The first author was teaching a "Mathematics for Liberal Arts" type class. In lieu of a quiz, students were assigned an essay on symmetry. We had been discussing reflections, rotations, translations, glide reflections, frieze groups, and wallpaper groups for the past week of an intense one-semester-fits-into-one-month class.
The students had used origami to explore isometries. They moved triangular figures around to study group structures. They had taken pictures of their feet to study frieze groups (thanks, John Conway!), and constructed several different wallpaper patterns using cut-and paste techniques to make tiles with unusual symmetry patterns. But, as usual, some students had difficulties with the ideas.
Several students came to speak with me about the essay, including the second author. He told me that he just didn't see any symmetry around him. At the time, we were standing in a building (the cafeteria) with sine-curve shaped ceiling beams arranged in parallel, supported by arms with 4-fold rotational symmetry. The light fixtures were hemispherical, with slits cut out in an 8-fold rotational pattern. Parallel sliced kiwi fruits were being served, with their near rotational symmetry. I merely pointed these things out, saying that we are surrounded by symmetry, all we need to do is look and listen.
"Listen?" Brandon asked. I hummed the first few stanzas of Beethoven's fifth symphony, and asked if he didn't hear translational and rotational symmetry. He looked at me like I was crazy.
A couple days later, in lieu of an essay, he sent me a piece of music that he had written, together with the following explanation.
Symmetry can be found in many things in nature. A mathematical phenomenon like symmetry can be used for many things where we would not initially think of finding it. Many mathematicians and artists alike have found inspiration in the beauty of symmetry and spend countless hours experimenting with symmetry to create magnificent art pieces. As an example of how symmetry can be used in an everyday setting such as music, I composed an instrumental piece that uses geometric concepts such as rotations, translations, and glide reflections.
The piece was written with an acoustic guitar by taking a simple I-V-vi-IV chord progression in the key of C major. Simple melodies using tones from those chords were created by translating a set of notes up in pitch so that the groups of notes were in the same shape. Translations were also found on the guitar as well. For example, the chirping sounds that can be heard in the song came from an original chord shape that was rotated around one of the notes on the fretboard. This method was applied to each chord from the progression.
After the writing process had been completed, the next step was sound design and composition of the song. This part of the writing process was very mathematical to me. During the design of some of the sounds that I was creating, I could see a lot of symmetry inside of the wave that was being displayed. I found that I could make some interesting shapes with the waves in order to make it sound good.
Lastly, the drums and percussion were also composed using symmetry. The program that I was using to record all of the instruments has a beat maker built into it. For the bass and snare drums I made a quick loop consisting of five drum hits and then translated those five hits in time to create a drum pattern. The percussion was made with the same beat maker but instead of using translations, I reflected the sequence so the next set of cymbal hits are the same as the previous ones except they start from a repeated last note.