1. Understanding Symmetries and Geometry through Dance
Laura Olliverrie

2. Motivation
The link between math and music has already been acknowledged, but lets take a step further to understand the connection between math and dance. I will explore the connections between mathematics and dance in both practice and theory.
Dancing Mathematics and the Mathematics of Dance by Sarah-Marie Belcastro and Karl Schaffer

3. Introduction
Every dance choreography incorporates the use of angles, geometric shapes, parallels, patterns, and symmetry in some way.
Dancers use symmetry and geometry to improve their performances and make them visually appealing.
Let us think of dance as an artistic endeavor and look at how deep mathematical concepts can be viewed through dance.

4. Superficial Links Counting Steps Noticing Shapes
Music is divided into counts and dancers use counting to mark the times at which movements are done
Noticing Shapes

5. Symmetry
The human body lacks rotational symmetry so it is impossible for a dancer to continue spinning forever. They will lose their balance.
Humans do have reflection symmetry
In dance, we notice the use of symmetry of motion for an individual dancer and that of a group

6. Klein Four Group
Dance choreography often displays pattern symmetry types such as reflection, rotation, translation and glide
Using just four symmetries—translation, mirror reflection, 180-degree rotation, and glide reflection—we can create what is called the Klein four group by combining the symmetries pairwise.

7. Try it Out!
If we are to fill in the table with compositions of translation (T), glide reflection (G), 180-degree rotation (R), and mirror reflection (M), we will end up with the multiplication table for the Klein four group.

9. Terminology of Symmetries in Time and Space
Unison (a set of movements performed at the same time)
Canon (a set of movements offset in time from each other)
Inversion (movements performed in reverse sequence)
Retrograde (each movement is reversed, in addition to the sequence being reversed)

10. How Math Sounds?
Rhythm is an important part of any dance, and some dance forms such as tap and clogging use sound extensively as well.
Let one person claps the eight-beat rhythm and the other clap the three-beat rhythm, each clapping loudly on the first beat of each cycle.

12. Basic Physics
Velocity is how fast and in what direction an object is moving
Momentum is mass times velocity
Force is basically a push or a pull
Gravity, support from the floor and friction from the floor are the forces that act on a dancer

13. Balance
No total force and no total torque (measure of how much a force acting on an object causes that object to rotate)
The center of gravity must remain directly above the area of contact with the floor

14. Turns/Spins The relevant quantities that describe turns/spins are:
Angular velocity is how fast an object spins
Rotational inertia is the inertia of a rotating object
Angular momentum is rotational inertia times angular velocity

15. Comparison Motion without spins Motion with spins Velocity Mass
Momentum
Force
Motion with spins
Angular velocity
Rotational inertia
Angular momentum
Torque

16. Other Dance Forms
Western Ballet and Bharatya Natyam - Use a strong sense of line Flamenco Duet degree rotational symmetry Hip Hop Dance - Angles and geometric shapes

17. Pedagogical Applications to Math in Dance
A number of programs have arised that are encouraging mathematical thought through movement.
Teaching math through dance
Bridge the gap
Helps kids combat math anxiety
Maths Dance: Triangular Squares by Corinne Wolfe

18. Further Links
In an effort to attract more girls to a STEM field of study, Virtual Environment Interactions (VEnvl) software allows girls top program 3-D characters to perform dance moves just by moving their own bodies.
The girls are then encourages to develop new computing strategies to improve their choreography.

19. References
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Schaffer, K., Stern, E. and Kim, S. “Math Dance” MoveSpeakSpin, Santa Cruz