Notes on Notation

Cyclic Groups

We name our cyclic groups with Cn notation, where n is a number that corresponds to the number of rotations in a group. For example, we illustrate our C3 group with shapes that have 3 rotations.

Dihedral Groups

We use Dn notation to name our dihedral groups, where by Dn we mean the group with n rotations and n mirrors. For example, we illustrate the D3 group with shapes that have 3 rotations and 3 mirror reflections as symmetries.

Note that while many books use this same Dn notation, others use the D2n notation, where they would call our D3 group D6. Neither notation is better, they simply differ by academic field or the backgrounds of the writers - so watch out if you read a book about abstract algebra! This book uses the Dn notation rather than the D2n notation for ease and clarity, and because this is the notation more commonly used by those who stare at shapes (geometers).

Pattern Groups

In this coloring book, we use orbifold notation to name each of the frieze and wallpaper groups with symbols, such as ∗2222. In this section, we describe what this notation means, and how to decode it.

(∗2222)

There are a number of ways that mathematicians use notation to classify the frieze and wallpaper pattern groups. For example, we could have used the IUC notation to call that same ∗2222 pattern pmm. The IUC notation names a group by its generators. This can be confusing because of the ambiguity it presents: As we saw, many groups have multiple choices for generators.

The orbifold notation names symmetry groups by their symmetries. The orbifold names can be read as descriptions of the symmetries we can find in the patterns they name.

Before we talk about how to read the symbols in the orbifold notation, let’s talk about what an orbifold is.

(∗2222)

We can think of an orbifold as a quotient of a surface divided by a symmetry group.

Imagine taking a pattern and folding it up along its symmetries until we come to the smallest piece that can no longer be folded.

(orbifold for ∗2222)

This piece is the orbifold. The symmetries of the original pattern are features of this piece, and they can be interpreted as instructions for how to unfold it to get our pattern again.

The original pattern (∗2222) has mirrors and four different ½ turn rotation points where the mirrors intersect. These mirrors are the bounding sides of the orbifold, and the ½ turn rotation points are its corners.

Reading the orbifold symbols

Groups are named in the orbifold notation by a string of the following symbols.

Positive integers and the infinity symbol 1, 2, 3, 4, 5, 6, 7, ... ∞ indicate a rotation point with that many rotations.

∗ is used whenever there are mirror reflections.

x indicates glide reflection that is not the result of other symmetries in the pattern group.

o indicates translations that are not generated by other symmetries in the group.

Whenever a pattern has mirror axes that intersect, there is a rotation point at their intersection. The number of rotations around that point is the same as the number of intersecting mirrors.

Patterns can also have rotations points that do not sit on mirrors.

Any number in a pattern name that comes before a ∗ symbol describes a rotation point that does not sit on a mirror, while any number that comes after a ∗ symbol describes a rotation point that does sit on mirrors.

For example, 442 names a pattern group that has two different ¼ turn rotation points, and a ½ turn rotation point, and no mirrors.

(442)

∗442 names a pattern that also has two different ¼ turn rotation points and a ½ turn rotation point, but this time all of those rotation points sit on the intersections of mirrors. The ¼ turns are where 4 mirrors intersect, and the ½ turns are where 2 mirrors intersect.

(∗442)

4∗2 then names a pattern with a ¼ turn rotation point that does not sit on any mirrors, and a ½ turn rotation point that sits at the intersection of 2 mirrors.

(4∗2)

The names of the wallpaper groups use only the numbers 2, 3, 4, 6 because these are the only rotations possible for patterns drawn on an endless piece of flat paper, or the euclidean plane. However, more rotations are possible on other surfaces, such as spheres and hyperbolic planes.

What about the frieze groups?

Orbifold notation describes frieze patterns as if they were wrapped around an infinitely large sphere rather than following an infinitely long line. For this reason, instead of using the o symbol to indicate translation, they use the ∞ symbol to indicate infinite rotations. These points of infinite rotation are at the poles of the sphere, while the frieze pattern wraps around the sphere like an equator.

(∞∞)

(∗∞∞)

However, if the frieze pattern has a horizontal mirror or a ½ turn or a glide reflection, then the poles are identical. We can fold the pattern along these symmetries so that the poles meet, and the orbifold has just one point of rotation, so only one ∞ symbol is used.

(∞∗)

(22∞)

For more about orbifolds and orbifold notation, read “The Orbifold Notation for Two-Dimensional Groups” by John H. Conway and Daniel H. Huson.