2. Introduction to Orbifolds

2.2. Euclidean 2-Orbifolds from Plane Groups

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There are 17 plane groups (wallpaper groups) defining the symmetry in all patterns that repeat by 2-dimensional lattice translations in Euclidean 2-space. We will derive the 17 Euclidean 2-orbifolds directly from standard crystallographic plane group drawings, but first we illustrate some basic surfaces of topology.

**Surface Topology**

Fig. 2.5 illustrates how
rectangles when wrapped up to superimpose identical edges give rise to
five basic topological surfaces present in the plane group orbifolds.
The other two surfaces needed are the the 2-sphere and 2-disk discussed
previously
in Sect. 2.1. The arrows on the edges of the rectangles indicate
directional specific patterns that are to be superimposed and glued
together. A surface illustration of the Klein bottle also is
shown in the header of these web pages. The projective plane surface
construction is illustrated in two steps. The intermediate stage is a
sphere with a hole in it that has an antipodal relationship along the
gluing edge of the hole; i.e., every point on the edge is identical to the
corresponding point 180^{o} from it along
the edge of the hole. This can be considered an antipodal operation
along the edge. The final step closes up the hole by puckering two
opposite points down while the two other points 90^{o}
from the first pair are puckered up, forming a pinched light
bulb type end called a crosscap. For graphical simplicity,
we will always draw the
projective plane using the intermediate stage which has the antipodal
gluing edge unmated.

**Plane Group Orbifolds**

Crystallographers have specific symbols for the geometrical symmetry operators corresponding to 2-, 3-, 4-, and 6-fold rotations (polygons); mirrors (heavy lines); and glides (dashed lines) along with conventions for how the symmetry drawings are laid out (Hahn, 1995). In Fig. 2.6. the double lines indicate where folding takes place, and the shaded lines are where cutting is done. After cutting, symmetry equivalent edges are pasted together to form the Euclidean 2-orbifolds at the bottom of each box.

The notation under the crystallographic drawing is the standard plane group name and that under the orbifold drawing is our notation for the Euclidean 2-orbifold with S, D, and RP denoting sphere, disk, and real projective plane, respectively. "Möbius" denotes a Möbius band with 1 silvered edge, and "Annulus" denotes an annulus with 2 silvered edges. S2222, S333, etc. are called pillow orbifolds and have the constraint that for Sijk..., (i-1)/i + (j-1)/j + (k-1)/k + ... = 2. Heavy lines and circles indicate mirrors, and a heavy dashed circle, arising from a glide, signifies a projective plane antipodal boundary. Primed numbers indicate the corresponding rotation axis lies in a mirror forming a dihedral corner, and unprimed numbers indicate cone points.

The spherical orbifolds in the third row are derived by using straight line cuts through 2-axes and appropriate angular cuts at other axes to leave some flaps which are then glued together to produce the 4- and 3-cornered pillow spherical orbifolds. The diskal orbifolds on row four simply require cutting along the double lines in the plane group drawings. The remaining diskal orbifolds (rows one and two) are derived by cutting along the double lines and along appropriate angles through the single axis pointed to by vectors perpendicular to the ends on double lines, then closing up the cut edges through the axis to form a complete silvered boundary.

The annulus and Möbius band in row one are derived from plane groups pm and cm by first cutting out an asymmetric unit bounded by those portions of the mirrors denoted by double stripes and matching the ends together as indicated in Fig. 2.5. The torus and Klein bottle asymmetric units require the whole and 1/2 unit cell, respectively, each folded as indicated in Fig. 2.5. The apparent self-intersection in the Möbius band is just a limitation of illustration techniques. The rules are that a manifold (or orbifold) can be embedded in whatever dimension Euclidean space is required. The Klein bottle (and the projective plane) can be mapped into 4-dimensional Euclidean space with no self-intersections.

**The Projective Plane**

For the projective plane orbifold, RP22, 1/4 of the unit cell is required for the asymmetric unit. At first we choose an asymmetric unit with a 2-axis on each corner and fold up as indicated in Fig 2.5. This places all four 2-axes on the dashed circle where the antipodal relationship holds so that it looks pictorially like the D2'2'2'2' symbol with the dashed boundary replacing the mirror boundary. However, we then note that by moving the asymmetric unit one quarter cell in either the x or y direction, there are now two 2-fold axes centered on opposite sides of the asymmetric unit as shown in Fig. 2.6. Folding about these 2-fold axes positions them in the interior of the orbifold as shown in the RP22 orbifold figure and there is still an antipodal relationship along the gluing edge. Thus, we can push two nonequivalent pairs of equivalent axes off the boundary to get two nonequivalent axes in the interior of the projective plane orbifold, or vice-versa, while still maintaining the antipodal gluing edge relationship. Only the projective plane has this amazing "sliding" gluing edge property.

**Lifting Plane Group Orbifolds to Space Group Orbifolds**

The ITCr lists the projection symmetry plane groups along three special axes for each space group. Different crystallographic families have different unique projection axes. For example a cubic space groups has special projected symmetries along (001), (111) and (011) while the orthorhombic special directions are (100), (010) and (001). Space group nomenclature used by crystallographers also follows this trend by listing generators for each unique axis with nontrivial projection symmetry.

Much of the orbifold topology literature (e.g., Bonahon and Siebenmann, 1985) uses a Euclidean 2-orbifold as the base orbifold, which is lifted into a Euclidean 3-orbifold using the Seifert fibered space approach (Orlik, 1972) while keeping track of how the fibers (or stratifications) flow in the lifting process. This works only for the 194 non-cubic space groups since the body-diagonal 3-fold symmetry axes of the 36 cubic space group violate the Seifert fibered space postulates. However, there are some work-around methods using 3-fold covers which let you derive the cubic Euclidean 3-orbifolds from their corresponding orthorhombic Euclidean 3-orbifolds. (See Sect. 6.)

Several examples are shown below for polar (orientable) space groups
derived from point group 422. Many orientable space groups have
underlying space S^{3} (i.e., a
hypersphere embedded in 4-Euclidean space) and are relatively
easy to draw in 3 dimensions since there are no symmetry
planes involved. Fig. 2.7 illustrates five different fibrations of Euclidean
3-orbifolds over the
2-orbifold D4'4'2',
corresponding to space groups I422 (97), P422 (89),
P4_{2}22 (93),
I4_{1}22 (98) and
P4_{1}22 (91).
The base Euclidean 2-orbifold is in
the middle of Fig. 2.7 and the Euclidean 3-orbifolds are in the top
halves of the boxes with singular set drawings in the bottom half. The
numbers of independent Wyckoff sets (i.e., elliptic 2-orbifolds) are
shown in the attached smaller boxes.

Note the correspondence between the 3-orbifold symbol and the singular
set drawing. In P422 we are looking down a trigonal prism fundamental
domain with vertical 4-axes along two edges and 2-axes along the seven
other edges and there are six trivalent intersections at the corners.
In I4_{1}22 the two 4-axes become 4-fold screws, one right
handed and one left handed, Also note that the twisted pair of 2-fold
axes has the opposite handedness to that indicated by the symmetry
symbol. In P4_{2}22 the P4_{2} axes become 2-fold
screw axes with 2-axis struts across the 2-screw loops since a
4_{2} axis contains both a 2-fold axis and a 2-fold screw
subgroup. The P4_{1}22 singular set diagram is called a link
since there are no connections between the three 2-fold axis Wyckoff
sets. If you have a copy of ITCr, check the close correspondence
between the symbols in Fig. 2.7 and the ITCr symmetry
drawings.

We do not currently use this lifted 2-orbifold convention since we now prefer to construct orbifolds from the full 3-dimensional fundamental domain. However most of the orbifold literature does use some variety of this convention and the existing 3-orbifold nomenclature usually is based on it. The reason is that the topological classification of 2-manifolds (surfaces) is classical and well understood, but 3-manifold classification is still incomplete. A manifold is an orbifold without a singular set.

3. Introduction to Critical Nets

2.1. Elliptic 2-Orbifolds from Point Groups

Crystallographic Topology Home Page

*Page last revised: June 3, 1996*