The dynamic nature of space on the pitch

Space and time are the two most commonly utilised words in the vocabulary of football. The game of football involves players with personalities and physical attributes who are constantly moving, and their positions affect the free spaces which the opposition looks to recognise and exploit in order to progress the ball towards the goal. Therefore, success in the game is attributed to the accurate interpretation of these spaces by players during the game. 

The coach has the responsibility of conceiving a game model through which all the players of the team can easily recognise those spaces. The interpretation is however not merely an individual objective. Success in executing the game model or plan also demands that a group of players, or ideally the whole team interpret the same spaces in the same way. This means that at any moment in time, two or more observers in different positions on the pitch must interpret the nature of the same space with fairly high consistency. As a coach or an analyst, your role is to see spaces in the same manner that players do, despite having a very different vantage point off the pitch. If you want to convey your ideas using the lexicon of space in football, it is imperative to understand the nature of those spaces to achieve similar consistency in their interpretation. 

Football analytics has progressed to the point that using tracking data and pitch control, it is able to formulate the mathematical space occupied by every player on the pitch to great accuracy. Nevertheless, when it comes to analysing how the game is perceived in the mind of a player, current mathematical models have a lot more variables still to consider to accurately interpret spaces the same way that players see them on the pitch. 

Abstractness of space beyond geometry

The issue with attempting to achieve an objective analysis of space in football is its abstractness. Space on a pitch doesn’t exist the way it is usually depicted by coaches on a tactic board. Every player who has played the game understands this. A coach’s tactic board represents the players as 22 identical circular objects and their positions define the space available to play. For example, space between the lines is represented by the area of a trapezium bound by the positions of the four defending players (two defenders and two midfielders). When a coach illustrates this space at half time in the dressing room, it is a static two-dimensional eucledian space determined only by the cartesian coordinates of four identical circular board magnets.

During the game, space in terms of its physical properties is not static. Space is dynamic. It is not only bound by the positional coordinates of the players at that particular instant but also their momenta. The speed and direction along with the body profile of each player influences each individual player’s momentum which subsequently influences the space held by the players. Furthermore, apart from the spatial variables, there is also the variable of time. Space in football exists only at a certain moment in time and is rapidly changing in relation to time. This dynamic property of space means that it is always transforming. Hence, to understand spaces, it is necessary to understand how the spaces are transforming with time and not just their geometry at any instant in time. 

There are other variables that also affect the space such as the momentum of the ball, and the weather conditions and the pitch conditions. Subsequently, if the player looks for a teammate to pass the ball, the momentum of the teammate and the body profile are also taken into consideration to calculate the space to play a specific type of pass. During the game, players constantly perceive spaces on the pitch taking into account all these variables intuitively and simultaneously.

We call this “phase spaces” and is defined by: where the ball is, what situation it is in, where the opponents are, the distances between the ball and the opponents and our own players, the trajectories made by each player and each opponent and the ball, the way the game is oriented, the organization that the game has. And all of this constitutes only a game situation that lasts a tenth or two tenths of a second. The moment the ball changes from site, change the players and a new situation appears. And likewise successively.

― Paco Seirul·lo,
on his theory of ‘Phase Spaces’ (Espacios de Fase)

In the first example with the help of tactical illustrations, we see a situation where the CB who has received the ball from the left side of the pitch is looking to switch the play to the right. Positioned in the inside channel, the CB has two players in the outside channel, the RW and the RB.

The game model requires the RB to create superiority behind the line of press by moving into a position between the lines. For the RB to offer a passing option to the CB, the space between the lines needs to be interpreted. In this situation, we commonly define the space between the lines as the area of the trapezium whose vertices are the four opponent players – a, b, c and d. This is how we expect the coach or analyst to illustrate this space to explain the game model on the tactic board.

However, during the game, the players do not perceive this space as a static geometrical shape. It is perceived dynamically by also considering the momenta of the four players – pa, pb pc and pd respectively. Note that the teammates, the CB carrying the ball and the RW are also in motion and have their individual momenta pCB and pRW respectively, which influences the behaviour of the opponents a, b, c or d and subsequently affects the space for the RB to receive the pass.

The momentum of an object is conventionally represented by the letter p. It is the product of two quantities, the object’s mass (m) and its velocity (v): p=mv

Thus, we see that the space between the lines the RB and CB perceives is a dynamic entity in motion that undergoes instantaneous transformation. For the RB to receive the ball successfully, we have to consider how the space changes in relation to the movement of the RB with a momentum of pRB and the motion of the ball, pball, when the pass is played. The space changes if the opponents react to the actions of the RB and CB.

Hence, these two need to pick the right moment in time to execute the action while the space remains relatively constant due to the inertia of the moving opponents. (Newton’s first law of motion or law of inertia states that an object in a state of rest or motion continues to do so unless acted upon by an external force). Note that the position and movement of the RW in this situation is necessary for partly maintaining the space as constant for a longer time period by pinning back the defensive line formed by c and d

Now let’s look at this tactical illustration one frame further, after the pass has been played. The manner in which the actions of the CB (the passer) and the RB (the receiver) influences the nature of the dynamic space and their perception of it, determines the actions that follow. Depending on how the RB perceives the space, he/she may decide to receive it with a touch and play it directly to the CF facing the same direction of momentum, or turn and look to play a deep pass into the space in behind on the outside channel for the RW. 

As with the spaces between the lines, bound by the positions of the opponents, the spaces in behind are also influenced by the momentum of the players. If the previous example demonstrated how the motion of players of the same team influences the space, here we see the case of space being influenced by two opposing players. For the space behind the opponent d to be viable, it is necessary for d to jump with a momentum pd (press leaving the position in the defensive line) and for the CF to be in diagonal momentum pCF to attack the space. The sum of these opposing motions (along with the position of the last defender d) impacts the viability of the space. Similarly, the space on the outside channel is influenced by the momenta of pc and pRW, both who are competing to win the same space. In this case, the one who is quicker to recognise the space and react by accelerating to it wins the space. For the sake of consistency of using the jargon of physics, here acceleration (or more accurately – Force, f) is defined as the rate of change of momentum with time. Players respond to changing situations by rapidly accelerating and decelerating using the forces generated by their musculature.

The representation of players as circular objects having the same physical properties leads us into believing that they are able to move in all directions with the same magnitude and quality. Although it serves for simplicity while explaining tactics, I posit rather that the players are better represented having an anatomical orientation. This way it is easier to understand through intuition that a player can accelerate in a forward direction faster than sidewards or backwards. It also factors the law of inertia which provides a better interpretation of spaces in tactical illustrations.

The representation of the same example as above, but with anatomical orientation of the players conveys more information about the nature of spaces during the game

I dive into another example to illustrate how body orientation and motion affects the nature of space. Let’s take a 2v2 situation of a CM progressing with the ball in attacking transition against defenders a and b, with a teammate LW on the left.

The CM targets to play a pass through the defenders targeting the wide space on the outside channel. He/she predicts that the LW, despite being onside while the pass is played, has sufficient forward momentum to meet the pass in the targeted space. Let’s assume that the CM, right footed, decides to play the pass on the ground with the inside of the foot that curls inwards into the path of the LW. 

Although the defenders a and b are separated by a physical distance of 10 metres, the space to play the pass through isn’t defined by the mathematical value of 10 metres. In the first case, we look at what happens when the defender a is tracking back with a momentum of pa and the defender b who is facing the oncoming CM decides to step forward with a momentum of pb. The space to play the pass through is closing in and effectively less than 10 metres considering the time it takes for the ball to pass through the defenders with a momentum of pball.

In another case, we see the defender b tracking back the run of a third player CF with a forward momentum of pb. Here, the space to play the pass is expanding although the distance between the defenders at this moment is still 10 metres. Considering the momentum of the ball pball and the defender pb, the pass will pass through with greater success because of a bigger gap. Thus, we see how the body orientation of the players and their directional momentum influences spaces on the pitch.

Contemporary football is familiar with the use of Voronoi diagrams to mathematically calculate the space occupied by each player at any instant. In mathematics, a Voronoi diagram is a partitioning of a plane into regions based on distance to points in a specific subset of the plane. Using tracking data of players positions during the game, coaches and analysts hope to objectively identify spaces in opposition structures that they can exploit and adjust their players positioning during different phases of the game better. 

Above is a Voronoi diagram of two teams, the blues in attacking phase playing entirely in the half of the reds who are currently defending. I would like to zoom into a specific part of the pitch on the left where the left winger from the blue is positioned on the shoulder of the right back from the reds. 

The mathematical Voronoi generated suggests that the left winger occupies a major area of the outside channel including the space behind the defence compared to the right back, a. But if we look at the body profiles of the players along with their directional momenta instead of viewing them as circular points (from which the Voronoi was generated), we see that the LW is only able to gain superiority in the space in the outside channel with a forward body orientation and an initial forward momentum pLW which gives the LW an advantage over the defender a with a back/sidewards momentum pa.

It is easy to understand this scenario intuitively from the perspective of the CM who is on the ball. For the CM, the space behind the defence is only viable if the LW is in forward motion towards that space. On the contrary, if the LW is oriented backwards without initial motion, facing in the same direction as the RB, the CM cannot play a pass to gain the same advantage as before.

In this case, the CM doesn’t view the space in behind in the same way, although the positions of all the players are exactly the same as they were earlier. As the body orientation of the player is different, the subsequent spatial configuration should also change so as to favour the defender a than the LW. Here I have manually adjusted the geometry of the Voronoi to portray the disadvantage for the attacking team purely because of the body orientation and momentum factor.

The Voronoi diagram does not take into account the position of the ball, the speed of the players or which way they are facing, which invalidates any conclusions that are drawn from it. Pitch Control, an alternative created by William Spearman, Lead Data Scientist at Liverpool FC, is much better at estimating the probability of successful passes, and evaluating the passing options of a player.

It is undoubtedly a great tool to view how the players occupy spaces in any instant and analyse their probabilistic pass options. Nevertheless I find it still far from predicting how players interpret spaces during the game. The science calculates the variables that take place on the pitch as they happen, but players anticipate what will happen in the future time. A forward can make a double movement and the space he/she wins as a consequence is not in the direction of the initial movement, but rather the second movement. The player has seen this space and anticipated the movement long before executing it.

How players conceive space-time models

So far I attempted to explain the dynamic nature of space and how players perceive them during the game. But the game is played by displacement of the ball. As the players are mostly occupied with tracking and technically manipulating the ball, they aren’t able to observe this dynamic transformation of spaces the way you have just visualized previously as an analyst. Instead, through periodically scanning their surroundings, they construct a panorama in their heads with multiple images. This panorama helps them conceive a space-time model. A space-time model is a prediction of how space-time will transform in the near future. The closer the model is to how the space-time is in reality, the better the player is able to choose the right action during the game.

One can never know with perfect accuracy both of the two important factors which determine the movement of an object — its position and its momentum.

— Heisenberg’s uncertainty principle

As a player if you scan something in your surroundings moving at a particular momentum p, even when you look away you are able to predict its position after a certain time t. A player is constantly predicting the positions of everything in the surroundings – teammates, opponents and even the ball – while not observing them. Assuming that everything on the pitch obeys the laws of classical mechanics, the simple kinematic equations of motion helps calculate the same results that players do intuitively without any background in physics! Furthermore, the results are not as important as the ideas of space-time that are conceived during the game.

Kinematic equations of motion in physics

Let’s look at an instance of the blues looking to progress the ball up the pitch with the CB carrying the ball out of the back.

The CM is currently a free option to pass the ball to. For the CM to recognise that he/she could be a progressive outlet for the CB, the CB needs to estimate the nature of the space by scanning the surroundings.

Upon scanning, the blue CM observes the opponent red CM starting to press with an initial momentum pCM. This pressing action in the surroundings effectively changes the space that blue CM initially had to receive the ball. 

Although the blue CM has scanned and identified the oncoming press from behind, while looking away towards the ball, the CM needs to predict how much space he/she will have by the time the ball arrives. This would depend on both the momentum of the marker pCM and the ball pball and their respective distances.

By conceiving a space-time model that predicts what will happen by the time the ball reaches the CM, he/she will decide whether to play the pass back first touch, or to turn and progress the ball further to the AM. The space-time model also gives a sense of where the pressure is coming from which helps orient the CM with the right body profile. The accuracy of this model ensures a successful action that is both safe in terms of keeping possession and advantageous to the blue team. This is also an example of contracting space as a result of markers closing in. 

As another instance of space-time modelling, I take the instance of the RB in possession of the ball on the right side of the field. The opponents engage in a ball-oriented press trying to forcing the RB to either play the long ball or play backwards. The CM on the right outside channel steps forward to provide a narrow outlet for the RB in the space bound by the surrounding markers. 

On the inside channel, the AM is man-marked by the red CM. So, when the AM tries to move behind the line of pressure in front of the red defence, the CM follows closely. This creates space centrally in the inside channel behind the red CF who angles his/her run to press the situation on the right. The blue CM on the right who steps up, scans and observes the motion of the AM clearing up the space and the other CM on the left who is moving forward into that space behind the red CF. 

When the blue CM on the right receives the pass from the RB, he/she can securely play the pass behind the press of the red CF into the space being created. The space-time model predicts the motion of the players affecting the space and presents a viable opportunity for the CM to play the pass to the fellow CM in the centre. The initial scan detects the momentum of the fellow blue CM and predicts his/her position in the next frame where the pass can be played to break through the press. Although in the previous frame the space appears to be smaller, the movement of the AM and the marker away from it and the angled motion of the CF shows how the space is expanding.

Players use predictive space-time models to manipulate the perception of space-time by surrounding players. Take the example of la pausa, where a player delays his/her action for the space to become more viable and advantageous. This skill is a demonstration of how space-time modelling enables a player to predict how the play will develop in the near future and thereby manipulate the positioning of the surrounding players and their subsequent perception of space-time.

The relative nature of space-time

Relativity of space-time accounts for the idea that space is subject to the perception of an observer. The players who play the game are the active observers and thus their respective interpretation of space has an element of observer bias. How much space and time a player perceives he/she has is influenced by a lot of factors, but most predominantly by his/her scanning behaviour. Each scan gives an opportunity to capture an image of the surroundings where a picture of the surroundings reaches the observer at a speed of light. 

Space and time are relative — they depend on an observer’s speed. But the speed of light is more fundamental than either.

― Einstein’s theory of special relativity

Since the observers (players) are in motion, the perception of space and time is relative to each observer. Constant, efficient and active scanning enables players to model their space-time predictions as close to reality as possible and consistent to one another. Ultimately, when a pass is played between two players it is necessary for both of them to interpret the spaces similarly for the action to be effective.

The technical abilities of the player also plays a big role in the perception of space and time. Technically better players perceive having more time and space than technically weaker players in similar situations during the game. Perception of space is also dependent on player profiles and positions. A central defender tends to perceive more time and space being on the outside of the positional structure than a central midfielder or a forward. Similarly wingers or fullbacks perceive the spaces on the outside channels differently. By virtue of players getting habituated to constantly perceiving familiar spaces because of their regular positions, they tend to shape their playing profile and instincts accordingly. A midfielder might sense the need to make a quicker decision and switch the ball to the other side because of contesting for tight spaces among opponents and focus on collective superiorities, while a winger tries to buy time and isolate opponents in wide areas to take them on in 1v1s more frequently. 

Fundamentally, I believe that the ball is the only thing that is absolute in the game. Everything else is relative, subject to the interpretation of the players who play the game. Although the geometric standards of the field, and it’s canonical elements like the dimensions of the penalty box or the boundaries of the goal are also absolute and irrefutable, the spaces within them generated by the players’ positioning and their subsequent actions are interpreted relatively.

This explains the technical challenge of scoring a goal versus playing a pass. A pass, even a long one, can be off by a couple of metres and still be counted as accurate because the receiver has the freedom of space to adjust and adapt to the margin of error and still receive it successfully. A shot on goal, however, if off by even millimetres results in rebounding off posts or a fingertip save by the keeper. Even the referees and linesmen cannot be a hundred percent objective in their interpretation of the players actions on the field, since they all are in motion, and are limited by their respective vantage points. Therefore I believe that only the ball is absolute, much like the elusive truth we constantly seek in our lives. Everything else ― Space, time, actions, tactics, justice or motivation ― is just an illusion. On the pitch, I, as a player, am trying to sell you the illusion that I have more space than you, the referee my perspective of justice, my coach a belief that the tactics are working and the spectators the motivation of collective triumph through a game of football.

Without the ball there is nothing; the ball is the mother, the source of life in football. What’s the goal there for? For the ball to go in it. Without the ball, nothing has any meaning.

― Juanma Lillo

Chaos or order is just a matter of perspective

I believe that the abstractness and relative nature of space should help us appreciate the power of perspective in the game, and in the universe. I reflect upon an anecdote a good friend of mine once shared: He said, “In our daily lives, we appear to be living in chaos and the nature of events manifest a great deal of unpredictability. But if we zoom out to the level of planet earth, we observe a relative order how the earth spins on its axis each day so consistently. Then we think about the geoclimatic changes, the great land masses and oceans moving on the planet and find another level of chaos, but if we zoom out to the level of the solar system, we find more order in the structured orbits of all the planets. And we can extrapolate this nature further on to the Milky Way or the expanding universe and notice the same phenomena. The variables of space and time are ultimately a question of perspective.” 

Similarly, playing the game of football is like being in the midst of chaos. The events happening around the player are so dynamic and unpredictable, and no player recounts having experienced the game in ‘real time.’ But sitting in the stands high above the level of the field, we question why the pass wasn’t played to the winger who has been positioned in free space for the last two minutes. Even further away, watching the game on a screen sitting on our couches we notice far less of what happens during the games, doused by the relative order of the same attempts to attack and score that our team has been attempting for the past hour.

The objective of a game model is to offer players cues to recognising the same spaces efficiently through common principles of play. To develop as a coach or an analyst, we must embrace the chaos that the game presents at the level of the players who play it and not hide behind the order of observing the game that a privileged perspective offers. Understanding spaces in the game as dynamic entities rather than static geometry helps us appreciate how the game is actually played and what makes it beautiful.