IntroductionI chose to investigate the relationship between mathematics and ballet since I first began dancing eight years ago. Annually, I make goals for myself to develop my flexibility in order to improve my dance techniques and movements. Ballet has been a major aspect of my life and has impacted the way I experience life and influences the way I think in a more attentive way especially about academics. Therefore, I wanted to incorporate ballet with the mathematical methods I have learned to further understand this unique relationship. I will investigate how various positions at the barre can have a positive effect on factors such as flexibility and posture to improve a dancer’s performance. I will take into account the factors that influence a dancer’s movements such as body weight, height, and experience. Therefore, the aim of this project is to find out which barre positions will improve a dancer’s flexibility based on the ideal ballet body type. Angles are especially important and emphasized when social dancing and solo performance. Dancers initiate positioning of arms and legs by the use of angles and body position by symmetry. For example, first position is attained when legs are rotated outward from the hip sockets so the feet can form a 180 degree line. Dancers are responsible for creating ways to exert choreography through angle and symmetry use when engaging with movement. Triangles are constructed in ballet with the placement of legs; single triangular shape with one leg in passé or double triangular shapes with two legs. Along with physical limitations, due to their genetic make-up, when putting forth their movements dancers have to have a great amount of strength to exert these positions gracefully and precisely. Dancers must accurately calculate the process when making use of limited space and effort to help them move into other positions. MethodI tested dancers from my dance studio from a range of flexibility, heights, and dancing experience. I studied the performance of ballet dancers when they completed warm ups at the barre in my classes. I collected and measured their time when they completed a position, angles, symmetry, type of position, height, and balance, and experience. I used this information to first see the influences that these simple characteristics have on performance and then further see how to improve balance and flexibility. A significant part of this information is how each dancer’s body type compares with that of the ideal ballet body and if they do not have the ideal body then what modifications are made to achieve the perfect position. The ideal body is slim, with a shortism to medium torso and long legs and arms. With reference to each dancer’s characteristics, each dancer completed a position. The barre is a horizontal pole leveled at a dancer’s waist for stability. In the “first position” the legs are turned out and the body is symmetrical with the heels of both feet placed against each other. From this position the right or left leg can move front, side or back. In “fifth position” the right heel is in front of and against the left toes which the legs are rotated outwards. I decided to put each dancer’s information in a table to organize and compare their measurements. I also decided to report the measurements to 4 significant figures since I wanted to have the exact numbers to have more accurate reportings when using the mathematical approaches I will use. Table 1DancersYears of experienceWeight(lbs)Height (cm)Calves circ.(cm)Quads circ. (cm)Arm length (cm)Biceps(cm)Leg length(cm)18127.4168.936.157.667.426.8107.228130.7170.839.459.373.929.0105.236143.1158.934.258.078.326.4111.345123158.833.851.268.025.3105.753104.5162.630.244.968.523.5106.067112.1173.336.658.273.927.4106.675106.8162.731.247.467.921.6105.386121.4157.330.445.668.422103.39611915530.844.367.321.3104104149156.331.744.865.620.9104.6115126.5155.832.854.363.322.6102.7124103.4160.531.246.963.423.3105.0137123.3172.334.657.376.326.4105.4147121.3157.330.445.668.422103.3I used a sit and reach test to measure each dancer’s flexibility especially to test the flexibility of their arms and legs:Table 2DancerHeight (cm)Sit and reach test, both legs (cm)1168.9522170.8423158.9364158.8435162.6456173.3427162.7388157.34991554610156.34411155.83812160.53713172.34514157.350Average height: height14= 2270.514=162.2Average flexibility: flexibility14= 60714=43.4I decided to measure each dancer’s flexibility by using a sit and reach test to have a simple way to measure the flexibility of their arms and legs as well as easily seeing how each dancer compares to each other in a simple test. However, this test only measures the flexibility of their arms and legs and with this test the results can vary depending on the effort the dancer put into reaching and how warmed up they were from doing the same simple stretches before participating in this test. From these positions, I also had the dancers do simple turns and jumps to further see the effect of flexibility changes by measuring distance, velocity and inertia. An example of a turn I had them complete was a fouette where they do a full pirouette followed by a pile on the standing leg and bringing the other leg to touch the knee each time.I measured the angular velocity to see how fast the dancer was spinning and the rotational inertia to see the tendency of the dancer to keep turning. Rotational inertia will depend on the dancer’s body mass and where the dancer is located on the rotational axis. The rotational inertia is proportional to the dancer’s mass and of the distance from where the dancer is located on the rotational axis.Gravitational force: F=mgwhere g is a constantConnection between force and momentum: p=mv=FtThe equations h=4t2 ;h=v0264; t=h2 where t is the duration of jumps in seconds, v0 is initial vertical velocity, and h is height measured in feet are ways that a dancer’s movements can be modeled and calculated in create a new viewpoint in how they are able to implement simple movements to further benefit their overall flexibility when transitioning into various positions. Table 4Height of jumpTime in the airInitial velocity 6 inches0.35 seconds5.66 feet/sec1 foot0.50 seconds8.00 feet/sec1 foot 6 inches0.61 seconds9.80 feet/sec2 feet0.71 seconds 11.31 feet/secSince h=4t2, jumps that last half seconds must raise the dancer’s center of gravity by one foot. The height of the dancer is a significant factor in this case. For example, a five foot tall dancer jumps ? of their height. A taller dancer will jump ? of their height will stay in the air for about 0.55 seconds, instead of 0.50 seconds. This is significant and influenced by the various heights of the dancers which is significant to how barre positions involving jumping will influence a dancer’s ability to improve their flexibility. Gravity only influences the vertical component of motion, not the horizontal:Height vs. time is a parabola, while horizontal distance vs. time is a straight line:A dancer’s process when jumping is also a parabola. When a dancer jumps includes a 2 foot rise in their center of gravity and includes 10 ft per second horizontal velocity, their process will model this:Along will a dancer simply jumping forward with one leg in front and the other behind them, having their leg out while turning also adds to the complexity of a dancer’s process when changing movements. When a dancer is turning with their leg out either to the front, side or behind them, this added movement increases the dancer’s technique and ability to calculate how much they are able to extend their leg in relation to how high and far they desire to jump. With this in mind, when a dancer turned in a complete circle, I measured their outstretched leg which is the radius of the circle to see the various factors coming into play when examining their flexibility. This circle below is a model of what the dancer turned and the bold line is the radius of 41 inches which was their outstretched leg parallel to the ground. The circumference of the circle is dWhere the radius = 41 inchesDiameter = 82 inches Circumference = 82 = 257.61 inchesA last way to model a dancer’s movements is by looking at their angular momentum which is the dancer’s angular mass and velocity whereL=lw. A dancer’s momentum is caused from a torque where t=Lt=dLdtdescribes over time, the change in momentum. The dancer’s movement when rotating is modeled by t = dLdt=ddt*(lw)=1dwdt=1a where a is the angular velocity. Having the dancer’s leg extended to one side results in having a larger radius and larger rotational inertia resulting in a smaller angular velocity. This causes for the dancer to slow down. Having the dancer’s leg pulled in to rest on her knee results in a smaller radius and smaller rotational inertia and a larger angular velocity. This causes for the dancer to speed up and complete more turns.From exploring the relationship between the dancers jumping and turning and this being modeled by the circumference and angular momentum helped me to further see another aspect of a movement that takes place during a barre position.This allowed me to understand its influence on a dancer’s ability to process the movement efficiently enough to recognize the length and height that they must stretch and raise their leg to achieve the most precise jump or turn. ConclusionFrom my exploration on investigating the relationship between mathematics and ballet, I have discovered that the factors of body weight, height, and experience are significant in the role of a dancer’s techniques and ability to create their movements. These components along with a dancer’s measurements allow them to perform their movements and sequences of these positions. By being aware of the limited space they have, the dancers are able to also calculate the specific movements and changes they have to make in order to be as efficient and precise in their sequences of positions. By using my results and analysis, I will share with the dancers who participated the ways that they can improve of including spacial and body awareness when executing movements specifically when beginning to warm up at the barre. In doing this, I will help them improve their overall performance when social dancing as well as solo performing. By further understanding this information, I am also able to implement this analysis in my own journey in enhancing my own flexibility and posture by becoming more self aware of my space and comprehending the influences that small changes to my movements will significantly aid me in achieving my dancing goals. Overall, the relationship between mathematics and dance is able to improve dancers and choreographers techniques and abilities to see creating movement and sequences of positions in a new perspective. Since using math has the ability to create new ways and ideas of how to think of dance. Choreographers using math when creating their next dance sequences will be able to come up with innovative ways to engage their dancers and improve their creative side. From using mathematics and art, people are able to think about usual and simple concepts in different viewpoints to become more inspired to explore these concepts.