A mathematical contribution to dance notation: Analysing labanotation with Euclidean geometry, computing matrices for dance notation, and choreographing with crystallographic groups.

摘要

Dances consist of bodies moving through space and time, a concept established by the great choreographer Merce Cunningham. Dance notation is the recording of these movements on paper. This multidisciplinary research aims at bridging the gap between the sciences and the arts. We mathematically investigate an existing system of dance notation, and use mathematical tools to generate new ones.; The arts of dance and dance notation contain numerous mathematical concepts, mostly relating to Euclidean geometry. The first objective of this research is to identify these mathematical structures present in Labanotation. The second is to characterize dances using algebra. In one section, positions of partners in contradancing are defined by matrices and calculated through matrix multiplication using Homogeneous Coordinates. In another section, body movements are encoded into 4 x 6 matrices; the rows represent the four-dimensional coordinate space, and the columns the different body parts. After raising into 5 x 7 matrices using the concept of homogeneous coordinates, summing a sequence of matrices provides a choreography matrix representing the final position of a dancer as dictated by the sequence. The third objective is to choreograph using crystallographic groups (or wallpaper groups). Geometric shapes are designed to represent the basic steps of certain ballroom dances, and each group is applied to each symbol using Artlandia's SymmetryWorks in Adobe Illustrator. A brief discussion explains why only five groups are relevant, and the ensuing results illustrate that these groups applied to the dance symbols generate mostly feasible choreographic routines.