The path to understanding Frieze Groups

Question

What is the "Path" for understanding what Frieze Groups really are?

Generally in mathematics, there is a is a path or "building blocks" approach to learning something. For example if I know how to count and I want to learn how to multiply you say the path is "Counting -> Adding (adding is fast counting) -> Multiplying (multiplying is fast adding) -> Exponents (exponents are fast multiplying)"

The starting point of your answer should be "From the basic axioms of a Group, you should study ConceptA, then ConceptB (with maybe a description of how it is related to prev concepts), etc..., etc..."

Group Theory seems to have so many branches, I get lost in the trees. I just want to know what to focus on one step at a time.

Please understand that I am just learning Group Theory (on my own, no school/university). I need basics. From what I have read, I understand that Frieze Groups are related to Wallpaper Groups and Crystallographic Groups, as you increase the dimensions and/or reduce restrictions. If it there is a corollary in a more simple group (Permutation Groups? i.e. permutation "cycle" structure?) then describe that.

It is hard to ask questions in "new" areas of study. Sometimes you don't even know what to ask, or the questions you ask simply reveal that you don't understand the basics of what you are studying. If my question reveals to you that I do not understand something, then please help me to know what I don't know to ask about :)

BTW, this is my first math.stackexchange post!

Answer 1

In a word: symmetry.

For example, you may have already encountered the set of all symmetries of the square: all the ways of placing a square on top of itself. There are 8: four rotations (including the 360 degree rotation), plus four reflections (flips). These form a group under composition (first do one symmetry, follow it with another).

The frieze groups are the same idea: we are given an infinite geometrical pattern, and ask for all the ways we can place the pattern on top of itself.

An excellent (and classic) refernce is Hermann Weyl's Symmetry, freely available on-line: Symmetry. Also there are many good websites devoted to the wallpaper groups and more generally to crystallographic groups. There's a fine set of notes available on-line, "Geometry and Groups", by T.K. Carne, particularly a propos to your question.

I should add that most of the groups studied in math are symmetry groups of one kind or another, although the kind of symmetry is often more abstract. Here's a semi-formal definition: let $S$ be a "structure" of some kind (e.g., a geometrical figure, or a algebraic object like a group or a field). Consider the set $\text{Symm}(S)$ of all mappings from $S$ to itself, $f:S\rightarrow S$, which "preserve the structure of $S$" and also have an inverses $f^{-1}$ which also "preserve the structure of $S$". The notion of "preserving the structure" has to spelled out precisely in any particular case. But the basic idea is that $f(S)$ "looks exactly like $S$". Then $\text{Symm}(S)$ is a group, and as I said, most of the groups studied in math are symmetry groups for some notion of structure.

One notational caveat: usually, $\text{Symm}(S)$ stands for all one-one, onto mappings of a set $S$ to itself. In other words, just the structure $S$ has as a set. Symmetries preserving a structure are often called automorphisms, so a better (and common) notation would be $\text{Aut}(S)$.

Plunging down a level (in response to the OP's comment): Just for the definition of a frieze group, there's not much more to say. Let's consider the more general context of wallpaper groups. We have a subset $S$ of the plane and we want to look at all isometries of the plane to itself that map $S$ onto itself. Depending on $S$, you can get all kinds of groups. Let's write $G$ for the group.

Further investigation involves both a lot of geometry and more group-theory concepts. Suppose there is a minimum distance $d$ such that every element of $G$ moves any point $x$ at least distance $d$, unless it leaves $x$ fixed. Then $G$ is called discrete. (So the group of all rotations of a circle is not discrete.) Suppose $G$ is discrete, and also there is a compact (closed and bounded) set $K$ such that applying all the elements of $G$ to $K$ covers the plane (i.e., $\bigcup\{g(K):g\in G\}$ is the whole plane). Then $G$ is said to be a crystallographic group.

The frieze groups are similar, but instead of the plane, you consider an infinite strip $\{(x,y):0\leq y\leq 1\}$.

The basic idea is to classify the sets $S$ by looking at the structure of $G$. If the $G$'s for two $S$'s are isomorphic as groups, that means that the $S$'s "have the same symmetries".

Historically, I think there were two sources for this theory. First, crystals are, at the atomic level, regular arrangements, sort of like 3d wallpaper. So there was strong motivation coming from physics to classify possible symmetries, especially since some of the symmetry is visible to the naked eye with well-formed crystals. This classification was achieved around the beginning of the twentieth century. It's a tedious problem, so it's natural to work on the simpler 2d and 1d cases. The frieze groups are kind of the 1-1/2 dimensional case.

Second, the huge number of examples in decorative art (everyone always mentions the Alhambra here) provided more motivation.

Three more references: Crystallography and Cohomology of Groups, an award-winning article from the MAA; Plane Symmetry Groups, another MAA article; and chapter 1 and appendix A of Sternberg's Groups and Physics.

So to come back to your original question: the path to understanding frieze groups (and wallpaper, etc. groups) is mostly to follow the geometry. Some general group-theory concepts do emerge, but a lot of the theory is rather specific.

Finally, if by "building blocks" you mean something like the way a finite abelian group is always isomorphic to a direct sum of cyclic groups, I don't believe the frieze groups (or the wallpaper or 3d crystallographic groups) can be classified that way.

Answer 2

Here is a more nuts-and-bolts answer to the question of a path to understanding frieze groups: a quick outline of the proof of the classification theorem(s). The structure of the proof is basically the same for frieze groups, and for 2d and 3d crystallographic groups, although of course there are many more cases to consider as one ups the dimensions. To keep the verbiage down, I'll just assume wallpaper groups below.

(1) Define the notion of a wallpaper group, that is, a discrete group of symmetries of the plane with a fundamental domain $K$ (see my previous answer).

(2) Prove that a wallpaper group $G$ possesses a lattice, that is, a subgroup $L$ of translations that is itself discrete with a fundamental domain, and has a basis of 2 vectors. Also, the basis can be chosen to be an integral basis. All this is surprising hard to prove. Often one finesses the issue by defining a wallpaper group to be one that possesses a lattice.

(3) Show that $L$ is a normal subgroup of $G$, and call $G/L$ the point group. Let's write $P$ for the point group.

(4) Show that $P$ is isomorphic (in a natural way) to a finite group of symmetries of the plane that leave the origin fixed.

Caveat: $P$ is not necessarily equal to the subgroup of $G$ consisting of all symmetries in $G$ that leave the origin fixed. Rather, a symmetry in $G$ might rotate the plane (say through 90 degrees) while simultaneously sliding the plane a distance. In $G/L$, we ignore the sliding aspect and just retain the rotation.

(5) Now classify all possible point groups. For the plane, the only rotations that can occur are through angles of 60, 90, 120, and 180 degrees. This is known as the crystallographic restriction. Based on this, it is easy to list all possible point groups. They are all finite, and are either cyclic groups of order 1,2,3,4, or 6, or so-called dihedral groups of order 2,4,6,8, or 12. (For frieze groups the story is a little simpler, for 3d crystallographic groups rather more complicated. Still, you end up with a finite list of finite groups.)

(6) Now we have an abelian subgroup $L$ of $G$, well-understood. Also we have a list of all the possible quotients $G/L$. Problem: use this data to classify all possible $G$'s up to isomorphism. In general, this is the basic problem of group cohomology. As it happens, this special case is one of the best entry points into that subject.

So far as a "path" goes, the following might work. (I'm not sure what your goal is: just frieze groups, or understanding symmetry groups more generally?)

Carry a running example or two along as you learn the concepts. I suggest the dihedral group $D_{2n}$, which is the group with $2n$ elements of all symmetries of a regular polygon with $n$ sides. $D_6$ (symmetries of the triangle) and $D_8$ (symmetries of the square) are common examples. (Note: some authors write $D_n$ instead of $D_{2n}$ for this group.)

(a) Definition of a group. A group with $n$ elements is said to have order $n$. Definition of an abelian (aka commutative) group.

(b) Definition of a subgroup. Example: the rotation subgroup $C_n$ of $D_{2n}$. Also, these $n$ subgroups of order 2 of $D_{2n}$: for each diagonal, there is a reflexion $r$ on that diagonal; the set $\{1,r\}$ is a subgroup of order 2.

(c) Definition of isomorphism. Two groups that are "basically the same" are isomorphic. Examples: $C_2$ is isomorphic to any of those order 2 reflexion groups just mentioned. (And they are all isomorphic to each other.) In fact, all groups of order 2 are isomorphic.

(d) Definition of coset. Related (not specifically group theory): definition of equivalence relation, equivalence class, partition of a set.

(e) Definition of a homomorphism, kernel, normal subgroup, and quotient group. Examples: $C_n$ is a normal subgroup of $D_{2n}$, the groups $\{1,r\}$ are not normal, and there is a homomorphism from $D_{2n}$ to a group of order 2 with kernel $C_n$. So the quotient group $D_{2n}/C_n$ is a group of order 2.

Now geometry:

(f) Definition of the group $\text{Isom}(E)$ of all isometries of the Euclidean plane $E$.

(g) Structure of isometries: every isometry is either a translation, or a rotation around some point, or a reflexion across some line followed by a translation. Alternately, every isometry can be expressed as $p\mapsto Rp+t$ where $R$ is either a rotation about the origin, or a reflexion across a line through the origin, and "$+t$ means translation by the vector $t$.

(h) Wallpaper groups: subgroups of $\text{Isom}(E)$ with the properties I listed earlier.

You will find a development somewhat along these lines in the references I gave, or in the reference another poster gave (Johnson, Symmetries). (I'm not familiar with Johnson's book, but the table of contents looked good.)