Regular Polygons

I have been wondering about spaces used in combat sports, almost all of which are formed by regular polygons: from the circle of sumo, to the square of boxing, to the octagon of mixed martial arts.

Regular polygons: circle, square, octagon

The circle is known in many cultures as the perfect shape or form: all lines of force radiate perfectly from the centre of the circle to its perimeter. In gladiatorial sports contested within a circle (which also include freestyle and Greco-Roman wrestling and various forms of animal combat) motion is continually assured by the degrees of freedom that this geometrical form allows; even when against the perimeter boundary there is plenty of room to maneuver.

The square competition area in organized boxing has existed at least since the institution of the London Prize Ring rules in 1743. Why the square (ironically referred to as a ring) instead of the circle for boxing? Is it because of the strong linear references characteristic of architectural forms in the modern age?

(As an aside, the circular area of competition in Greco-Roman and freestyle wrestling is inscribed on a square mat, which is then rolled up into a cylindrical form for storage. Similarly, the circular sumo dohyo is inscribed on a raised square platform, which presumably is meant to honour the historical traditions of the sport while fitting into the rational rectilinear spaces of modern Japanese arenas. Was this always the case for spectatorship of sumo?)

With the square, we cannot make the same claims to maneuverability mentioned earlier, despite its perfect symmetry on all axes that bisect the centre point. Lines of force are fairly constrained along horizontal and vertical dimensions, which leaves dead spots of motion in the corners — the last thing a boxer wants to do is get trapped in a corner with no line of flight to escape. This is not to say there is a dead spot in action; to the contrary, the constraint on motion often yields to a violent increase in action.

As for the octagon of mixed martial arts, it seems to exist primarily as a form of differentiation from other combat sports that serves an important role in the brand strategy of the Ultimate Fighting Championship organization. But beyond this pragmatic association, the octagon seems to offer us, by way of superficial observation, a hybrid of the two spaces mentioned already. Geometrically, this makes sense to us: a circle is nothing more than a regular polygon with an infinite number of sides, so the fact that the octagon doubles the number of sides of the square suggests that it will be more like the circle in the way it structures movement possibilities within.

Interior AnglesWe notice the difference particularly in the corners of each polygon: the interior angle of a boxing ring (black line) measures 90 degrees, while the interior angle of a mixed martial arts octagon (red line) measures 135 degrees, giving combatants in the latter space 45 more degrees of freedom to maneuver should they become trapped in a corner. What does this mean in terms of practical consequences? It suggests a competition with more mobility, more movement, and more action.

But we must qualify this last term: what do we mean by "action"?

In the glory days of prizefight boxing last century, action often meant a flurry of punch combinations being rained down on a boxer trapped against the ropes or in the corner; large, lumbering, powerful boxers, relics of the industrial age. Action today, by contrast, is far more about speed, with the goal being to sacrifice as little of the earlier gains made in power as possible. Thus, the newfound mobility offered by the octagon creates new kinds of strategic challenges as part of the action. How does one engage an opponent without sacrificing too much power in the more open space at the middle area of the octagon, or when retreat by the opponent is more easily possible, particularly in a lateral sense?