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Playing safely with sticks and stones

People who train Capoeira claim that it is a game, but I cannot completely agree.  It is like gymnastics or fighting sports, and both of those have a way to win.  For the former, the winning "player" or gymnast achieves the highest average score from a group of judges, and for the latter, the fighter wins by knocking down his opponent or at least by landing a few hits in the allowed places.  On the other hand, Capoeira is never much played with any formal scoring system.  The players don't even need to make contact during a "good" game.  How to win isn't clear, and in fact, what winning would even mean isn't clear either.

If I were to guess why people insist that it is a game, then I would guess that the reason is because the two players are basically pretending to fight, and as such they are not engaged in a real fight.   If I think back to the real fights that I had when friends or classmates lost their tempers, there was dignity at stake, and the fighters didn't have to win clearly in order to keep it; they just needed to avoid losing badly.  We would try to hurt or overpower each other, and one way to win--or at least to seem not to lose--was to make a fool of the other guy or get in a few good hits.  Capoeira may make use of attitudes like this also, and it is just a game because no one should take it too seriously.  However, the lack of serious fighting does not imply that it isn't a dangerous sport or that its players aren't basically training in the same way that real fighters would.  A Capoeira school is not a militia group because the schools typically have a much more playful attitude.  

Hopefully the sense of the word game as it applies to Capoeira seems clearer now, but still I wonder if there is a way to play the game with a few rules, transforming it into more of a game in the sense of there being a competition with winners and losers.   Recently I've been working on a set of rules that might function as such without losing the attitude of play-fighting. 

Roda divided into four quadrants.

The first rule fixes the available positions within the circle of play to a total of eight different ones.  Players would have to stay in one of the spots, but from one spot to another the players are allowed to move but under certain constrains.  Another rule would constrain the players to a predetermined subset of playable moves.  To score a point, a player has first to flank the other player and then play a move (such as a kick). 

The arrows represent the eight possible positions that players in the circle can take, and only one player is allowed to occupy any single quadrant at one time.   Hence there are four different lines where the two players can face each other, and there are eight different arrangements where one player flanks the other.  For example, the two pictures below represent a transit in which player one (orange) performs the au and player two (blue) takes a forward step.

Figure showing the movements of two players during one measure

The two players start off facing each other, but after the turn player two is on the right flank of player one. Depending on who arrives late or early, player one can still evade an attack because he is likely to have arrived first (since player two must wait for him to leave).

The three Ginga positions shown in a code

Thus the division of quadrants and the reduction of movements may make the game seem a bit too rigid, but it at least has the advantage of notation.  The three symbols represent the three ginga steps interpreted such that the outline circle is the right foot, the solid circle is the left foot, and the line just indicates the forward direction.

Roda symbol showing two players

 As an example application of the notation to a game scenario, suppose that from opposite corners, the red player and the blue player are both facing the same quadrant, but their ginga phases aren't the same.  For instance the blue player has both feet side by side in the central ginga position while the red player is in the left ginga position.  The circular illustration attempts to represent the situation.

The system might serve the purpose of defining a kind of kids' game with sticks and stones laid out on the ground in patterns.  The kids would take turns each moving their Capoeirista around the roda.

Also, the game circle played this way incidentally exhibits the symmetry of a structure known as the dihedral group in terms of abstract algebra. The permutations of moving from one of the eight positions to any other form a subgroup of the symmetric group (S8) on eight letters, and that subgroup is isomorphic to the dihedral group of the square (D8), which is known to be generated by two elements, particularly the quarter-turn rotation and a reflection. Hence, we can consider every movement of the capoeira player from one position to another as a combination of rotations and reflections.