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#### Mathematics of dancing: Creating a perfect choreography

Mathematics and dance are deeply intertwined. Mathematical concepts can be used to understand dance at a more profound level and to create better choreographies, and at the same time dance analogies can make math lessons more vivid and accessible for the students.

Geometry – with its shapes, patterns, angles and symmetry – is perhaps the most apparent field of mathematics present in dance. Looking at a solo dancer frozen in one position, we can see the lines of the body, their angles and directions in relation to each other and to the room. In a moving group of dancers, we notice the lines and shapes created by the ensemble, their change with the music, and the patterns of beats that cause those changes.

 Look at this dancer in a grand plié in the second position on relevé, hands straight above the head. Her legs and the floor form a rectangle, meaning that the thighs are parallel to the floor and the shins are perpendicular to the floor. Thus, the angle at the knees is ninety degrees. Her arms form a V-shape, so that an imaginary line connecting her palms forms a triangle. The dancer’s body is symmetrical around her spine. Looking at her from above, the dancer’s body should follow as straight of a line as possible (that of course depends on how well the dancer can turn out her hips). Now, consider a dancer coming out of an attitude. Her spine is perpendicular to the ground. The extension of the front leg forms the axis with respect to which the back leg and the upper body mirror each other. As a consequence, the line passing through the front leg bisects the angle formed by the back thigh and the spine. In addition, the line tangent to the curve of the upper body at the hips dissects the angle formed by the dancer’s thighs. Finally, the arms form an ellipse with the dancer’s head being the lower focus.

In couple dances, apart from individual lines of each dancer, we also have shapes and patterns caused by the interaction of two bodies.

 In this tango pose the man’s and the woman’s bodies are in similar arrangements; in fact, one could construct the woman’s pose from the man’s using three simple actions: reflection, rotation and rescaling. The rescaling causes the woman’s pose to become shorter and wider – her back leg reaches farther than the man’s back leg. She is exaggerating the movement to get lower, into the position of surrender typical for argentine tango.

Pieces involving more than one dancer often use the idea of translation. Translation of a pose is when several or all dancers of an ensemble perform the same movement at the same time. The geometry of translation, i.e. the location of each dancer, is independent of the pose and only subject to choreographer’s wish.

 The dancers form two parallel lines; the simplicity of the formation’s geometry emphasizes the beauty of proper ballet technique. However, depending on the feel of the piece, the choreographer might choose to place his dancers in a pyramid or a differently organized formation. Since dancers are three-dimensional creatures, their movements and poses exhibit different geometrical relations depending on the angle at which we are observing the piece.

In addition, with groups, sometimes the formation has to be taken apart in order to see the geometric relationships – a challenging but very interesting task.

 Two dancers on the outside are in the same translated pose; their bodies define a splitting line for the remaining two ladies. Those fill in the vertical levels while bringing the ensemble together through the shapes their bodies create. The dancer in the front has the same leg arrangement as the outside ladies; she is exaggerating the knee bend to get into her position. The dancer in the back is opposing the other’s downward action. The dancers’ positions form a zig-zag line on the dance floor.

Dance is always dynamic, and the changes in formations and shapes are the icing on the cake of dance geometry. The choreographers – often intuitively, sometimes knowingly – use the rules of mathematics to create pieces that look light and fluid. One of ways to ensure that is to consider all dancers together and look at the path travelled by the center of attention mass (CAM) of the ensemble. To calculate the CAM, instead of recording the body masses of the dancers, we would assign the weights based on the type of movement performed and how likely the moves are to attract the audience’s attention. For example, dancers that are off-stage would have zero weight, and a dancer leaping across the stage would carry more weight than a dancer frozen in a pose somewhere on the side. Or, depending on the atmosphere of the dance, a dancer crouching down and being still could have more weight than dancers moving around him. Thus, the weight of each dancer would vary throughout the piece, and so would the position of the CAM.

Geometry in dance is unavoidable. The moment a dancer enters the floor, his or her body and moves create shapes and patterns that simply wait to be noticed by the audience. Mathematics provides a helping hand in making these shapes perfectly aligned and therefore most pleasing to the eye.

Full text: Wasilewska, K. (2012). Mathematics in the world of dance. Proceedings of Bridges 2012: Mathematics, Music, Art, Architecture, Culture, 453-456.

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