Soccer is a beautiful game of people and sciences: psychology (see post: mind-bend it like beckham), biology, and among others, mathematics. In particular, geometry can be found in almost all aspects of the game. To the standard viewer, these applications may not seem obvious. It probably looks like a bunch of people kicking a ball towards a common goal trying to keep another team from scoring on their own. That’s true, but it’s the science and language of movements that make one team beat another.

First, the simple applications of geometry and topology in soccer. The field is rectangular in shape with other shapes creating boundaries, halves, penalty boxes, corners, and goals. The field is geometrically symmetrical. The ball is semi-spherical adding it’s own element in shot curvature, reflections, etc. The shot itself must take on a very calculated velocity (angle + speed). Passing with a give-and-go, long ball, or off the wall indoors are all derivatives of geometric shapes. Using a wide-angled pass is less probable to be intercepted than a “square” pass. The wall aims to accomplish much of what a goalie does against a breakaway: take away the angle. Setting up a wall at 1o yrds or a goalie coming off a line reduce the target for a shooter. The cross must be a calculated trajectory, usually bending in or out to give more advantage to the goal-scoring team. Bending out allows for a more forward angle versus a completely perpendicular cross which is much tougher to head towards a desired target. These are some simple components of soccer in which geometry “takes shape”.

Let’s also think about team formation and spatial distribution around the field. Formation is not random, and it’s known that having the optimal formation at any one time greatly increases the chance of achieving an objective, whether that’s scoring a goal, playing stiff defense, maintaining possession, or slowing the game pace.

Connectedness: How many neighbor nodes does the person with the ball have? How passing options are there currently? The more interconnected the 11 nodes might be, the more optimized the network.

Compactness: The team should be like an accordion that can open and close, adapt and react, become dense and spread out. Localized compactness assists in defense, spreading out opens up for offense.

In passing, triangular formation creates multiple opportunities to give-and-go. Moving without the ball to create shapes builds the passing support network that creates scoring chances. Thinking ahead of reflected or return passes based on complementary angles off your teammate will get you the ball back in stride. Combine this geometric intuition with a good mental game is key.

In all, having a locally-connected space and visualizing complementary angles is advantageous for any team. The name of the game is adaptation and proaction. Thinking 3 passes ahead, using all sciences in your toolbox both mentally and physically, and anticipating the faults of your opponent will surely increase the probability of a win. Having a kick ass shot doesn’t hurt either.