The Most Famous Shape In Football - The Triangle

Football is a beautiful game when it is played in the beautiful way that it should be played. It is a sport which combines the strategic thinking of a chess master, the touch of a Formula 1 drivers' foot, the balance and poise of an Olympic gymnast and the creativity of a master painter. It pleases our senses, stimulates our impulses, shapes our feelings. These are the reasons that this sport of ours is unique, unmatched by any other. Sure, you get those who say you can get all those qualities in basketball, tennis or rugby. For those who do say that, I have some bad news for you - it's not the same. Nowhere close.

Art is everywhere around us. Have a look at the world around and tell me what you see. What do you see in your building-on the outside of your building, the landscape around you, the cars, roads, and man-made structures. The answer is shapes. Virtually every man-made structure is based on the fundamental mathematical division called geometry. It is the shapes of geometry which allow our architects, engineers, designers, builders and artists to create the buildings, tunnels and bridges of our world. Just like in our physical world around us, our beautiful game is based on geometry and just like the physical world, the shapes that are created by football are disguised by our inability to see the deep aspect of football, or profundus. With such a huge amount of our attention invested on the actual ball, we miss the shapes that would seem obvious to a coach or one who is looking them. Such shapes are created by the relationship of players to each other, the relation between the balls' position on the pitch to the position of the players, and of the balls' movement on the pitch. It are these shapes which occur thousands of times in a single game and are the ones which spectators are oblivious too. Quite often, the only time a spectator (especially a spectator who is not at the ground) will only be prompted to notice these patterns when they are pointed out by the commentator. Even then, the average spectator will soon forget about these patterns and resume their narrow-sighted attention on the ball and not much around it. So why does it seem so difficult to concentrate on these patterns? It is because they are so complex.

If we have a look at a 433 formation-the most fantasised of triangulated formations-it is clear to see what's on offer in terms of the potential passing routes. Graphically, there is a potential to have every player to be part of at least one positional triangle in relation to his teammates. Since this is the case, it is also possible for the ball to make a triangle on the pitch in three passes with three players. Another option is for only two players to complete the triangle if one of them moves to compensate for the third player not being there. Furthermore, there are thousands of possible combinations of passes that the players can make and the path that the ball can travel. It are these possibilities which people tend to associate with possession football.

The potential triangles in a 433 and the lack of in a 442. Image from futbolforgringos.com

The next question we must ask is whether this association between triangular spacing and possession is in fact a correct one. To answer this question, we can look at the actual geometry to reveal the answer. Firstly, ask yourself this: can you create a shape on the pitch other than a triangle using only three players. The answer is no. The only way you can position three players on the pitch without forming a triangle of some sort is by placing them in a straight line. In football, straight lines are the enemy of footballers because this is the worst type of spacing that three players can make. The reason is because only one player in this line has the option of passing the ball to both his teammates assuming that there are no defenders in between. The players on either side of him can only pass to the teammate in the middle. The only way that a player at the end of the line can pass to the other end of the line is by chipping the ball in the air or curving the ball around the middle player. This is bad for accuracy and requires greater skill to perform, thus reducing the efficiency and probability of a successful pass. To sum up- only one of the three players has the option to pass to the other two player, while two players can only pass to a single teammate.

This nicely leads to the principle of triangulation, or, spacing players such that they form the triangle shape. The first thing to note about a triangle is that a triangle cannot, by the law of physics/mathematics/geometry, exist on a flat plane (the playing surface) with having some sort of latitudinal and longitudinal spacing. In other words, at least one of the points of the triangle must have at least one of either horizontal deviation or vertical deviation. If this does not happen, the three points would essentially be in a straight line. If you still don't understand, try thinking of it like this. Imagine that you draw two points on a paper (the paper represents the football field). If you connect these two dots with a line, you will have a straight line between the dots. Now extend this line so that it goes beyond the dot. Now draw another dot on the extended part of the line. You should now have three dots on a single straight line. Because you drew the last dot on the extended line, you still have a straight line and not a triangle. However, if you drew the third dot off-centre from the extended line, you have created a triangle because you have deviated the last point so that it is no longer directly in line with the first two dots. Think about it, to deviate the third dot off-centre from the straight line, you either moved the dot left, right, up or down. That, essentially, is what I mean by latitudinal and longitudinal spacing.

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Now that we understand a basic property of a triangle, we can now explore how triangles make it easier for a team to keep possession. Before we do so, we must firstly understand a basic requirement that allows a team to keep possession. This requirement is teammates for the ball possessor to pass to. If a player with the ball has zero available teammates to pass the ball to, it is virtually impossible to keep possession of the ball. Triangulation helps to solve this problem by utilising a brilliant geometrical property of the triangle- the hypotenuse.

The hypotenuse is a property of the right-angle triangle and it is the key as to why triangles are so effective to keep possession. It has to do with distance. If we imagine a right-angle triangle, we know that the hypotenuse is the longest of the three sides. If we had three players who formed a right-angle triangle on the pitch, the two players who are stationed on either end of the hypotenuse will have the longest pass to each other in terms of distance. The other two possible passes are both shorter in distance. This means that a defender would have more distance to run along the hypotenuse and less distance to run along the other two sides. This has important implications for football because if we can can figure how to manipulate the triangle so that we can make more use of the longer hypotenuse and less use of the shorter sides, possession football will become easier because the opponents would have to cover more ground to reach reach the player with the ball and his teammates. This effectively means extra space is created for the team in possession and consequently increases the time they have to make correct passing decisions and makes it easier for his teammates to find space to receive passes.

The way to achieve this is to ensure that no two consecutive players are on the same longitude or latitude as each other. l.e. no player is directly vertical or horizontal to his nearest teammate. In the following diagram, we see a midfield four that are all on the same horizontal plane, meaning there is no vertical deviation at all. In the second midfield, there is a clear vertical and horizontal spacing between the players, i.e. vertical and horizontal deviation.

What this does is create triangles which maximise the distances between each point of the triangle. In effect, it creates triangles entirely made up of hypotenuses. The knock on effect of this is that, since the distances are maximised by utilising both vertical AND horizontal spacing, it is harder for the defending team to intercept passes, close down opponents with the ball, and mark the players without it. The other effect it has is spacing the players of the defending team away from each other so that they cannot keep a compact shape or press in numbers. This reduces more of the game to 1v1 battles and favours the team who has better technique and tactical organisation to pass the ball and keep possession. All these favourable outcomes are due to the triangle, and cannot be achieved with any other shape.

As triangles are crucial to the design and construction of real world structures such as buildings and bridges, so too are they crucial in the design of a football team. When you hear a comment such as "passing in nice triangles" in reference to a nice piece of passing by a team, they are referring to the concept of triangulation. Even though people tend to ignore the deep analysis of geometry to explain why triangles are so important in football, they are nevertheless correct in making the connection between triangles and effective possession. Perhaps the team that are currently most famous for the use of triangles is Barcelona. One term which sums up perfectly this style of play is "strangulation by triangulation". When you have players such as Messi, Xavi and Iniesta playing in triangles across the whole field, it makes for a lethal combination.

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In the end, triangles in football are not just something that is thrown around by managers and the media when referring to possession. It is a fundamental geometrical part of football, something which makes football beautiful. Just like the most famous artists, the brightest mathematicians and the engineers of this world use triangles to create and innovate new methods, styles and discoveries, so too does football. When we start to truly look for the profundus, the deep side of football, we can truly appreciate how important and fundamental the hidden world of geometry is to football. Perhaps it is only appropriate to label the humble triangle as the the most famous shape in football.

Are Systems Overrated? (Bonus Article)

Are systems overrated? What do I mean by that. I am talking about systems of play and formations that the average fan talks about e.g. 442, 433, 4312 and so on. My question is whether we talk and compare formations of two teams playing against each other too much. Do we overrate the way a team is written on paper in the sense that the formation a team adopts on the field has a large effect on the flow of the game and ultimately the result.

To give you a prime and recent example I point you to the recent game between Italy and Spain in the 2012 European Championships. the major tactical talking point was Vicente del Bosque's decision to not start with any of his three recognised strikers and instead play with Cesc Fabregas in a false nine position. While this was a surprise to many observers, it wasn't a major shock by any stretch of the imagination as Spain can comfortable play without a striker because their ability to possess the ball is unmatched by any other national team currently. In theory, playing a midfielder in the number nine position should enhance the team's ability to possess the ball for even longer periods of the game. After Italy scored first through Antonio di Natalie, Fabregas equalised soon after and the game ended in a draw. While Fabregas scored a goal in that game, the general consensus from journalists and supporters was that the striker-less formation used by Spain restricted Spain's goalscoring chances. The media suggested that a striker must play the next game against Ireland and indeed the inclusion of Fernando Torres in that game yielded two goals for him and a 4-0 win for Spain. It seemed to confirm in many people's eyes that by playing a natural striker who would play on the last man, Spain had a focal point to aim for and created many more chances to score.

You could argue that this is the case but you could also equally argue against it. Was it the change in formation that allowed Spain to win? Possibly. However, it is even more probable that it was just a case of Spain's opponent being of a lower quality than Italy. It is natural that you would expect Spain to score more goals against Ireland than you would against Italy. Consequentially, the argument that Spain's change in formation was the largest contributor to them being more successful in the second game than the first suddenly makes less sense and logic. It is by this example where I question whether tactical formations are over-emphasised in the media. I am not entirely dismissing the notion of formations entirely, far from it. I am simply pondering if the result of games are attributed to the formations of the two teams. To show you an example of what I mean, I would like to point your attention to a common stereotype of a 433 vs 442 formation battle.

Dream team vs Fantasy team. 433 vs 442. Who wins?



The standard perception is that the team playing the 433 will outnumber the team playing 442 in midfield and will therefore dominate possession. On the other hand, it is said that the team playing 442 will deliver a greater threat up front via the twin striker partnership and can also dominate the wings by utilising the space between the opponents wingers and fullbacks. This stereotypical attitude is common throughout the world of the football commentator. You often hear a commentator complain that,

"the striker is isolated upfront and has no support. I think the manager should throw on [strikers' name] at halftime and change to two upfront. The gaffers' got to shake things up a bit, otherwise they'll never get back into this,"

or something to that nature. Naturally, the solution always seems to throw on another striker and change formation and somehow that is always the way to fix the problem on the pitch. The problem I have with this attitude from pundits and fans alike is that it seems that they are not trying hard enough to properly think and analyse the situation at hand. To often I find myself in a position of watching a match and virtually predicting word-for-word what the commentator is going to say next. I find it a rather unique skill to be honest.

The main thing to take out of this is that too often in football it is the micro-tactics which are neglected; that is, the tactics which look into really detailed analysis of the individual interactions between players and the movements and spaces these create. It is these aspects (or "finer details") that ultimately make the difference between who wins and who loses. By being to broad and too general is our analysis of the games, we are missing the key substances which are the real difference makers. It is perhaps a symptom of a lack of expertise in the field of micro-tactics which causes commentators and fans to become so predictable and boring in their communication of the tactics of a match. Not to mention that it takes an incredible amount of data, time and work to compile some resemblance of a usable and useful dossier on micro-tactics. However, this is slowly changing as more data and statistics becomes available to the public in the form of television, internet, communications technologies and modern performance data capturing capabilities.

While the trend is starting to go in the direction of micro-tactics, there is still a huge misconception to those unfortunate enough not to have caught up to modern trends, that believe that a 433 will always dominate in midfield against a 442. Unfortunately, you may be hearing that from a commentator who is voicing his opinion in the next game you watch on your television.