We begin by introducing names for certain features that may occur in symmetrical patterns. To each such feature of the pattern, there is a corresponding feature of the quotient orbifold, which we will discuss later.
A mirror is a line about which the pattern has mirror symmetry. Mirrors are perhaps the easiest features to pick out by eye.
At a crossing point, where two or more mirrors cross, the pattern will necessarily also have rotational symmetry. An n-way crossing point is one where precisely n mirrors meet. At an n-way crossing point, adjacent mirrors meet at an angle of /n. (Beware: at a 2-way crossing point, where two mirrors meet at right angles, there will be 4 slices of pie coming together.)
We obtain a mirror string by starting somewhere on a mirror and walking along the mirror to the next crossing point, turning as far right as we can so as to walk along another mirror, walking to the next crossing point on it, and so on. (See figure 19.)
Figure 19: The quotient billiard orbifold.
Suppose that you walk along a mirror string until you first reach a point exactly like the one you started from. If the crossings you turned at were (say) a 6-way, then a 3-way, and then a 2-way crossing, then the mirror string would be of type *632, etc. As a special case, the notation * denotes a mirror that meets no others.
For example, look at a standard brick wall. There are horizontal mirrors that each bisect a whole row of bricks, and vertical mirrors that pass through bricks and cement alternately. The crossing points, all 2-way, are of two kinds: one at the center of a brick, one between bricks. The mirror strings have four corners, and you might expect that their type would be *2222. However, the correct type is *22. The reason is that after going only half way round, we come to a point exactly like our starting point.
In the quotient orbifold, a mirror string of type *abc becomes a boundary wall, along which there are corners of angles /a, /b, /c. We call this a mirror boundary of type *abc. For example, a mirror boundary with no corners at all has type *. The quotient orbifold of a brick wall has a mirror boundary with just two right-angled corners, type *22.
Any point around which a pattern has rotational symmetry is called a
rotation point. Crossing points are rotation points, but
there may also be others. A rotation point that does
NOT lie on a mirror is called a gyration
point. A gyration point has order n if the
smallest angle of any rotation about it is 2/n.
For example, on our brick wall there is an order 2 gyration point in the middle of the rectangle outlined by any mirror string.
In the quotient orbifold, a gyration point of order n becomes a cone point with cone angle 2/n.