So today I was thinking about an algebra for [lexicon]Poi[/lexicon] and I think I came up with a neat system. So the basic idea behind a [lexicon]Poi[/lexicon] algebra is a compact representation for [lexicon]Poi[/lexicon] movement that allows one to fully express a "move" and allow for the easy creation of new moves via a combinatorial scheme.
Terminology:The basic unit of movement for [lexicon]Poi[/lexicon] is a unit circle rotation. There will be actions that can be placed upon the unit circle that allow for pattern creation. These will be divided into driving actions, translations, and rotations.

Driving actions - For the purposes of this algebra there are 4 driving styles. a unit circle rotation (U or understood), [lexicon]isolation[/lexicon] (I), extension (E), and pendulum (P).

Translations - Translations are movements in space. I will define a few common types of translations that will allow the algebra to be more compact.
L - linear translation
W^x - [lexicon]weave[/lexicon] translation( x stands for the quantity of beats e.g. 2,3,5,7)
A - anti-spin translation
T - generic translation all other translations can be thought of as subclasses of T

Rotations - Rotations occur when the [lexicon]Poi[/lexicon] changes its [lexicon]plane[/lexicon] of motion with no translation. The distinction between a translation that changes planes and a rotation is that a translation that changes planes has a smooth [lexicon]Transition[/lexicon] while a rotation has a sharp change. consider the difference between the graph of an absolute value curve
Initially I won't talk much about rotations but they would be how you would describe lots of atomics.
Another important aspect of this algebra is the idea of homeomorphism. For [lexicon]Poi[/lexicon] two movements are said to be homeomorphic if there exists an isotopy between them, this practically means if the difference between the two movements is the [lexicon]timing[/lexicon] of your arm rotating via your elbow in a circle (as opposed to a piston, this ex or if the movements can be rotated such that they become one another. This means for example all anti-spin flowers are homeomorphic since the difference between an n-petal and a k-petal [lexicon]antispin[/lexicon] [lexicon]flower[/lexicon] is merely the [lexicon]timing[/lexicon] of your hand.
Another interesting thing that comes from this is that linear extensions and anti-spin are homeomorphic, they are different types of translations.
The use of Charlie's 9-square theory comes in handy for further describing [lexicon]Poi[/lexicon] movement. While we can move our [lexicon]Poi[/lexicon] into arbitrary places Charlie makes a very pertinent point in that these 9-squares do a pretty good job of defining movement in one [lexicon]plane[/lexicon] (or alternatively an 27 cube version for defining movement by a stationary person in all the planes.

so let's make some moves! let's start with a standard 4-petal diamond anti-spin motion. Since this is an [lexicon]antispin[/lexicon] translation of the unit circle that is composed of well 4 loops or 3 beats. A(mid_top,right_mid,mid_bottom,left_mid). This describes a standard "diamond" [lexicon]antispin[/lexicon] [lexicon]flower[/lexicon] with your [lexicon]Poi[/lexicon] moving in the forward [lexicon]direction[/lexicon].

A chain of moves can be created easily with this methodology
A(mid_top,right_mid,mid_bottom,left_mid)L(Left_mid,right_mid,mid_mid,mid_top,mid_bottom)^2

this describes a 4-petal [lexicon]antispin[/lexicon], into doing a cross linear extension twice note the exponential means you do it twice since multiplication in this algebra means number of full cycles of the movement should be done. This leads naturally into how this algebra can be used for CAPs
CAPs:CAPs are Continuous Assembly Patterns. CAPs are movements that combine different driving styles and combinations of translation/rotation within a single cycle of movement. For instance the standard [lexicon]CAP[/lexicon] can be represented by

A(mid_bottom,left_mid,mid_top)^(1/2)E(mid_top,left_mid,mid_bottom)^(1/2)

I would like to point out that although not often considered there are in fact two types of CAPs I call stable CAPs and unstable CAPs. Stable CAPs are CAPs where the exponent is a whole number, this indicates that you can continuously cycle through the [lexicon]CAP[/lexicon] without transitioning into something else. Unstable CAPs, however do not have a whole number exponential and do not use a full cycle. This brings us to one important point.
Stalls are unstable CAPs. When you consider a [lexicon]stall[/lexicon] it's typically a quarter of an [lexicon]isolation[/lexicon]. so stalls could be represented by say I(left_mid,mid_mid)^(1/4) which represents either a downward or upward [lexicon]stall[/lexicon] from the left into the middle. Incorporating stalls into some sort of comprehensive [lexicon]Poi[/lexicon] description has always been tricky because they are in some ways so different from other movement. This algebra gives some illumination as to why this is so.
TransitionsOne major issue with this system at the moment is as always transitions. If in my previous example:
A(mid_top,right_mid,mid_bottom,left_mid)L(Left_mid,right_mid,mid_mid,mid_top,mid_bottom)^2 (umm [lexicon]wave[/lexicon] is being weird and not autowrapping so it's just going off screen so imagine that the equation above is all one line.

imagine if you now want to go from that to a W(left_mid,right_mid) . Your hand is currently at the mid_bottom position (6'o clock) to do the above [lexicon]weave[/lexicon] you first have to [lexicon]Transition[/lexicon] from mid_bottom to left_mid, this can be done via an unstable [lexicon]CAP[/lexicon] T(mid_bottom,left_mid)^(1/2) the 1/2 exponent indicates that you only do 1/2 of the pattern which is going just from mid_bottom to left_mid, if this had been T(mid_bottom,left_mid) the understood movement would be to move from mid_bottom, to left_mid, back to mid_bottom, which could be for instance a linear extension at an angle or a weird at angle cat-eye. Notation like this has roots in group theory.
I plan to build a [lexicon]wave[/lexicon] robot that integrates a lot of this notation into sort of emoticons so you can create equations that are much easier to read and quicker to write.
This notation can also be expanded into the 27 cube, cube system as well which will give you the ability to fully express all motions in relations to your body but will require a bit more thought as to building emoticons that can be used for equations.
The notation for distinct commonly used types of translations can and should be expanded and revised. for instance perhaps a reel translation should be added.
Lastly I haven't even considered rotations really but say movements like atomics can be thought of as CAPs built up by translating and rotating (generally) the unit circle.
I think this system gives a compact atomistic way of representing [lexicon]Poi[/lexicon] movement that allows for much of current [lexicon]Poi[/lexicon] and I'd love to hear feedback from everyone (hopefully constructive :P).