Arena design and mathematics

Summary: Modern arena designers consult mathematicians to determine the effects of design on play and crowd behavior.

A sports arena is essentially an enclosed area consisting of a large open space where a sport is played, surrounded by seating for spectators. It may also include various facilities for athletes, spectators, and the press. Many sports use specific terms for arenas, like “park” for baseball and “stadium” for football. Some sports arenas are open-air while others are roofed. The word “stadium” comes from stadion, an ancient Greek unit of length. Mathematics plays a significant role in the design and maintenance of modern sports arenas, including not only the geometrically shaped playing surfaces but also the optimization of seating, sightlines, acoustics, lighting, spectator traffic flow, and placement of restrooms and concessions. Features such as retractable roofs and convertible forms to accommodate multiple sports require careful design as well. Mathematicians also analyze and model features of sports arenas to determine their potential effect on the game play.

The rules of each sport dictate dimensions for the field of play. Some such as hockey, football, basketball, and soccer specify exact dimensions for the playing surface and delineate areas for specific activities, like the rectangular key in basketball or the half-circle goal crease for amateur hockey. Baseball, on the other hand, standardizes the dimensions of some features such as the distance between bases and the distance between the pitcher and home plate, but the outfield varies depending on the positions of the outfield walls. Further, aspects of game play can be affected by design choices. Fenway Park’s outfield wall known as the “Green Monster” is notorious for stopping home runs, yielding more doubles and triples. When the new Yankee Stadium produced a higher rate of home runs, there was speculation about a “wind tunnel” effect. Statistical analyses suggested that curvature and height of the right field wall were more important than wind speeds or patterns. Statistician George “Bill” James developed the concept of park factors, which attempts to measure how park characteristics influence game outcomes.

Robert F. Kennedy Memorial Stadium in Washington, D.C., which opened in 1961, was the first multiuse stadium. It was widely decried for being a “concrete donut.” Some critics suggested its wavy shape and curvature optimized it for baseball seating, though the widely replicated design has deficiencies for baseball and football. Some critical seats were too low for football and too high for baseball, resulting in poor sightline angles.

The baseball configuration was also more symmetrical than most baseball-only fields. Modern designers use mathematical techniques and tools (such as Mathcad software), simulations, and three-dimensional modeling for their designs, resulting in unique facilities like The Float in Singapore, which is literally floating on Marina Bay. Similar methods are involved in the design of arena roofs or domes, some of which are retractable. Calculating the amount of material needed to construct a curved dome, as well as calculating the weights, forces, and stresses, typically involves the use of calculus. These calculations, in turn, partially determine the type of support required.

Geometry and graph theory also contribute to dome design. R. Buckminster Fuller suggested that domes are strongest when the edges lie along great circles. Triangles are often used to give great strength with minimal weight, while other support structures resemble the latitude and longitude configuration on a globe. Fibonacci sequences and plane tilings also are used in the design of some domes. Veltins Arena in Germany uses features like hinged columns with ball-bearing edges that move in three dimensions. Both Veltins Arena and University of Phoenix Stadium in the United States feature sliding roofs and retractable natural-grass playing surfaces weighing millions of pounds. These were mathematically modeled extensively before construction. Transformative structures of this type have become known as “kinetic architecture.”

Mathematicians continue to investigate questions related to sports arenas, some of which have wider applications. Researchers have considered the impact of sports arenas on land values using hedonic regression models. Mathematical analyses of crowd sequence videos (frequently taken from sports venues) benefit research in areas including surveillance, designs of densely populated public spaces, and crowd safety. In some cases, people are conceptualized as a “thinking fluid” to which fluid dynamic and stochastic models may be applied. Unusual events like emergency evacuations are fairly rare, and there are legal barriers to obtaining extensive live footage. As such, computer scientists and mathematicians have developed detailed simulations for both “normal” behavior and unusual crowd events. Some have suggested that topology optimization would be beneficial for investigating arena evacuation plans.

Bibliography

Puhalla, Jim, Jeffrey Krans, and Michael Goatley. Sports Fields: Design, Construction, and Maintenance. Hoboken, NJ: Wiley, 2010.

Winston, Wayne. Mathletics: How Gamblers, Managers, and Sports Enthusiasts Use Mathematics in Baseball, Basketball, and Football. Princeton, NJ: Princeton University Press, 2009.