Mathematics of ballet

Summary: Ballet uses geometry to create captivating moving art.

Ballet can be considered mathematics in motion from basic counting (keeping time with music, and doing demi-pliés in childhood dance classes); making lines, angles, and geometric shapes in space via basic positions and choreographed routines of principal dancers and the corps de ballet; communicating stories in ballet productions (like the classic Swan Lake, or a seasonal favorite The Nutcracker); conversing visually among dancers (as in a pas de deux with Margot Fonteyn and Rudolf Nureyev); and by representing general emotions, moods, and abstract themes (as in George Balanchine’s Serenade). Words from the French language may be common in ballet terminology, but concepts from mathematics abound as well. These representations, communications, and geometric creations can all be achieved and evidenced through the dance figures and ballet movements.

Ballet distinguishes itself from many other forms of dance through its use of the “turnout” (an outward rotation of the legs in the hip sockets to form a 180-degree line with the feet in first position). This turnout gives the dancer a strong base and the ability to move in any direction while allowing a more open body presentation to the audience, yet holding the graceful curves and shapes of the dancer’s body to preserve a svelte “line.” Other standard positions of the feet, carriage of the arms, or basic movements of the body produce angles such as a 135-degree arabesque, a 90-degree attitude, or a 45-degree battement tendu. The rond de jambe à terre or en l’air utilizes circular movements of the leg to trace semicircles or arcs, on or off the ground. These geometric lines, circles, and angles continue when basic steps become building blocks to more complicated movements. Meanwhile, dancing on the tips of the toes (en pointe), another distinctive ballet feature, heightens the dancer’s lines in a vertical fashion. The linear extension, from head to toe, fingertip to fingertip, does not end at the extremities but continues as if through an imaginary line into the space around the dancer.

Ballet as Geometry

One of the earliest ballet performances was the sixteenth-century Le Balet Comique de la Reine by Balthazar Beaujoyeulx, commissioned by the court of France. During that elaborate production, the dancers performed dozens of geometric figures involving triangles, circles, and squares for their geometric proportions and spatial configurations. These beginning ballets were influenced by the writings of Pythagoras and Plato and represented the cosmic and heavenly significance of numbers and geometry. A twentieth-century choreographer, Frederick Ashton, however, was inspired by mathematics for its sheer beauty in his creation, Scènes de Ballet. Working from a book of Euclid theorems, he specifically used geometry to create floor patterns and dance movements that could be viewed from any angle to see the geometric figures and “symmetrical asymmetries.” Combined with the strong rhythms and counts of Igor Stravinsky’s music, and geometrically patterned costumes and set details, Ashton’s work was said to have beautifully combined mathematics and ballet for its visual imagery.

Notation Systems

To preserve these choreographed works of art, dance notation systems were created to symbolically represent the positions, steps, and movements of the dancers. Early seventeenth- and eighteenth-century systems, such as Feuillet notation, recorded mainly floor patterns and feet positions, whereas the twentieth-century notation systems, Labanotation and Benesh Movement Notation (written on vertical and horizontal staffs, respectively), corresponded to the scores of accompanying music. These notation systems detailed the entire body movements from head to toe of every dancer. Even with the advent of video recording, it is these symbolic notations showing graphical representations of the step details that best preserve ballets for future generations.

Bibliography

Cooper, Elizabeth. “Le Balet Comique de la Reine: An Analysis.” http://depts.washington.edu/uwdance/dance344reading/bctextp1.htm.

Greskovic, Robert. Ballet 101: A Complete Guide to Learning and Loving the Ballet. Milwaukee, WI: Limelight Editions, 2005.

Minden, Eliza Gaynor. The Ballet Companion. New York: Fireside, 2005.

Schaffer, Karl, and Erik Stern. “Math Dance Bibliography.” http://www.mathdance.org/MathDance-Bibliography.pdf.

Thomas, Rachel. “Scènes de Ballet.” http://plus.maths.org/issue24/reviews/ballet/.