# “[T]ransversely isotropic and mirror-symmetric (space group:$D_{\infty h}$)”, its orbifold notation?

I am trying to understand this frieze pattern $D_{\infty h}$ aka its orbifold notation. This describes spider's silk. The authors call it a space group, some sort of generalization from orbifolds.

Please, explain the term "[T]ransversely isotropic and mirror-symmetric (space group:$D_{\infty h}$)" in Conway-and-Thurston notation aka orbifold notation.

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This frieze group ($D_{\infty h}$) is called $*22\infty$ in orbifold notation or $p2mm$ in the crystallographers' notation. Like other frieze groups, it is a group of plane symmetries which contains a one-dimensional lattice of translations. If you take the generator of the translational lattice to be in a horizontal direction, then the group can be generated by this translation together with a reflection about a horizontal line and a reflection about a vertical line. This is why it is being called transversely isotropic and mirror-symmetric. The group also contains other symmetries generated by these such as additional reflections about vertical lines, $180^\circ$ rotations, and glide-reflections.

The orbifold notation comes from taking a quotient of the plane by the symmetry group. After doing this, you no longer have a $2$-manifold (something locally homeomorphic to the plane) but an orbifold, which is locally homeomorphic to quotients of the plane. The $*$ denotes the presence of a boundary crease where the orbifold has been folded upon itself by reflectional symmetry. The two $2$s denote the presence of two corner points in this crease where vertical mirrors meet the horizontal mirror. The $\infty$ symbolizes the open end of the orbifold at vertical infinity.

For a picture of a plane figure with this symmetry, see for example here, where the group is called $F_7$, or here, where it is called Class IV.

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