Gymnastics and mathematics
Gymnastics is an athletic sport that emphasizes balance, flexibility, and strength, allowing participants to perform graceful movements in various forms such as artistic, acrobatic, and aerobic gymnastics. Mathematics plays a crucial role in gymnastics, particularly in understanding the mechanics of motion, choreography patterns, and competition scoring systems. Key mathematical concepts include angular momentum, which governs how gymnasts rotate during their routines. A gymnast can manipulate their speed and rotation by adjusting their position relative to their center of mass, showcasing the relationship between physical movement and mathematical principles.
Moreover, the scoring system in artistic gymnastics is quite intricate, combining difficulty ratings for routines with deductions for execution errors, resulting in a comprehensive score that evaluates both technical execution and artistic performance. In disciplines like trampolining, energy conservation principles come into play, as the gymnast transitions between kinetic and potential energy with each jump. Factors such as body type and mass also influence performance, as different styles of gymnastics favor different physical attributes. Overall, the interplay between gymnastics and mathematics reveals the underlying calculations that enhance athletic performance, making it a fascinating area of study for both sports enthusiasts and mathematicians alike.
Gymnastics and mathematics
Summary: Performing gymnastics depends upon an understanding of geometry and forces.
Gymnastics is an athletic performance activity that depends on balance, flexibility, and strength for producing graceful movements. Gymnastics can be recreational or competitive. There are also numerous forms of gymnastics, including artistic, acrobatic, and aerobic. The main mathematical topics involved in gymnastics include the mechanics of motion, patterns in choreography, and competition scoring systems. Mathematics has sometimes been described as “mental gymnastics.”
Rotations
Many gymnastics routines include rotations. The key mathematical characteristic of a rotating body is its angular momentum, which is equal to the product of the mass, the velocity, and the distance between the center of mass and the axis of rotation. When there are no external forces, the angular momentum is conserved—it does not change. Gymnasts cannot change their mass, but they can reposition their center of mass relative to the axis of rotation, making the speed change to preserve the momentum. When a rotating gymnast tucks in closer to the center of rotation, the speed increases. For example, a gymnast can hold onto a bar by the hands and keep the body straight, making a relatively slow rotation around the top uneven bar called “giant swing.” As the gymnast tucks his or her limbs in closer to the bar, the center of mass becomes closer to the axis of rotation, and the gymnast spins faster. Mathematics also helps determine the optimal angle at which the gymnast should release from the bar in order to perform subsequent transitions and maneuvers.
While simpler routines can be performed intuitively, through trial and error, competitive gymnasts develop complex sequences of moves that involve detailed calculations of mass, momentum, velocity, position of the apparatuses, and so on. Conversions between rotation and moving along straight lines are a part of many routines, with speeds and directions determined by conservation of momentum laws. For example, a gymnast runs to a springboard, accumulating momentum. As the gymnast jumps off the springboard, the vectors of the momentum generated by the springs and the momentum of the run are added together, propelling the gymnast forward at about a 45-degree angle to the floor. The gymnast can then push off a horse ahead, converting momentum into the angular momentum of rotating the body around the horse. At the highest point of this rotation, the gymnast can tuck the limbs in, moving the mass close to the axis of rotation and accelerating for a flip in the air. For example, a triple back somersault involves two and three-quarter body rotations before landing. Before landing, the gymnast straightens out, moving the limbs farther from the axis of rotation and slowing down the rotation, allowing for a soft, safe landing on the feet.
Scoring of Artistic Gymnastics Competitions
The current system of scoring in artistic gymnastics is relatively complex. It assigns a difficulty score to the attempted routine and then subtracts from that score for mistakes in execution. The score is analytic—it is based on decomposing gymnastic routines into individual elements. Existing elements are summarized in the illustrated Table of Elements, and given difficulty ratings from A (0.1 points) to G (0.7 points). Additions to the Table of Elements are frequently named after gymnasts who first performed them successfully. Such new elements are submitted by the competing gymnasts ahead of the competition event, to be evaluated by an international committee.
Eight highest difficulty values of the routine, added together, form the difficulty value (DV). Skills from five required Element Groups are awarded 0.5 points each, for the maximum 2.5 points in composition requirement (CR). Finally, an additional 0.1 or 0.2 points are given for each element if elements are connected, which adds to connection value (CV). The difficulty score (D) is the sum of these points: D=DV+CR+CV.
In addition to the difficulty score, there is an evaluation of the artistry and execution called “E-score.” The judges take away points from the perfect 10.0 E-score for technical or artistry mistakes. Each fall costs 1 point.
Trampolining and Conservation of Energy
Many gymnastic apparatuses are somewhat springy. Trampolining is a type of gymnastics that occurs entirely on trampolines and uses flight-like moments between contacts with the surface for striking routines. Trampolining involves the accumulation of energy. First, the kinetic energy of the gymnast’s limb flexes and motions is converted into the potential energy of the stretched trampoline fabric. Then, the gymnast is thrown in the air, converting this potential energy into the kinetic energy of the motion. As the gymnast gains height, the kinetic energy is converted into the potential energy again. Gravity pulls the gymnast down with acceleration, converting to kinetic energy, which converts to the potential energy of the stretched trampoline upon contact, and so on.
From the point of view of mechanics, the trampoline is a device for storing the gymnast’s potential energy between jumps. This view can explain, for example, why gymnasts cannot jump infinitely high, adding more and more energy to the trampoline. The maximum stretch of the trampoline limits the amount of energy stored in it. This can also be used to compute the theoretical maximum height of a trampoline jump.
Different types of gymnastics are easier to perform with different body types. A lower body-mass-to-height ratio makes it easier to twist during movements and to hide momentum transitions in the twisting, so tall, skinny people are better suited for artistic gymnastics. In trampolining, twists and transitions are not as crucial as higher rotation speeds and are easier with a higher body-mass-to-height ratio. Also, both take-offs and landings on trampolines require significant bursts of energy and muscle strength. Therefore, shorter, stockier athletes are better suited for trampolining.
Bibliography
Jemni, Monem, ed. The Science of Gymnastics. New York: Routledge, 2011.
Roper, Tom. “Mathematics and the Motion of the Human Body.” The Mathematical Gazette 74, no. 467 (1990).
———. “Mathematics and the Motion of the Human Body, Continued.” The Mathematical Gazette 74, no. 468 (1990).
Sommer, Christopher. Building the Gymnastic Body: The Science of Gymnastics Strength Training. Mesa, AZ: Olympic Bodies, 2008.