Third-order Motions (three circles composition)

Hi everybody !

I would like to share some stuff about what could be called third order motions.

Currently, we usually know and use second order motion, which are basically relations like Antispins and Inspins : Composition of 2 circles. Notice that this set also includes first order motions, when one of the 2 circles does not spin (a relation i call Nospin).

Consequently, third order motions contains second and first order motions. This lead to a level which enable to describe almost all patterns that we could draw.


Before ... let me explain my vision of what a pattern is (even if it is a paste of one of my previous posts):

To my mind a pattern is a cyclic curve that can be define in a unique way by its frequencies or number of tours ("harmonic" component) and by its radius or lengthes ("modulus" component).

In the field of the 2 circles compositions ... which is for instance the most commonly used ... the "model" is the following :



O is the shoulder, M is the hand ("Main" in french), E is the Extremity of the object (poï, club staff) and Ebis is the other extremity (in the case of the staff).

Theta1 and Theta2 (frequencies or number of turns) define the harmonic part of a pattern whereas Rho1 and Rho2 (radius or length) define the "modulus" part.

Rho=1 when the arm is stretched (when the poï is unwrapped).

The patterns will be defined in the following way : Theta1 Theta2 ; Rho1 Rho2.


And now examples :



The first example is 1 4 ; 1 1 (if E would have run the other way : -1 -4 ; 1 1)
The second example is 1 -6 ; 1 1 (if E would have run the other way : -1 6 ; 1 1)

As we can see these patterns are in a particular case of "modulus" component ... as the 2 radius or lengthes are equal (in this case ... the simplest ... the values are 1) ... and called "rosettes".

There is another particular "modulus" part mode which give curves called cycloïds.

This case happen when the speeds of M and E are equal in the earth (or audience) reference.
It means in terms of maths that Theta2/Theta1=abs(Theta1/(Theta1+Theta2)) (Theta1.abs(Theta1)=Theta2.abs(Theta1+Theta2)).

With the first example (1 4 ; 1 1) above it would give us :



And it would be called 1 4 ; 1 1/5.

With the example of a 3 foils "rosette" antispin (1 -3 ; 1 1) :



It would be called 1 -3 ; 1 1/2.


All the cycloids are not feasable ... i mean if we keep the arm stretched (Rho1=1) ... because the differents cases of wraps (thumb excluded) can only take the following values :

Rho2 -> Wrap

1/5 -> 2h+1f
1/4 -> 2h
1/3 -> 1h+3f
2/5 -> 1h+2f
1/2 -> 1h+1f
3/5 -> 4f=1h
2/3 -> 3f
3/4 -> 2f
4/5 -> 1f
1 -> 0f

(h for hand and f for finger)

There is also patterns with more than one arm turn (Theta1>1) ... the most popular example of these kind of pattern would be :


AKA 2 -5 ; 1 1 and 2 -5 ; 1 2/3 (if we keep the same sense of running as the examples above)

But there is a lot of other examples like:



AKA 3 2 ; 1 1 and 3 2 ; 1 3/5 (rotated by 90° in this image)

[img:center]http://www.mathcurve.com/courbes2d/epicycloid/epicycloid13.gif[/img]

AKA 3 4 ; 1 1 and 3 4 ; 1 3/7 (rotated by 45° in this image)

In general ... with poïs ... antispin patterns have this rule of feasability : abs(Theta1)<abs(Theta2)/2 ... whereas Inspins patterns have this rule of feasability : abs(Theta2)>abs(Theta1)/2.

Now let's talk about these third order motions ... which have the following "model" :



We still have O as the shoulder, M as the hand ("Main" in french), E as the Extremity of the object (poï, club staff) and Ebis as the other extremity (in the case of the staff).

However this time we interpose a new center I (for Intermediary) between O & M ... this point is virtual but can be occasionnally identifed as the elbow (for example).

This time the patterns will be defined in the following way : Theta1 Theta2 Theta3 ; Rho1 Rho2 Rho3.

Thus, we can enable the hand to draw all the antispins, inspins and obviously nospins ... and extend substantially the index of patterns.


A word on litteral patterns designation :

This new model lead to new type of naming. In term of "harmonic" part, our classical Antispins are Theta1 0 -Theta3 (simplified in + 0 -) cases or the opposite -Theta1 0 Theta3 (simplified in - 0 +).
Our Inspins are the + 0 + cases (or - 0 -) and the Nospins are cases 0 0 + (or 0 0 -) & + 0 0 (or - 0 0).

Moreover we also have all the new third order stuffs :

In-Inspins + + + (or - - -)
In-Antispins + + - (or - - +)
Anti-Inspins + - - (or - + +
Anti-Antispins + - + (or - + -)
No-Inspins 0 + + (or 0 - -)
In-Nospins + + 0 (or - - 0)
No-Antispins 0 + - (or 0 - +)
Anti-Nospins + - 0 (or - + 0)


And now a few examples :

A first example is within a video of Mel "Red Pants" :

From 6:30 to 6:36, Mel does this pattern :

One hand does this 1 -4 8 pattern ... and the other one does the opposite aka -1 4 -8 pattern ... so left & right draw the same shape but one is the horizontal symmetry of the other in term of motion (left and right patterns are in butterfly mode).

More examples to come ...

Awesome, can't wait for more. Just to add some resources:

http://www.mathcurve.com

Wherre is the last image from?
What tool was used?

I, WANT, MORE !

The "red curves" images and animated ones in the first post are from www.mathcurve.com.

This the referent website i used when i started my researches ... and it is a good one (unfortunately it all written in french) !

The last image is a screenshot taken from a personal excel sheet. This is just the plot of some mathematical formulas i improved little by little to best describe spinning motions.

I will post tomorrow a short index of patterns which are IS and AS (the first ones) hand-path based.

A word on starting configuration :

The 3 circles composition model allow more starting configuration than the previous one.

1. 2. 3. 4.

In these picture we have points O, I, M & E from the previous model ... respectively sorted by dot size.

1. All the segment [OI], [IM] & [ME] are in an "open" configuration (each is an extension of the previous one). Thus we will simplify this in "oo" (for open open).

2. [OI] & [IM] are in an "open" configuration ([IM] is an extension of [OI]), but [IM] & [ME] are in a "close" configuration ([ME] is "closed" on [IM]). Thus we will simplify that in "oc" (for open close).

3. This time [OI] & [IM] are in an "close" configuration ([IM] is "closed" on [OI]), but [IM] & [ME] are in a "open" configuration ([ME] is an extension of [IM]). Thus, we will simplifiy that in "co" (for close open).

4. Finally, we case the case were all the segments [OI], [IM] & [ME] are in a "close" configuration (each is "closed on the previous one). Thus, we will simplify this in "cc" (for close close).

Notice that in the second order motions we had already two different starting configuration which were obviously "c" (for close) and "o" (for open). For a given "harmonic" part Theta1 Theta2, the patterns are the same whatever the starting configuration choice.

However with three circles composition, the patterns started in "oo"(or "co") configuration is not the same than if we start in "oc" (or "cc") configuration, for a given "harmonic" part Theta1 Theta2 Theta3.


Let's illustrate that below with a few examples of anti-antispins (AAS) patterns (the simplest and maybe prettiest) :

In all the following part, we will found different antispin (2,3 & 4 foils) hand pathes and always the same 2 foils antispin pattern for the relation between [IM] & [ME].

AKA 1 -2 4 (or its opposite -1 2 -4) started in "oo".

AKA 1 -2 4 (or its opposite -1 2 -4) started in "co". The pattern is the same as previously ... just 1/4 of a turn rotated.

AKA 1 -2 4 (or its opposite -1 2 -4) started in "oc" this time : the pattern is not the same.

AKA 1 -3 6 (or its opposite -1 3 -6) started in "oo".

AKA 1 -3 6 (or its opposite -1 3 -6) started in "oc" this time : the pattern is not the same.


AKA 1 -4 8 (or its opposite -1 4 -8) started in "oo".

AKA 1 -4 8 (or its opposite -1 4 -8) started in "oc" this time : the pattern is not the same. Notice that this pattern is the same as my first example taken from Mel's video above (just 1/8 of a turn rotated).


Finally to describe completely a motion pattern we "only" need the "harmonic" part (numbers of turns : Theta1 Theta2 Theta3) ... eventually the "modulus" part which rarely evolve (lengths or radius : Rho1 Rho2 Rho3) ... the starting configuration ("oo"/"co" or "oc"/"cc") ... and finally specify how much of the pattern we choose to take/draw.

Using the notation in this arrangement : Startpos Th1 Th2 Th3 (Rho1 Rho2 Rho3) d ... the complete description of our last example would be -> oc_1 -4 8_(2/5 3/5 4/5)_1 -> oc_1 -4 8_1 if simplified.

A litteral transcription could be : oc AAS(4,2) ("open-close" started Anti-antispin, the first with four foils and the second with two)

And of course, it is possible to build new CAPs with these new elementary patterns ... play with it !

Later, examples of a few patterns from the In-Inspins, In-Antispins, Anti-Inspins families ... and so on.

Oh my gawd! It was nice to see a drawn out version of Zan's Triangle. I think that was the first one but correct me if I'm wrong. These are just awesome I'm gonna have to work on this stuff and show the idea during practice

Edit: I mean Zan's Diamond. Excuse my moment of stupidity

A little update with a few examples of In-Antispins (IAS) patterns family :

AKA 1 1 -3 started in "cc".

AKA 1 1 -3 started in "co".

AKA 1 1 -5 started in "oc".

AKA 1 1 -5 started in "oo".

AKA 1 2 -8 started in "oc".

AKA 1 2 -8 started in "oo".

okay I am a bit confused at the moment...
I think I got the idea of this but just some more questions about it...

Wasn't Zan's Diamond a way to get the 3 timing positions (same, split and quarter) in one pattern?

Isn't the first image just a idea of making a pattern out of triquetras and uses as a transition point 2 cateye?
But isn't it, that most of this picture could be made of a Linear Movement for example (illustration 2,3 and 4)


Also I think that I don't really know what exactly would be a "Anti-Nospin"
Could you may post just some picture of what it would look like?

What I think I figured out is that the mumbo-jumbo at the bottom right corner is what the poi would be doing if your hand was just moving in a circle. And I looked at Mel and the anti-antispin thing and that would be Zan's Diamond. And think of it this way, that IS how you get all the timings in the same pattern if that is how it works. Because I never noticed the quarter time thing.

I remember someone on Tribe talking about this a while ago. I got as far as trying to get a flower with horizontal poi and anti-spinning them. That was a while ago though.

Thanks for reminding me.

Even I don't fully understand your theory I think there is a de-
finitely something very nice. I will write Drex what is he thin-
king about it, becease he like your previous theory too.

The mumbo-jumbo you talk about is somehow the definition you have given. Actually, it shows the pattern drawn by [IM] and [ME] from [OI] point of view or reference. It is a local, relative second order pattern (it refers to the two last digits of the "harmonic" part)... so yes the hand path in this case is consequently a circle.





I did not hear of Zan's Diameond before so maybe i am not the guy who could answer that question.

If Zan's Diamond is the move that i took from Mel's Video ... one side does 1 -4 8 and the other does the opposite -1 4 -8 (kind of butterfly mode of the same pattern between the 2 side) ... and both side start in "cc" in a vertical position ... one above and the other below (1/2 turn shift between the 2 side) as you can see below :



Thus, our side patterns are in butterfly, starts with the same configuration and we have a 1/2 turn position phase between side ... so to my mind we are in same timing and horizontal symmetry during the whole motion.



Could you just explain me before what exactly a triquetra is please ? This notion looks very recent and i did not get into it ... maybe this is something i call differently.



Yes i can be ... but the linear hand path is a very particular case of 2 foils antispins (or ellipses) and it would happens only when OI=IM.
For upper number of foils, linear hand pathes such as triangle or square does not give strictly the same patterns but it can be a good approximation i guess.



Here are some examples of Anti-Nospins :

AKA 1 -2 0 (or -1 2 0) started in "oo".

AKA 1 -2 0 (or -1 2 0) started in "oc".

AKA 1 -3 0 (or -1 3 0) started in "oo".

AKA 1 -3 0 (or -1 3 0) started in "oc".

AKA 1 -4 0 (or -1 4 0) started in "oo".

AKA 1 -4 0 (or -1 4 0) started in "oc".

As you can guess, those started in "oc" are far more difficult to execute (i would say nearly impossible) than those started in "oo".

The "mumbo-jumbo" (^^) at the bottom right corner describe a Nospin started in "o" (also known as an extension) or a Nospin started in "c" (also known as an Isolation when OM=ME).


Let me know if have other questions ... or if you need more explanations or deepening !

I thought Zan diamond is this 0:22-0:28 http://www.youtube.com/watch?v=CFez03GHpTE ..

For me a triquetra is a 3 Petal Antispin Flower.

at each petal you can transition into a cateye and between two petals you can transition into a cateye.
So for me this first pattern would be
Triquetra on the left side with petals at 2,6 and 8 o'clock.
Then you use a vertical cateye connecting the petals at 2 and 6 o'clock and then you transition into another triquetra with the petals at 2,4 and 8 o'clock.
Then at the petal at 4 o' clock you can transition via a horizontal cateye and end up at a petal at 6 o'clock.
And then you can start do it all again.

Zaltymbunk: I am having trouble deciding what the different trajectories in some of your diagrams are in real terms.

Extremity is logically the poi head, but what would main and centreinter be?

For the most part I can follow the maths/diagrams, but can not actually spin this....

I've only glanced over it, really, but centreinger seems to be the unit circle, if you do x pattern following that you get the "main" which is the usual pattern.

Extremite is differentiation from the usual pattern. Have a look at the relative functions in the bottom right of each one.

I can spin all these, but the maths up the top just baffles me.

Edit: Hmm wait... in some cases it appears to be the norm version, in others it appears to be the hand path... I may have to re-examine my hand paths in linear isolations... at the moment the hands just kinda do their thing.

Thanks for reminding me to revisit cycloids, though.

Wanna swap? Ah well, will have a closer look when I am not so rushed.

Nice to see you again by the way.

Can you show your pattern generator sheet ?

Better yet, Aston, you should come here and show me the maths and I'll show you how to spin em.

Damian, you've reached your monthly bandwidth limit on imageshotel.org and I can't see your images. =(
It would be great to find another place to host them without bandwidth restrictions. Perhaps we could (at least temporarily) host them at alienjon.com/Zultymbunk, if you like, or maybe Drex can host them? One way or the other, can I download them from you?

Here are some other examples:
Zan's Diamond at 0:22


Hand-to-Head tracing Starting at 0:20


My "Antispin Weave Fountain"

Concerning the name "no spin". I just want to point out that this makes sense from a particular frame of reference. Specifically if the spin is observed from a frame of reference oriented to the 2nd Center of Rotation, for example a fly sitting on the hand of the poi spinner. If on the other hand, we look from the frame of reference of a stationary audience, then there is spin, but it is "concentric spin", ie concentric to the hand path.

I think Zaltymbank tends to use the local frame of reference, and I tend to use the audience. I'm not saying that one name/concept is better or worse, just want to make sure people know they are the same phenomenon viewed from different frames of reference.

Concerning Zan's Diamond:
Whether you are using circles and smooth curves, or linear moves along polygons, you are doing different flavors of the same thing.
It's just Yin or Yang flavored.

Zan was using linear moves tracing along the "diamond", wich can also be thought of as a hand path that traces 4-petal antispin of the "box" variety (diagonal petals).
Zan however had reduced the curves to polygons, using the minimum number of linear movements between the minimum number of nodes to create a sharp polygon version of said antispin.

This yields a very sharp/yang flavor to the pattern.
If on the other hand you try to smooth the hand path out until it is much closer to the smooth curves of the iconic 4-petal antispin flower, then you will have the same base pattern with a much smoother curved Yin flavor to it.


Explaining Yin vs Yang "flavors" a bit more:
A circle can be though of as a polygon with infinite segments and points.
If you take n number of points around the circle, and connect each point to the next, you get a polygon with n sides.
The fewest points you need to make a polygon is 3, which is the sharpest polygon.



We could imagine each point of the triangle is the location of the hand during the apex of each antispin petal, for instance. If your hand traces along the lines of the triangle, your antispin will go from a smooth yin flavor to a sharp yang flavor. This will yield a pattern that can be seen as a series of linear extensions (arcs) along the lines traveling between the petal (loop) at each node.

On the other hand we could place each petal (loop) in the middle of each line segment, and the arcs centered at each node. This can be seen as a series of linear isolations (of the loopy variety) along each line segment, with arc passing each node. It may feel a bit like isolated antispin. Your hand will trace one triangle, wile the poi head's antispin pattern will insinuate the reciprocal triangle.
These ideas are further explored in Cyrille's paper Spinning & Geometry (mirrored by Drex).


Between curved/circular and sharp/polygon renditions of the same patterns, the difference in flavor is most pronounced between triangle and circle.
As we increase the number of sides/nodes on the polygon get closer to approximating the curve of the circle, so that the difference in flavor is less and less.
















... and so on until
you can't tell
a polygon
from a circle.

Said more verbosely:
At each petal (loop) the poi is in a similar state as it is in the tip of a cateye.
Therefor you have a seamless transition point between the 2.
You can lockout from your flower into a cateye that is pointing radially from the center of your original flower.

On the other hand, in between petals you have an arc that connects 2 loops.
This state is similar to that of the elongated part of a cateye.
Therefor you can make a seamless transition, locking out in a cateye that is perpendicular to the center of your original flower.

So yeah, wherever there is a loop you can lockout into a cateye, and wherever there is an arc you can lock out in a cateye.
But for that matter, wherever there is a loop you can lockout in an isolation, and wherever there is an arc you can lockout in an extension.

From flowers (inspin or antispin), and for that matter any patterns that contain loops and/or arcs in general, you can use 1 of 3 types of unit circle lockouts:

• extension
• isolation
• cateye

Extension has the 2 ends of the poi in a together time - same direction relationship.
This is related to translational symmetry (in this case along the radius in a concentric manner).

Isolation is a split time - same direction relationship of the 2 ends.
This is related to radial symmetry.

Cateyes have the 2 ends relating in opposite directions, in any possible phase, ie, together, split, quarter, and anything in-between for that matter.
This is related to mirror symmetry. The phase dictates the angle of the mirror axis.



These lockouts raise an important point:
We should be conscious of the the distinction between closed figures and open figures.
With a closed figure, if you start at time T1 at position A you will end at time T2 at position A, with a similar poi/momentum state. This means you can repeat the same pattern, in it's entirety, n times (however many you want).

With an open figure if you start at time T1 at position A you will end at time T2 at position B, with a poi/momentum state that may or may not be dissimilar.

You can take a closed figure and chop it up into a series of open figures.
For example a circle is a simple closed figure. You could cut it into 4 quarter circles.
However you don't have to put these quarters back together to only make a circle.
You could use some of them combined with other open figures, to assemble another pattern.
This assembly might be a new open figure, or it might be a new closed figure.
It might even be a more complex closed figure that returns to the same poi position/momentum state as the start of your original pattern (ie your circle).
This allows you to transition cyclically between these 2 patterns in a more complex assembly pattern.

One of the ways you can use these transition principles is using some form of lockout, as stated above. If you lockout from the complex pattern for too many cycles, the cohesiveness of your assembly may be lost.
But if you for example, lockout at each petal of a triquetra with one cycle of cateye, you make a pattern that approximates one of the above mentioned 3-order curves.
Notice that I said approximates!



Is there a difference between lockout assemblies and complex curves with perfect harmonic ratios?

Well, yes and no. What the heck do I mean by this anyway?
Let's take a simple example: 2-lockout inspin flower VS 2-loop inspin flower.

Many people learn inspin flowers using lockouts.
For a 2-petal flower they make a longarm half-circle, then lockout into a static circle.
During that time their hand is static.
Then the small circle completes, and their hand makes another longarm half-circle, followed by another static circle...
Finally they have completed 1 cycle of 2-petal flower, and they could continue to do so if they like.

This uses 2 open figures (the 2 half-circles) and 2 closed figures (the 2 static circles).
This presents an interesting problem:
At the 2 points where the poi position/momentum state are the same in the different figures, we could choose to omit a lockout, or lockout for n number of spins, not just one. So this pattern is actually an assembly of simpler harmonic ratios, or parts of them.

Now lets look at a 2-petal inspin that uses 2 loops. It is an accurate recreation of the mathematical curve that has a 2:1 ratio between the 2 Centers of Rotation. You will only come back to a similar position/momentum point once per repetition of the pattern. If you deviate from it, you are then using part or it to create an assembly.

Zaltymbunk is looking at the figures that are irreducible closed figures in themselves. These figures arise out of relatively simple harmonic ratios. From these figures as a base, we can chop them up and make assemblies.

So yes there is an important difference between assembling lockouts, and irreducible harmonic curves... but we as humans will never quite recreate them perfectly.
Further more there are plenty of assemblies that approximate the perfect curves nicely enough for an audience... for that matter, making cool assemblies is a lot of fun.

So no, it isn't all that important to use only perfect harmonic curves, and think in those terms. It is however useful to understand the differences and the relationships between the 2.

Final point:
Wether you think of a 3rd-order curve, or an assembly of cateye lockouts connected by the open figures you chopped out of a triquetra, you will be getting a similar pattern. Understanding the implications of each thought model will give you a variety of creative ways to explore poi.

I can definitely host them, not sure for how long, but at least for a few months.

Going to try and grok this stuff on the weekend, thanks.

why don't post it on wikipedia?

Other idea would be to back up it at picasa or flickr

I can't see the reference picture. Is there anywhere else to find it right now. Everything looks great and I love all the concepts but it's hard to figure it out without a reference.

SoD, I made these pictures a while ago, they were discussing head patterns whereas your video discusses hand patterns, though they can be applied to hand patterns.