Symmetry and "Old Time Dances"
Traditional "old time" or country dances incorporate a large number of symmetry elements in their dance forms. In crystallography, and in "old-time" dancing, most symmetry elements (rotation axes, mirrors, center of symmetry) have been successfully completed if all the atoms (dancers) occupy the same positions before and after the operation. Some symmetry operations involve a translation (glide planes or screw axes) in which case the atomic/dance positions are replicated at a prescribed distances and directions from the original position. In this exercise, we will use a dance set as a metaphor for a crystal structure, and we will illustrate the locations of the symmetry elements as they appear throughout the dance form:
- Each dance set (e.g. a square) can represent a unit cell of a crystal.
- Each dancer represents an atom. (We will deal with the question of whether or not male and female dances represent different types of atoms later–this does affect the symmetry content).
- The various dance moves we will use will demonstrate mirrors, rotation axes, a center of symmetry, a screw axis, glide planes and translation! You will see that the symmetry elements come together in very definite combinations, and are only present in specific locations in the crystal structure/dance set.
- To extend the metaphor, if the dance is done correctly and all dancers end up at their appointed positions then you have a "perfectly ordered" crystal; if the dancers get partially lost along the way, this is a crystal with "defects;" and if the set entirely breaks down and everyone is lost, you have a meltdown, and the result is an "amorphous"material–without order!
- As you watch the videos of the dance forms, you'll see that the dancers don't always do the move at the same rate (despite the strong beat from the music). Obviously, there are kinetic effects that affect the ordering of the dancers/atoms in the dance set/crystal.
- You'll also see that in the move "balance and swing," the dancers/atoms are demonstrating vibrational energy via simple harmonic oscillation.
- Equivalency of dancers/atoms (aka gender issues): In most of our dance representations of symmetry elements, we assume that all dancers/atoms are equal. This allows for the highest symmetry to be represented for each dance form. For example, in an allemende left (left hand turn) if the two dancers represent the same type of atom then this operation is a 2-fold rotation; but, if each dancer is unique or special, then this symmetry is lost because each dancer can only replicate him/herself after one full term, i.e. no symmetry at all. Any time a special condition is applied to the motifs (male/female, cation/anion, etc.), symmetry will be decreased because there are fewer identical points. Only in the contra dance form do we make a distinction between male and female dancers to represent a glide plane.
The lattice Pg, showing a glide plane, and considering male (square) and female (circle) positions as being unique. This is the starting configuration for an "improper" contra dance. The lattice Pm, showing the left side mirrored to the right (male=female). This is the configuration for the "proper" triplet dance. The P4mm lattice, showing the positions of symmetry elements represented in the square dances.
Look for these symmetry elements in the following demonstration videos. Allow us a bit of artistic, creative license as we use these dance forms to demonstrate symmetry operations. Enjoy the dances, and try them out yourself!