aFootball: The Football Universe
Charles R Paez Monzon - 2019 - aNatureTechnologies
- Mein Fuβball Weltanschauung
2.1. A Relational Dynamics Model for Football Universe
In this work, the arquetipical mechanics will dominates, but with persistent intromission of the alternative view [Smolin] of relations between things, agents, events in the unbounded landscape once we have a model of the internal structure of the world, the ‘world of football’ in our case of study. Science needs mathematical theory to model the world phenomena and to discovery the laws of relational dynamics based on a preminence of the ideas of observable of relations and state of relations. Let us explore this mathematical perspective of the world in order to have an idea about all possible world that can happens in the relational Footbal Universe or aFootball1.
This relational football is already present, unconsciously, in the middle of football data science departments and enterprises like StatsBomb, Opta, Wyscout, BarcaInnoHub, etc. Their dataframes are events driven and columns pairs are relations between relevant observables inside the pitch. Figure 2.1 shows a dataframe of all the events in the range (2526..2537) of the effective possession phase 152 of 12 events, that corresponds to the 98th possession brake ‘PB’. Phase possession of ‘Barcelona’ starts with from a throw in ‘play pattern id’ 4 at xy-location (13.9,1.0) in the minute 68. The sequence of events ends up in event 2637 at xy-location (63.9,36.7). This manner to describe football has adopted the physical-mathematical model of observables and states relations that marks time as an event-driven variable, that adopts the manner footballers experience their football universe on the pitch. I am mathematical modeling such an experience, that is all.
2.1.1. Relevant Observations
In football, the relevant observation of Gon is measure change in ball position and game-time, together and yields an event-coordinate tuple (α, β). This relevant observation is an event. Each event has relations with the rest of the football universe and, that set of relations constitutes its view of the universe. Each view has to be unique.
The ‘actual’ in a CHAS complex hierarchical adaptive system, as a football game, on average keep expanding into the ‘adjacent possible’ [S Kauffman]. By doing so, they increase the diversity of what can happen next. Increase the ‘novelty’ of events. Each player maximize the rate of exploration of adjacent possibles as fast as they can get away with it.
If in the ‘actual’ there are two events, let us explore their ‘adjacent possible’ 1-step neighborhood away from the ‘actual’. And sooner than later we can identify n-step neighborhood where they start to diverge. If n is smaller than m, these two events views are more distinguishable from one another in n than in m. Therefore, we most expect with the latest mindset of science that football dynamics maximizes ‘variety’. Two players associated with these two events do not need to be near to have very similar views because they are in a closed space. So they could be happen far away and not necesarily near by. So, synergy can create ‘entanglement’ of their views and intelligent and synergestic decisions can be simultaneously adopted.
Let be C the 2D relation-space with event-coordinates (α, β) on the unbounded landscape.
Let us look forward for mathematical laws that rule sequence of events. Call an unparametrized curve γ in C a motion γ of the system - a football motion -. Perform a sequence of measurements of pairs (α, β), and find that the points representing the measured pairs sit on a motion γ. Then we say that γ is a football motion. We express a football motion as a relation in C
f(α, β) = 0 (1)
Thus a motion γ is a relation (or a correlation) between partial observables. Let say, between the player on the ball and his teammate that moves to open space to receive the pass. Or, between the player on the ball that decide a pass and a rival player who guess his action and intercepts the ball re-gaining possession.
2.1.2. Space of Space of Any Possible Football Flow
Then, disturb the system - with a football action like receiving or intercepting the ball being passed - and repeat the analysis all over again. At each new disturbance, a new football play - motion γ - is found.
At the end, The space of sequences of football motions or trajectories in the unbounded landscape is limited. It is just a 2D subspace of the space of spaces:the infinite dimensional football space of all football motions. Only a **2D space Γ of γ curves - _trajectories- ** can be realized in the unbounded landscape.
Bingo. If we fix the place of the goal archeries, and patch the unbounded landscape; we get a bounded landscape and we can know about all possible football flow dynamics that can happen in the bounded landscape. ! We can compute - the all possible football flow dynamics - without any player. ! We have foresee all aFootball plays.
In a small perturbation, we can coordinatize a football motion γ by - a rotation - two real numbers A >= 0 and 0 ≤ φ < 2π, and (1) is given by f(α, β; A, φ) = α − A g(ωβ + φ) = 0 (2)
and gives curve γ in C for each couple (A, φ). Let us call it (2) the matemathical law of football relational dynamics.
2.1.3. Relativistyic State
As Γ is the 2D football space of football motions, coordinatized by A and φ. Γ is the relativistic phase space of the bounded landscape - the space of football motions -. And a point in Γ is a relativistic state, the second relevant concept that we have been looking for.
A relativistic state is determined by a couple (A, φ). It determines a curve γ in the (α, β) plane. That is, it determines a correlations between the two partial observables α and β, via (2).
When a team re-gain possession during a match, let me say at alpha α-point relativistic state, through an alpha-event open the adjacent possible of a sequence of intelligent, cognitive and synergestic event pipeline of football flow dynamics towards the rival goal. Such a sequence is a protopian-trajectory of incremental progress of ‘expected goal xG’ accomplishing xG football motion by football motion in the sequence that in case of end in an omega event with a pass to the net of the rival goal we call it a Levy flight, an actual football flow dynamics from alpha α-point to omega w-point where a teammate player scores a goal.
The most important relativistic observable will be the scored minus conceded goals or ‘goal difference’ of each team and in terms of analytics, the ‘xG_difference’ product of the effective action porcentage executed at the collection of omega- events that happened during the match. A relational view of a Levy Flight or an effective phase possession that ends up in a goal scored has been described and illustrated in relational drawings produced with ‘matplotsoccer’ library [Decroos] like show in Figure 2.2.
Now Gon has been learning and starting to talk a new ‘language game’ for aFICS football and the local community - HC technical team and synergons - that now on will begin to speak it fluently. But,any how the mathematical mindset push me to formalize it.
2.2. The New Language Game for aFootball
The triple (C,Γ,f) called language game is sufficient to describe all the analytics and predictions of classical mechanics and broad enough to describe possible adjacent relations between relevant observables in the football flow dynamics.
The language game for aFootball is based on a set of concepts
- The relativistic configuration space C, of partial observable
- The relativistic phase space Γ, of the relativistic states, and
- The evolution equation f=0, where f: Γ X C -> V a linear space.
The state in the football phase space Γ is in stasis until the system is disturbed. Each relativistic state in Γ determines (via f = 0) a football motion γ of the system, namely a relation, or a set of relations, between the observables in C. Once the adjacent relativistic state is determined (or guessed), the evolution equation predicts all the possible events, namely all the allowed correlations between the observables, in any subsequent measurement.
In the football language, notions like instantaneous state, equilibrium state, evolution in time, observable at time t have no role to play and have no sense in this context. May be, we must take into account:
- The thermal time hypothesis. In nature, so in football, all variables are equivalent; we can find the system in an arbitrary state ρ; if the system is in a state ρ, then a preferred variable is singled out by the state of the system. This variable is what we call time and has no meaning in the structure of the world and the flow of time is the flow in which football happens to be, when described in terms of the macroscopic parameters we have chosen. So, the language game and football flow dynamics are event driven.
This new ‘language game’ to be used about the new football universe in which all are immersed today. It is an innovation and a disruption to be assimilated as culture or dominant mindset for academies, clubs, associations, players, head coaches, trainers, agents, journalists, fans, enterprises and, any alien in the aFootball Universe.
An aFootball of rotors and multivectors. But, this comment is material for another special blog publication.
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In geometric calculus disciplines any 2D motion is a rotation and any trajectory is a sequence of rotations. ↩