# The Aerodynamics of New Design Soccer Balls Using a Three-Dimensional Printer
## 1 Introduction  
Soccer balls have changed considerably from the traditional 32-panel ball, which consisted of 20 regular hexagonal panels and 12 regular pentagonal panels. The panels in those traditional balls were arranged as in a truncated icosahedron. When Adidas unveiled the Teamgeist ball for the 2006 World Cup in Germany, a ball with 14 panels, a new era of soccer ball design began. The number of panels on World Cup soccer balls then dropped to eight for the Jabulani ball used in South Africa for the 2010 World Cup, and then to six for both the Brazuca ball used in Brazil for the 2014 World Cup and the Telstar 18 ball used in Russia for the 2018 World Cup. The Europass ball that was introduced for the UEFA Euro 2008 had a prestressed concrete (PSC) creeps surface design. That ball's surface was roughened with various protrusions, such as squares, triangles, and hexagons. The Al Rihla and Oceauz balls, used for the 2022 men's World Cup in Qatar and 2023 women's World Cup in Australia and New Zealand, respectively, each possessed 20 panels. Previous studies [1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 11; 12] reported on the effects of surface properties on a ball's aerodynamics and flight trajectory. Other studies [13; 14] evaluated and assessed the variations on aerodynamics and flight properties due to changes in the panel orientation that are attributed to the reduced number of panels. Research on how the number of panels, shape of surface structures (convexity and concavity), and surface design (such as protrusions) of modern soccer balls affect their aerodynamic properties have been published [15]. Three-dimensional printing is a technique that fabricates complex structures and geometries from 3D model data. The performance of 3D printers has improved significantly in recent years, allowing 3D printers to be used extensively in the research and development of modern sports equipment [16; 17; 18; 19; 20; 21; 22; 23]. Studies on the dimple structures on golf balls using 3D printers have also been conducted [24; 25; 26]. Surface materials used in the construction of soccer balls vary greatly between ball types and manufacturers. Seam structures between panels may be machine-stitched, hand-stitched, or thermally bonded, and panels may be smooth or textured; those features are found to influence the aerodynamic characteristics of soccer balls [3; 8]. There does not appear to be a canonical parameter that researchers use to characterize surface features. For this study, eight 32-panel soccer balls with various surface features, such as differing dimple structures, were created using the same material, polyamide-12, with a 3D printer. Computer-printed soccer balls allow for the creation of controllable surface variations. Computer-determined areas of smooth and roughened parts of balls are much more easily found than with real soccer balls. After fabricating the soccer balls, their associated aerodynamic parameters were determined with wind tunnel experiments. Though the current study determined side and lift coefficients, which could potentially extend the previously published work [27] on manufactured dimpled soccer balls, those coefficients will not be presented here because of a desire to keep a tight focus on the correlation between surface roughness and critical Reynolds number. To build on what is known about how a soccer ball's surface influences its aerodynamic coefficients, this study introduces a new surface parameter that determines a soccer ball's critical Reynolds number. The critical Reynolds number is the Reynolds number associated with the minimum in the drag coefficient. That minimum occurs when separation of the boundary layer of air transitions from turbulent to laminar as the air's flow speed is reduced. Parameterizing surface roughness is challenging because of the myriad ways in which a surface may deviate from being smooth. Efforts to do so date back half a century to the seminal work performed by [PERSON] [28].