Yesterday I wrote about the many holes that had been poked in the theory of hybrid construction I posted a few weeks ago, among which are its incompatibility with any timings other than split-time or same time and the fact that it can't account for a static spin versus extension hybrid. Thus begs the question of how exactly we can define hybrids in a way that is extensible (ie, that works at any size shape we can image).

To get to the bottom of this, let's take a look at what makes each combination of timing and direction tick in the first place. Fundamentally, what we describe with each timing and direction combination are the interactions of three or more points. The circular motion of poi around the hand or another point of interest, such as mid-leash if one is performing an isolation is our basic element here. Fundamentally, with poi in hand I can think of only two basic types of motion: either the poi orbits the hand or it switches back and forth under it in pendulum. We can add the additional variable of a center of rotation for the hand as well, but it doesn't change either of these basic properties. Even when hand and poi share the same center of rotation (extension), relative to the hand the poi still completes an orbit of it.

When we add one more point to the mix (ie, a second poi head--right now I'm working on the assumption of the hands remaining together as in the case of classical poi motion of butterflies and weaves), then the two poi heads are moving in such a way that we can contrast their motion. This is where our ideas of timing and direction come from--we arrange them according to points at which they are most divergent or most convergent. In opposites, the point of furthest opposition occur either at the sides (same time) or at the top and bottom (split time). In same direction they are either eternally convergent (same time) or eternally divergent (split time). As Charlie helpfully pointed out in his 9-square poi theory series, when we add the additional variable of the two hands moving to this equation, they are able to make transitions without changing the direction of the poi and thus the hands can travel through all four combinations of timing and direction with the hands by switching between antispin/extension and polyrhthym hybrids in which one hand is in antispin and the other in extension.

But this begs a fundamental question: we assign a limit of 4 timing and direction combinations according to these four points of reference (top, right, bottom, left)--why? As noted yesterday, this type of framework makes a poor fit for triquetra vs extension hybrids as at either bottom corner while they appear to flash for a moment through same direction/same time, this is ultimately an illusion--they're in opposites but the timing is now in base three rather than base two. What if we attempt five petal flowers or higher? These truly don't break down in any way that easily fits into the four basic timing and direction combinations framework--which probably means that the framework is itself flawed.

The second nail in this coffin comes from watching Charlie's 9-square videos. I mapped through the transitions of all four timing and direction combinations to realize something that should have been obvious from the outset: while one could use it to switch seamlessly between the T & D combinations with the hands, the poi were a different matter entirely. Indeed, without a stall or pendulum, one was locked in either same direction or opposite direction the entire time. There were really only two modes and they were mutally incompatible.

Enter a cool TED Talk:

Viewed in this context, poi spinning takes on a slightly different context. Throughout the video, Sautoy is referencing a kind of symmetry that is based upon rotating a given figure. This isn't the only type of symmetry that we as human beings find aesthetically pleasing: there is also the type where a figure is seen to mirror itself over a given axis.

Viewed in this context, we can see each of the different classical combinations of timing and direction as expressions of these two types of symmetry. When both poi or both hands are moving and we frame their motion relative to each other, the variants of same time or split time can be seen to be subsets of either type of symmetry. When the two hands are moving in opposites, they reflect over either a horizontal (split time) or vertical (same time) axis, whereas same time is subject to the same rules of symmetry as the shapes Sautoy visits in his TED Talk: when producing two shapes in same time/same direction, they exhibit the same properties of symmetry when rotated 90, 180, and 270 degrees as the figures he mentions at the mosque. When hands or poi move in same direction/split time, we can rotate the figure draw by either hand 180 degrees and likewise find that they are tracing out the same shape.

This by itself isn't revolutionary, Cyrille arrived at the same conclusion in his paper on poi geometry, but what he only began to cover is what happens when we work through patterns that are symmetrical along degrees of motion in increments other than 90 degrees or what happens when we explore asymmetry between the two hands or within a single cycle.

Given that same direction/split time movement will yield shapes that are symmetrical in both respects upon completion of one 180 degree increment of movement, it would be most useful to view this symmetry in the context of any motion within an infinitesimally small range of 180 degrees. This is appropriate given that it overlaps the description of degrees of twist also occuring in 180 degree increments. One possible extension of this framework would be to see these symmetrical comparisons occuring over degree of twist increments.

To explore this idea, I've worked up two diagrams that display how poi or hands moving either same direction or opposite direction can be seen to work in multiple timings when one opens up the degree of symmetry to be in points of reference other than simply up, down, right, and left. Indeed, poi continue to display radial or axial symmetry regardless of how many degrees of separation there are between the two poi/hands or what angle the axis of reflection happens to be between them.

Sautoy or "Radial" symmetry:

Reflective or "Axial" symmetry:

Viewed in this context, there are not four combinations of timing and direction, but two degree permutations of both types of symmetry. While there are few people who can spin in either quarter or triplicate time, this breakdown covers not only their styles of spinning, but a conceivably infinite number of variants based upon altering the degree of inclination of the axis of symmetry or degree of separation between the hands, depending upon symmetry type.

The same rules of symmetry continue to exist as we add iterations to this idea. For example, if both hands are performing antispin flowers in axial symmetry at zero degrees (traditional opposites/same time), we find that the same property of mirrored symmetry continues to exist even though each hand/poi combination appears to be moving against each other. The same is true of extensions, inspin flowers, and any other motion derived from each hand/poi combination performing deliberately symmetrical geometric patterns. But what of when they are not performing deliberately symmetrical patterns?

Take, for example the pattern Tank gave me of a static spin versus extension hybrid: in this case, no matter how much I rotate the degree of origin for either hand/poi combination or reflect each's motion, they will never draw out the same shape. A static spin will always create a circle that is not as wide as an extension and an extension will never create a circle in which the center of rotation is the same as the hand. When viewing the motion of each combination of each hand/poi combination across a degree of twist, the shape of this hybrid will never be axially nor radially symmetric.

The same is true of polyrhythm hybrids. Even if we take the old standby, the extension versus triquetra (otherwise known as the "mercedes"), we find that given the fact that one hand is creating an antispin flower and the other an extension, they will never be symmetrical at any degree of radial rotation from the origin. We can test this across every variant of this hybrid in the original combinations of timing and direction that we are familiar with and still it will not produce a figure that is either axially or radially symmetric.

It should be noted that this breakdown makes an important assumptoin: that both hands will complete a single cycle of the type of movement they exhibit. The reason for this is that breaking from a pattern before completing a cycle is a trait exhibited by none of the patterns traditionally viewed as hybrids.

If we accept this methodology, the only conclusion that can be reached is that our traditional ideas of timing and direction are in fact narrow ranges of possible combinations of symmetry rendered in different degrees of separation or reflection, that all flowers or compound circles are in fact iterative extensions of this concept that can be applied at any range of circle or shape sizes, and that hybrids should then be classified as any two-hand movement in which each hand completes at least one cycle of motion and the shapes rendered are both axially and radially asymmetric when compared to each other.

Traditional combinations of timing and direction represent comfortable expressions of degrees of radial or axial separation for these two types of symmetry as they line up with comfortable points of reference for us: the furthest right, left, up, and down points that we can reach, but framing poi motion in this way restricts our capabilities for engaging in a plethora of poi movement that might otherwise produce an abundance of new and exotic types of movement.

There are other places this rabbit hole goes, for example how CAPs can be viewed within it and how it applies to things like weaves and the like, but I think this is a good start. Please send feedback on this.