The geometry of attacking football

Football tactics are about creating and controlling space. Before I started properly analysing the game, I had underestimated this key point. I thought football was primarily about controlling the ball. It’s not.

While individual players need to be extremely good at passing and receiving, this is an individual level skill. It is important for the one player out of 22 who has the ball at that moment. At the collective level, for the other 21 players, the skill is in how they use space.

To identify the mathematics behind space creation, I started with the most perfect example of a team using space: the Barcelona team of 2010–11. At the centre of that team was Lionel Messi, but around him were other brilliant players. Xavi, Andres Iniesta and Pedro were all part of what may have been the best club team of all time. It was these players who created the space in which Messi operated. In Soccermatics I used the following player position diagram from a match between Barcelona and Panathinaikos to illustrate the teams style of play.

The solid circles are Barcelona players, the open circles are the defending Panathinaikos players

The lines connecting the players is called the Delaunay triangulation. It provides a way of seeing the passes available to the players. And what we can see from this snapshot is that Messi has a number of options available to make a pass, most notably Iniesta and Xavi. In this particular situation, the pass went to Xavi, a shown below.

This diagram, known as a Voronoi diagram, shows how Barcelona break down space. The dotted line marks the zones for each player, the areas of the pitch that each player is closest to. When Messi makes the pass to Xavi the Panathinaikos are stranded between the two Barcelona players, making it easier for them to complete the pass. It is not only their skill on the ball, but also their geometrically accurate positioning that allows them to make the pass.

All clubs now have access to tracking data from their matches, where the position of all the players and the ball is collected at least 10 times per second. This allows them to make Voronoi diagrams for every point in time: to see which attacking players are creating space and which defenders are narrowing the attacking space down.

One example of this approach is shown below.

This video was produced by Jaime Sampaoi and his colleagues at the CreativeLab in Vila Real, Portugal. Here we can see from kick-off how the red team presses, forces a mistake and gains control of the ball. The red teams left midfielder then opens a lot of space for himself, making a couple of passes before creating a chance and a goal. The question for the blue team is why they let him have so much space.

The Voronoi diagram, and other automated methods for measuring space, suits the way modern managers see the game. As a manager, Pep Guardiola is constantly hunting for new opportunities to exploit the areas on the pitch that the opposition has left open. After oppositions started to close down his Barcelona team in the middle of the pitch he found new ways of creating space out on the wings with Bayern.

It is still early days, but the dream is that in the future, these computational tools could help Guardiola and other coaches find new spaces in which to attack and identify areas in defence that they are leaving open.

Delaunay triangulations and Voronoi diagrams

These two mathematical structures are intimately related. Given a set of points, i.e. player positions, the Voronoi diagram divides the space in to zones. Each player’s zone is the area of the pitch that is closer to that player than to any other. In other words, the zone is the areas of the pitch the player would get to first (assuming all players run at the same speed). It is the area of the pitch they control. The Delaunay triangulation is then the lines connecting up players in neighbouring zones.

This story originally appeared on Nordic Bet’s Blog.



Professor of Applied Mathematics. Books: The Ten Equations (2020); Outnumbered (2018); Soccermatics (2016) and Collective Animal Behavior (2010).

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David Sumpter

Professor of Applied Mathematics. Books: The Ten Equations (2020); Outnumbered (2018); Soccermatics (2016) and Collective Animal Behavior (2010).