Symmetry in Two Dimensions

Abstract

Here we simplify matters by considering only patterns in two dimensions, in which case there are just four planar systems, five lattice types, and 17 plane groups.

Symétrie, en ce qu’on voit d’une vue, fondée sur... la figure de l’homme. D’où il arrive qu’on ne veut la symétrie qu’en largeur, non en hauteur, ni profondeur. (Our notion of symmetry is derived from the human figure. Hence, one wants symmetry only in width, not in height, nor in depth.)

—Blaise Pascal

Review Questions

• 1. Define the term plane group.

• 2. A planar pattern with a geometrically square unit cell containing two lattice points can best be described in which planar system?

• 3. The pattern below belongs to which planar system?

  

• 4. The pattern below belongs to which planar system?

  

• 5. The planar stone lattice-work pattern pictured below is an example of a jaali and forms part of a screen outside the Sheesh Mahal above the Aram Bagh gardens in the Jai Mandir part of the Amber Palace, near Jaipur, Rajasthan, India. In this picture:

  Identify with the appropriate symbols all of the nontrivial proper rotation axes.

  Use appropriate lines to indicate the positions of any mirror or glide planes.

  Sketch in the smallest possible primitive unit cell.

  To which plane group does this pattern belong?