythoff constructions can result in same tilings; however, if you superimpose the resulting tilings onto the original tiling by Schwartz triangles, you can get different pictures, and even equivalent tiles can become inequivalent this way (another way to look at it is to say that they originate from different parts of the original Schwartz triangles), cf. plane tilings "3 | 6 2" and "2 6 | 3" from the table from the beginning of your question. Apparently they even write vertex figures (see below) in different ways to highlight it. Although it doesn't matter if you are only interested in resulting tilings (both are simply hexagonal, in that case).
• A Coxeter diagram also describes the tiling. It can describe even tilings that cannot be described by a Wythoff symbol. If you ignore the difference between common, ringed, and empty nodes, then it only describes the symmetry group of the tiling. See its Wikipedia article.
• p4m, [4,4], (*442) -- these are different notations for the symmetry group of the tiling. p4m is the "crystallographic name", ​[4,4] is the Coxeter's bracketed notation, ​(*442) is the orbifold notation. For the case of Euclidean plane (both the symmetry groups themselves and all these notations), see the article on the wallpaper groups.

The vertex figure describes what tiles we encounter if we travel around a vertex (each of the inequivalent vertices, if there are many). In particular, $[3^34^2; 3^26^2; (3464)2; 3446]$ means the following:

• $3^34^2$, or $33344$, or $3.3.3.4.4$ (from the most concise to the most explicit way to write down the same information) means that there is a kind of vertex such that when we travel around it, we encounter 3 triangles and then 2 squares;
• $3^26^2$, or $3366$, or $3.3.6.6$ means that there is another kind of vertex such that when we travel around it, we encounter 2 triangles and then 2 hexagons;
• $(3464)2$ means that there is yet another kind of vertex such that when we travel around it, we encounter a triangle, then a square, then a hexagon, then another square (I don't know what do brackets followed by a $2$ mean);
• $3446$ must be the vertex figure of the last kind of vertex: a triangle, a square, another square, a hexagon.

Schläfli symbol is another way to write down a simple vertex figure. It is best suited for the generalization for regular (where all vertices and facets are the same) polytopes in higher dimensions.

---

How to comprehend it all? Well... First of all, focus on some particular domain. For example, (temporarily) ignore all hyperbolic stuff, ignore stellar polyhedra, ignore higher dimensions, ignore Coxeter's bracketed notation, ignore everything unrelated to some chosen thing you're studying. However it is important to understand that polyhedra and tilings of the sphere are the same thing in this context. Wythoff construction is a thing that can be understood in an isolated way, so it can be a good start. Symmetry groups are an important concept, so it would be useful to study the platonic bodies and the wallpaper groups. Coxeter diagrams and (very closely related) Dynkin diagrams are a very big deal in many other contexts, so it may be a good idea to look at them, too, although this way it's easy to quickly go down the rabbit hole and get frustrated.

It's all for now, I am leaving the opportunity to recommend comprehensible books about tilings to others. Perhaps The Symmetries of Things is a good choice.