Soccer and mathematics
Soccer, a globally beloved sport, intertwines with mathematics in various fascinating ways, reflecting both individual skills and team dynamics. The historical connection to mathematics is exemplified by mathematician Harald Bohr, who, in addition to his contributions to the field, was also a talented player and Olympic medalist. The role of statistics in soccer, much like in other sports, is significant as it helps in analyzing player performance and game strategies.
The biomechanics of soccer, particularly in executing moves like the instep kick, showcases how physical qualities and technical skills converge, with researchers employing advanced mathematical techniques to examine player movements. High-speed video analysis allows scientists to capture and quantify the dynamics of kicking, revealing how joint and segment angles influence performance.
Team tactics also involve complex calculations, as players must intuitively assess speed and direction to create scoring opportunities amidst simultaneous movements on the field. Notational analysis, aided by video technology and software, allows for the dissection of player actions to inform strategies. Furthermore, mathematical modeling has even been applied to understand the aerodynamics of the soccer ball, enhancing the grasp of its flight patterns. Overall, the intersection of soccer and mathematics provides deep insights into the sport's mechanics and strategy.
Soccer and mathematics
Summary: Mathematical modeling and statistical analysis can help inform individual techniques and team tactics in soccer.
Soccer is a sport that has been enjoyed worldwide for more than a century by both players and spectators. In the early part of the twentieth century, mathematician Harald Bohr, founder of the field of almost periodic functions and brother of famed physicist Niels Bohr, was a skilled player and a silver medalist on the 1908 Danish Olympic soccer team. He was reported to be so popular that his doctoral dissertation defense was attended by more soccer fans than mathematicians. In general, the sport is often cited for its equal emphasis on individual skills and team tactics. As in other sports, statistics are frequently cited by sports commentators. In addition, technically demanding individual actions, as well as masterfully executed plays, can all be described and analyzed using statistics and mathematics, which is done worldwide by numerous sports scientists. One could even say that the players, perhaps unconsciously, use or display “mathematics in motion.”
Individual Technique
The effectiveness of any of the various moves a player uses (kicking, heading, or dribbling) depends on a combination of physical qualities and technical skills. This idea can be demonstrated using the instep kick as an example; the instep kick, with the aim to kick the ball as hard as possible, is by far the most studied soccer movement by sport scientists. In order to maximize the forward swinging speed of the shank, physical qualities (such as strength and speed of contraction) of the knee extensor muscles and the hip flexor muscles are important. However, research has shown that technical skills are equally important. The specific technical skill required for optimal kicking is coordination—how the shank moves relative to the thigh.
Coordination is one of the topics studied in the scientific field of biomechanics, which relies heavily on mathematics. Biomechanics researchers use high-speed cameras in their laboratories to record kicking performance from top level players. From the video footage, the researchers can obtain the three-dimensional position in space of selected points on the kicking leg. Using mathematical concepts from vector algebra and trigonometry, joint and segment angles can subsequently be calculated. These data, in turn, allow calculations of a number of kinematic parameters of the foot, shank, and thigh, comprising linear velocity and acceleration and angular velocity and acceleration.
In mathematics, the most common way to calculate velocities and accelerations from position data is to use calculus. This method, however, requires the position data to be specified as a mathematical function. This is not the case with position data obtained from video footage, which are discrete in nature—they consist of thousands of numbers, specifying the three-dimensional position of numerous points on each video frame. From the cameras’ frame rate, the elapsed time between frames can be calculated, which instead allows numerical differentiation of the position data using a computer. Finally, by combining the kinematic data with data for each segment’s mass and moment of inertia (a measure of a segment’s inertia when rotating) and using the principles from Newtonian mechanics, the researchers can calculate how the movement of the thigh affects the movement of the shank and vice versa. The forward swing of the thigh generates a force at the knee that causes the shank to swing faster forwards. The force is larger, the faster the thigh moves, while the effect of the shank is larger, the closer the knee angle is to 90 degrees. Top players instinctively coordinate thigh and shank movements in order to take maximum advantage of these intersegmental forces, although science so far has failed to determine precisely what optimal coordination is.
Team Tactics
When a midfielder executes a beautiful play that a forward picks up between defending opponents and scores, a lot of “hidden” mathematics is occurring. The midfielder’s team members and opponents are all moving simultaneously in different directions with different speeds, yet the midfielder still manages to precisely calculate the required ball speed and direction to execute his play, so the ball and forward meet at the intended spot out of reach of defending opponents. Situations like this are analyzed by sport scientists and coaches using the methods of notational analysis. With video footage and specialized software, the various actions (sprinting, moving sideways, tackling, or heading) of each player from both teams can be registered. Statistical calculations can reveal which situations are most likely to lead to a certain outcome, such as scoring a goal, and which general tactics lead to most of these situations. Digital representations have also been used to assist with tactics and analysis. Researchers from the University of Sheffield digitized a soccer ball (including even the stitching) and computed airflow around the ball. They found that the specific shape and surface of the ball, and its initial orientation, are significant in determining the ball’s trajectory through the air. Measurements on actual balls in a wind tunnel at the University of Tsukuba verified these mathematical simulations.
Bibliography
Chartier, Tim. “Math Bends It Like Beckham.” Math Horizons 14 (February 2007).
Putnam, C. A. “Sequential Motions of Body Segments in Striking and Throwing Skills: Descriptions and Explanations.” Journal of Biomechanics 26 (1993).
Reilly, T., and M. Williams. Science and Soccer. New York: Routledge, 2003.