Generating sets and fundamental regions

A generating set for the symmetry group of a plane pattern P is a set S of symmetries of the pattern such that every symmetry of the pattern is a product of transformations which are either elements of S or inverses of elements of S.

A pattern can have many different generating sets. In Kaleidomania one particular generating set is chosen for each type of plane symmetry group.

For example, what are possible generating sets for the "zigzag" strip pattern below? The symmetry group of this pattern contains the identity, translations, rotations (half-turns), vertical reflections, and glide reflections. In other words, in the notation introduced in class this pattern has type TRVG. It has crystallographic symbol mg, and Conway symbol 2*∞.

The shortest translation vector is from one top vertex to the next top vertex. The shortest translation vector is the vector AB in the figure below. The rotation centers occur where the center line intersects the figure. The center line is the dotted line below. Two of the rotation centers are the points O and P below. A vertical reflection mirror passes through each of the vertexes of the figure. Two such mirrors are the blue lines shown below. The shortest glide vector is half the shortest translation vector. It's the vector CD below.

What are some possible generating sets for the symmetry group of this strip pattern?

Let v be the vector AB, and let w be the vector CD. Let T_v be the translation with vector v, let G_w be the glide with vector w, let R_P be the half-turn with center P, let R_O be the half-turn with center O, let V_m be the reflection with mirror m, and let V_n be the reflection with mirror n.

Some generating sets for this pattern are:

{T, V_m, R_P}

{V_m, V_n, R_P}

{R_P, R_O, V_m}

{G_w, V_m}

The set of generators chosen by by Kaleidomania for this pattern is {T, V_m, G_w}.

A fundamental region for a plane pattern P is a region R of the plane such that:

(1) The whole plane can be obtained from the fundamental region R by moving R around using the symmetries of the pattern. If R' is obtained from R by a symmetry transformation of the pattern, then R' is called an "image" of the fundamental region R.

(2) No region contained in R (except R itself) has property (1).

There are two marvellous consequences of properties (1) and (2):

(a) The images of the fundamental region do not overlap. Thus these images form a "tiling" of the pattern by congruent "tiles"! (The outlines of this tiling form what in Kaleidomania is called a "grid.")

(b) There is a one-to-one correspondence between the images of R and the symmetries of the pattern. More precisely, for every image R' of R there is a unique symmetry F of the pattern P such that F(R) = R'.

Fundamental regions in Kaleidomania are chosen to be as simple as possible. A pattern can have many different fundamental regions.

For example, a fundamental region for the stip pattern above is the yellow strip (thought of as extending to infinity) illustrated below. This is the fundamental region used by Kaleidomania for this type of strip pattern.