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 |

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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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.

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.

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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