Mathematics of contra and square dancing
The Mathematics of contra and square dancing involves a fascinating exploration of movement, symmetry, and spatial arrangements within dance formations. At the core of this style is the basic square, comprised of four couples, which exhibits symmetries under various rotations and reflections. Dancers can engage in numerous movements, with each male-female pair capable of ten possible interactions, leading to a total of 240 different movement combinations. This complexity is rooted in the idea that dance is fundamentally about movement rather than static positions; the roles of initiator and responder in dance actions create distinct experiences, even when the final arrangement appears the same.
In addition to the basic square, other configurations enhance the dynamics of square dancing. For example, each male-female pair can dance around one another in a formation called a Do-Si-Do, while more intricate arrangements involve inscribing circles within squares or forming smaller squares within larger ones. These variations showcase the versatility of dance movements while adhering to the mathematical principles of symmetry and spatial organization. Understanding the mathematics behind these dance forms can offer deeper insights into the beauty and structure of traditional dancing practices.
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Mathematics of contra and square dancing
Summary: Square and contra dancing employ many mathematical principles, including symmetries and permutations.
The Basic Square
The basic square consists of four couples. A square is symmetric under rotations of 90, 180, 270, and 360 degrees. Some or all of the dancers in the basic square can rotate in a circular movement according to these symmetries. Including the mirror reflections about each of the two lines of symmetry passing through the center of the square and parallel to an edge, there are six different targets of movement for the dancers. Further, in respect to each male (m)-female (f) pair, there are 10 possible movements. Thus, f1 could be directed to replace either f2 or m2, m1 could replace either f2 or m2, or both f1 and m1 could replace f2 and m2. Since there are four pairs, there are 240 possible movements among the dancers (6 × 10 × 4 = 240). Dance is about movement and not positions; thus, dance movements are not transitive. A movement of f1 to f2 is not the same as a movement of f2 to f1, although the outcome is the same arrangement. The two cases differ in respect to who initiates the action and who must react to the other’s actions.
Secondary Squares
Besides the basic square, several other squares are part of square dancing. First, each m-f pair is a square. Several calls direct the movements of these dancers relative to one another. Thus, in a Do-Si-Do, the two members dance a square around one another and return to their initial positions. Alternatively, the basic square can be divided into a square within which a circle is inscribed. Four of the dancers constitute the square, while the remaining dancers move inward so that they are contained by the larger square. These can then be instructed to move according to the four symmetries. This arrangement can be inverted. The pairs can move toward a center point and form the radii of a circle, while the square that contains the circle is implicit. Again, the four symmetries constrain these movements. Instead of being expanded, the square can be constricted. The larger square can be divided into two smaller squares, each with four dancers. The dancers can be instructed to form smaller squares with the pair on the right, the pair opposite, or the pair on the left.
Bibliography
Mathematical Association of America. “Square Dancing Takes a Mathematical Spin.” http://mathdl.maa.org/mathDL/pa=mathNews&sa=view&newsId=230.
Mui, Wing. “Connections Between Contra Dancing and Mathematics.” Journal of Mathematics and the Arts 4, no. 1 (2010).