An Analysis of Eongatubabo and Burbot

I originally intended to go through the endings in the same order I listed them in my "Presenting a Transverse Exchange" article, which is why I started with the "Apache Door Ending" a few days ago. However, I've been doing some thinking and experimenting with Eongatubabo, and some ideas are starting to gel. I figured I'd do this one next.

Eongatubabo. It's a Nauruan name for a technique known in various places throughout Micronesia. Honor Maude also documented it as the "Banaban Movement". Done as originally shown to Maude, the movement has a smooth elegance to it, although it is tricky to describe because it is such a fluid maneuver. Beginning with a three-loop configuration (loops on 1, 2, and 5):

1. 1 hooks down 2n and holds it to the palm.
2. 2 enters the 5 loop from above, hooks 5n and 2f back over 1n, and then picks up 1n, rotating up and back to position.
3. 1, still hooking down 1f and 2n, moves under 5f and returns with it, releasing 1f and 2n in the process.
4. Release 5.
5. 5 removes upper 2 loop from above.
6. Release 2.
   

If you started with Opening A, at this point you'll have a figure the Nauruans call the "shuttle"--basically a mostly-uninteresting hourglass figure.


You'll get something more interesting if you give the index fingers a full twist away from you; that's the figure called "Two Noddies".


Note that Eongatubabo is usually finished with a Caroline ending, for a stronger presentation:

1. 2 removes 1 from above.
2. 1 moves under 2 loops and picks up 5n.
3. 1 removes upper 2 loop.
4. Caroline extension.

Murphy's Power Lift or the two-diamonds extension also work well.

Done right, this all flows quickly, and the strings seem to dance on the fingers. However, it can be difficult to analyze, because the entire sequence is executed as essentially a single maneuver.

Now, it might surprise you to learn this, but the Inuit "Burbot" figure (from Jenness) is actually very closely related to the Eongatubabo maneuver, though you'd probably never know it to see the two techniques side-by-side. Here's the Burbot:

1. Opening A.
2. Transfer 2 to 1, then 5 to 1 (so you have three loops stacked on the thumbs). Keep the loops separated, and in order.
3. 5 enters 1 from below, then moves to the far side of the upper loop from below and hooks it down, holding it to the palm (this will be the transverse string that was originally 5f).
4. 2 enters 1 from above, then moves to the near side of the lower loop and hooks it up, returning to position (this will be the transverse string that was originally 1n).
5. Release 1.

At this point you'll have two double-walled diamonds, known as the "Burbot" (a kind of fish).


Now, aside from looking similar to "Two Noddies", how exactly is this related to Eongatubabo?

Try this excercise. Do Eongatubabo, starting with Opening A, and with the index loops rotated a half turn away from you.

What do you get? The Burbot!

Now, try this. Do the Burbot, but with the index loops turned a half turn toward you. The result? Two Noddies!

Mathematically, you could represent this relationship like:

2ba + burbot = eongatubabo
2ab + eongatubabo = burbot

(Where 2ba is mizz code for a half rotation of the index toward you, and 2ab is a half rotation away from you.) Setting those equal to zero:

2ba + burbot - eongatubabo = 0
2ab - burbot + eongatubabo = 0

And then reducing:

2ba + 2ab = 0

Which, since 2ba and 2ab are additive inverses (a half rotation away from you cancels out a half rotation toward you), this is what you'd expect. Burbot and Eongatubabo are the same! Mostly. The difference is only in how the fingers move the strings.

Scott (a member of the ISFA mailing list, and I realize I don't even know his last name. Throw me a bone, here, Scott!) shared a thought with me in a recent email. He pointed out that although the Burbot, as it is generally executed, doesn't exchange transverse strings (the original 5f and 1n end up on 5f and 2n), it could be done that way. Instead of starting with Opening A on 1, 2, and 5, try doing Opening A on 2, 3, and 4.

1. Place loops on 2 and 4.
2. R3 picks left palmar string.
3. L3 picks up right palmar string through R3 loop.
   

Then:

1. 1 moves under 2n, over 2f, under 3n, over 3f, under 4n, and returns with 4f.
2. 5 then does the same from the other direction: 5 moves over 4f, under 4n, over 3f, under 3n, over 2f, and returns with 2n.
3. Release 2, 3, and 4.

The result? The Burbot! And the transverse strings exchanged sides. Essentially, this is what you're doing when you stack the strings on the thumb. By moving the fingers through the stacked loops, you're basically threading your finger over and under the near and far strings, and returning with the transverse strings. Try the original Burbot instructions and pay attention to where your fingers go, relative to the strings, and you'll see what I mean.

Now, if you do Eongatubabo about a million times, in slow motion, and give yourself a splitting headache by trying to watch the strings, you'll see a similar thing happens. However, it's easier to see if you break Eongatubabo into two discrete steps. Following James Murphy's naming pattern in his "North American Net System" article, I call these two steps "First Micronesian Weave" and "Second Micronesian Weave".

Begin with Opening A, with the index loop rotated a half turn. Then do the First Micronesian Weave:

1. 1 moves through 2 from above, and returns with both 5n and 2f.
2. 3 picks up 1f.
3. Release 1.

Look familiar? It should. I talked about a family of similar maneuvers in my article titled "Variations on the Transverse Exchange". Murphy has also written extensively about three weaves in this family in his "North American Net System" article, where he calls them the First Inuit Weave, First Klamath Weave, and First Navaho Weave.

Now, do the Second Micronesian Weave:

1. 1 moves through the 2 loop from above, into 5 loop from below, and then continues into the 3 loop from below.
2. 1 hooks down 3f through the 5 loop.
3. 1 picks up 5f and returns, releasing 3f in the process.
4. Release 5.

Lastly, finish Eongatubabo:

1. 5 removes 3 from above.
2. Release 2.
3. Display (via Caroline ending, or other).

By breaking Eongatubabo into those two separate weave steps, I found it much easier to consider what the loops are doing. Let's take a close look:

First, 1 pulls 5n and 2f toward you. This creates a "window" in the 5 and 2 loops, and the 1 loop is essentially pulled through that. This is similar to what happens with the Burbot, when you pull the near transverse string through the stacked loops. Note, though, that because 1 opens the window by grabbing 2f, the 2 loop is "flipped"; the 1 loop is then pulled through the 2 loop from above, instead of below. This is why you need to give the 2 loop a half turn in order to get the same Burbot figure, while Burbot needs no rotation.

The second weave does the same, in the opposite direction: it pulls the 5 loop through the old 1 loop (which is now the 3 loop), and the 2 loop. This isjust what you do when, in Burbot, your index finger enters the stacked thumb loops from above and pulls out the lower near transverse string.

The only bit that had me bothered by this analysis was the necessity of giving the index finger a half rotation when doing Eongatubabo, in order to get Burbot. What I wanted was a transverse exchange that was equivalent to Burbot. And, having broken Eongatubabo into pieces, I was able to see my way clear to it.

The solution was in analyzing the first step of the first weave. As I mentioned before, by grabbing 2f, you introduce a half rotation toward you into the 2 loop. What if you pick up 2n instead? Go ahead and try it, starting at Opening A (with no loop rotations). Move the thumb under 2n, over 2f, and return with 5n and 2n.

If you complete that first weave and then do the Second Micronesian Weave, you'll see that this is not quite the solution. It's almost there, but not quite. The problem is that you wind up with a knot of strings intersecting the far transverse string.

What's the difference between the modified version and the original? Only a half rotation of the index loop. Because the original weave introduced a half rotation of the 2 loop (due to how it was picked up), we need to reconsider the first move of the second weave. If the 2 loop is no longer flipped, and we enter it from the same side as we did when it was flipped, then we're essentially introducing a half-rotation again, later in the system. So, instead of entering the 2 loop from above, let's try entering the 2 loop from below.

In other words, try these modified weaves:

Modified First Micronesian Weave:

1. 1 moves under 2n, over 2f, and returns with 5n and 2n.
2. 3 picks up 1f.
3. Release 1.

Modified Second Micronesian Weave:

1. 1 moves through the 2 loop from below, into 5 loop from below, and then continues into the 3 loop from below.
2. 1 hooks down 3f through the 5 loop.
3. 1 picks up 5f and returns, releasing 3f in the process.
4. Release 5.

And it works! Starting with Opening A, we can use this modified Eongatubabo to exactly mimic the Burbot (with the exception that Eongatubabo will turn the figure upside down, because it exchanges the transverse strings).

All that's left is to put the pieces back together into a maneuver that flows as fluidly as the original Eongatubabo. Here's my attempt:

Modified Eongatubabo:

1. 1 moves under 2n and hooks down 2f, holding it to the palm.
2. 3 moves into 5 loop from above, hooks back 5n and 2n, and hooks up 1n.
3. 1, still holding down 1f and 2f, moves under and returns with 5f.
4. Release 5.
5. 5 removes 3 from above.

This modified Eongatubabo retains the same fluidity as the original, but does not introduce a half rotation into the index loop. It is the same as the Burbot maneuver (excepting only the near/far swap of transverse strings).

So, hopefully you stayed with me through all of that! To get this back on the topic of presenting a transverse exchange, let's use Burbot, Eongatubabo, and our modified version to present the Inuit Net Opening (see the Apache Door Ending article for a description of that).

If you start with the Inuit Net Opening on your hands, and then you do Eongatubabo, you'll get the following figure.


Now, try our modified Eongatubabo:


Significantly different! However, it should be the same as what we'd get if we did the Burbot, only upside down. And, sure enough, if you take the Inuit Net Opening, stack the loops on the thumbs, and then finish the Burbot figure, you get:


Success! (Depending on how you extend the figures, your results may look dramatically different. In the above I'm using the Power Lift and Caroline Extension.) I encourage you to experiment, applying these to other three-loop configurations. Let me know if you find something interesting!

Postscript:

In case anyone is interested in mizz code, here is my attempt at describing the various maneuvers described in this article, in mizz code.

Eongatubabo:

1. 1.6,2a
2. 2adh,(2b&5a)1a
3. 1,(2ah&2blj&5aj)5b (auto-off 1.6)
4. 5
5. 5T2h
6. 2
   

Burbot:

1. 1,2
2. 1,5
3. 5.6,(1bl)1bh (1bl=two strings,1bh=transverse string)
4. 2.3,(1ah)1al (1ah=transverse string,1al=two strings)
5. 1

First Micronesian Weave:

1. 1,(2b)5a
2. 1,2b
3. 3,1b
4. 1

Second Micronesian Weave:

1. 1,(2b&3a&5a 3b*)5b
2. 5

Finish Eongatubabo:

1. 5T3
2. 2
3. SPR & arrange (caroline ending, etc.)

Modified First Micronesian Weave:

1. 1,2a
2. 1,5a
3. 3,1b
4. 1

Modified Second Micronesian Weave:

1. 1,(2a&3a&5a 3b*)5b
2. 5

Modified Eongatubabo:

1. 1.6,(2a)2b
2. 3adh,(2a&5a)1a
3. 1,(3bj&2b&5aj)5b (auto-off 1.6)
4. 5
5. 5T3

Apache Door Ending

Alright, I'm going to start going through as many endings as I can, applying them to a "three loop" configuration: loops should exist on thumb, index, and little finger, and the near thumb string and far little finger string should be transverse. I'll begin with the Apache Door ending.

Because "Apache Door" is an attractive figure in its own right, I think it's easy to forget that it's simply a unique ending technique applied to "Opening A". But what happens when you apply it to a different three-loop configuration?

I'm going to be using as the basic figure what James Murphy calls "First Inuit Weave", "Second Inuit Weave", and "Continuation Move". (See his North American Net System article, here.) For the sake of brevity, I'll just call it "Inuit Net Opening":

1. Opening A.
2. 1 moves through 2 from above and returns with 5n.
3. 3 picks up 1f.
4. Release 1.
5. 1 moves through 2 from above, under 3, under 5n, and returns with 5f.
6. Release 5.
7. 5 removes 3 from above.

If you're interested in learning "Mizz Code", the same opening is described like this:

1. base
2. 1,(2b)5a
3. 3,1b
4. 1
5. 1,(2b 3 5a)5b
6. 5
7. 5T3

(Lovely how concise that is, no?)

Alright, so you've got the "Inuit Net Opening" on your hands. It should look something like this:


Now, just do "Apache Door":

1. Transfer 2 loop to the wrist (inserting hand into 2 loop).
2. 1 gets 5n.
3. 5 gets 1f.
4. L1 moves under all strings (so all strings should rest between 1 and 2).
5. R1 and R2 grasp both L1 loops (NOT the left wrist loop) and remove them from L1.
6. L1 returns to position, moving under all loops.
7. R1 and R2 replace the grasped loop on L1.
8. Repeat 4-7 on the right hand.
9. Release the wrist loop over each hand, and extend.

Or, in mizz code:

1. 0,2
2. 1,5a
3. 5,1b
4. 1,(s)1
5. 0
6. SPR & arrange

The result should look something like a double-walled diamond with wings. (It kind of reminds me of Pippi Longstocking!)


Now, that figure alone is attractive, but it's hiding a little secret. If you poke your fingers into the little knots on either side of the diamond, they'll open up, and you'll get this beautiful mesh figure:


Try this with other three-loop configurations! I described several possibilities (and hinted at many more) in my "Variations on the Transverse Exchange" article; see what patterns emerge when you apply the Apache Door ending to them.

Mizz Code quick reference

Along with James Murphy's articles that I mentioned in my last post, I've also been studying "mizz code" (http://home.p07.itscom.net/nenemei/v2/index.html), a novel approach to describing string figures that focuses on where the strings move, rather than how they move. It's a challenging notation to learn!

As I was going through his code guide I took notes, and tonight I finished formatting those notes. The resulting "Mizz Code quick reference" is now available here:

http://stringfigures.info/mizz-quickref.html

I'll be updating it and clarifying it (and fixing it!) as I receive feedback, but hopefully this will help others get started with Mizz code!

Investigating String "Systems"

I've been silent here for a couple weeks, but not because I've been neglecting string figures. On the contrary, I've been studying the writings of James Murphy (a.k.a. Inoli), a luminary in the string figure world. His ISFA articles were written as a means of demonstrating how string figures can be used to help students build a solid foundation for studying mathematics, but they are fascinating simply as a study in how to really investigate string figure systems.

His articles are all available as PDF's, here:

http://www.torusflex.com/torusflex%20project1/isfa%20articles.html

I can't recommend these enough. If you have any interest (at all!) in learning how to really explore permutations on a string figure, these articles are full of incredible insight. He presents techniques such as his amazing "power lift" extension, as well as tricks like "rolling", inverse weaves, vertical nets, and more. Along the way you'll learn how to make such beautiful figures as his "Inuit Bowl", "Cherokee Seven Stars", "Lightning Across the Middle", and many others, as well as how to discover your own variations.

Extremely inspirational reading. Well worth the time spent!

Presenting a Transverse Exchange

The structure of a "transverse exchange" was discussed in my previous article. However, a transverse exchange is ultimately only interesting insofar as it is results in an interesting presentation. Fortunately, the final configuration lends itself well to a multitude of different endings and extensions. For the purpose of this article, I'm going to assume you're using the "three finger" version, with loops on 1, 2, and 5.

Apache Door Ending

This is the ending used by the well-known "Apache Door". This gives you a very stable extension, with an attractive frame on all four sides, but the resulting presentation is small. Also, this ending can be difficult to achieve for tightly wound loops.

Many Stars Ending

The Navaho "Many Stars" figure uses this ending. In this one, 1 and 2 both remove the 5 loop from below. Then, you navaho both 1 and 2, and then extend by hooking the distal segment of the former 1 loop down through the 1 loop (either with 1, or 5, or whatever is handy). The extension here is large, but slightly unstable, with a tendency to collapse in the middle. In can benefit from the Caroline ending (by transferring 1 to 2 from above, picking up 5n with 1, transferring 2 to 1, and then doing the caroline extension), though this turns the figure upside down.

Caroline Ending

The Caroline Ending is a good, wide one, though it can result in a lot of empty space to either side of your pattern. Just transfer 1 to 2 from above, and then move 1 underneath, pick up 5n, and then transfer (upper) 2 to 1 from below. Then, do a Caroline extension.

Eongatubabo

Since this movement is, itself, a transverse exchange (see Variations on the "Transverse Exchange"), you're effectively just chaining this onto whatever other transverse exchange you did, and capping it off with the Caroline Ending. Still, it's a quick way to present a pattern, and it has the benefit of not altering the internal weave--it just adds diamonds to either side of the "main" pattern.

Amwangiyo

This is another Nauruan maneuver, although it is known elsewhere in the Pacific Islands. It's basically just the Gilbertese Movement, followed by a thumb twist, followed by the Gilbertese Extension. (See my article titled More on Mataka for a description of the Amwangiyo maneuver.) This is a really pretty and robust extension, and can be further extended by doing the Nauru Ending (which is basically a way to do a Small Amwangiyo "in place", without inverting the figure).

Small Amwangiyo

This is a good extension for forcing an otherwise narrow figure to fill out the left and ride sides of a Caroline extension. It creates a few tight loops on the ends of the figure which pull the figure left and right. In brief, the Small Amwangiyo works like this:

1. 1 moves through 2 from above and returns with 5n.
2. 5 moves through 2 from below and returns with 1f.
3. Release 2.
4. 2 moves under the "inner" (non-transverse) 5f string, hooking down the "outer" (transverse) 5f, and rotating down, toward you, and back to position.
5. Release 5.
6. 5 removes upper 2 loop.
7. Double navaho 1 (transverse loop over non-transverse loop, then release non-transverse loop over transverse loop).
8. Caroline ending.

"Two Kick" ending

I'm not sure if this ending has a published name or not. I call it "two kick" because it uses the "kick" maneuver (from "Mizz Code") to free up 1 and 5 for a Caroline Ending. It's actually very similar to the Small Amwangiyo; essentially:

1. 1 picks up 5n.
2. 5 picks up 1f.
3. 2 hooks down over palmar string, rotates toward you and up, allowing original 2 loops to slip over knuckle.
4. 1 removes 2 loop from above.
5. 2 and 3 move under the "inner" (non-transverse) 5f string, pinch the "outer" (transverse) 5f between them, and return to position with 5f on the back of 2.
6. Release 5.
7. 5 removes upper 2 loop from above.
8. 2 and 3 move under the "inner" (non-transverse) 1n string, pinch the "outer" (transverse) 1n between them, and return to position with 1n on the back of 3.
9. Release 1.
10. 1 removes 3 from above.
11. Caroline ending.

This doesn't give you as wide (horizontally) an extension as Small Amwangiyo, but the "double kick" (steps 5-7 and 8-10) can add some complexity to your final figure. Also, there is some variation possible just within this ending; steps 1 and 2 can be varied to pull the target string through the 2 loop, or under it, instead of over it.

Gilbertese Extension

The Gilbertese Extension requires that there be at least two loops on 1, and that 1 have both a transverse 1f and a transverse 1n. This is an easy requirement to fullfil if you have a transverse 1n and a transverse 5f—simply let 1 remove 5! (There is a lot of variation possible here, too, since you can choose between moving 1 under the 2 loops, or over the 2 loops, or through the 2 loops.)

Once, you've got the thumb loops set up, the Gilbertese Extension is easy to do. It's the same maneuver used at the end of Amwangiyo. I'll save myself some bandwidth and just point you at my "Mataka Series" article, where I describe the Gilbertese Extension.

"Tree" Ending

This is one that I came up with on my own, though it is very likely published somewhere. It's basically the Caroline ending, but with a navaho maneuver thrown in:

1. 1 picks up 5n.
2. Navaho 1.
3. 1 picks up 5n.
4. Caroline extension.

I call it the "tree" ending because the affect of the navaho in step 2 is to create some loops that depend from the upper frame string, but do not touch the bottom, like fruit hanging from a tree.

Incidentally, experimenting how 1 picks up 5n in steps 1 and 3 gives you a lot of room for variation, too. Whether 1 moves over, through, or under 2 changes the presentation, as you'd expect.

Northwest Coast Intertwine

This is a maneuver used by the Native Americans of the Northwest Coast. It is used in (among other figures), the Klamath Indian Owl's Net figure, as well as the Kwakiutl "Butterfly" figure. It is used to create mesh-like figures, but can be applied wherever you have loops on 1, 2, and 5:

1. 3 removes 5 from above.
2. Rotate 3 away and down, holding the 3 loop to the palm.
3. 4 and 5 enter 3 loop proximally, helping hold the loop to the palm.
4. 3, without leaving its loop, moves under 2 loop and enters 1 loop distally, hooking back 1f. 1f is pulled through the 345 loop.
5. 45 release their loop and help 3 hold the new 3 loop to the palm.
6. There is a loop around the palmar string. This loop has a distal and a proximal string. 3, without leaving its loop, enters that loop from the far side and hooks back the proximal string, pulling it through the 345 loop.
7. 45 release their loop and help 3 hold the new 3 loop to the palm.
8. 3 releases its loop, proximally enters 2 loop.
9. 2 and 3 bend over 1 loop and pinch 1n between them.
10. Place 1n on back of 2 by rotating 2 and 3 down, away, and up, returning to position.
11. Figure is extended between 1 and 2.

It's a complicated maneuver, and the resulting extension is unstable (tending to collapse easily) but it gives some attractive results. In particular, I've found that applying a Caroline ending after the extension (by transferring 2 to 1 and reversing the direction that 5 is inserted into the figure) gives a very robust extension, while preserving delicate meshes that the intertwine produces.

Conclusion

This is hardly the end. Rather, it's just the beginning! There are undoubtedly many, many more presentation techniques that could be applied, and I didn't even talk about other kinds of maneuvers that could be applied before, after, or between transverse exchanges (like rotating fingers to achieve more complex twists in the final pattern). If you find this article interesting, and come up with some attractive patterns as a result of playing with trasnverse exchanges, please let me know!

I intend to write some follow up articles in the coming weeks that explore some of the permutations described in this article. We'll see where I actually get with that. :)

Variations on the "Transverse Exchange"

I've been playing with a particular class of string figures a lot lately. However, I've not been able to find any existing published nomenclature for many of the maneuvers that I've been playing with, so I wanted to describe them here and see if anyone else knows what they may be called. (This is not to say that I've invented the maneuvers! Far from it. It is only that the maneuvers have no specific name, as far as I am aware.) For now, I'm calling this class of figure a "transverse exchange".

The reason for the name is that the figures all begin with an exchange of the transverse string on 1n with the transverse string on 5f. There are many ways this exchange can happen; thus the name is a general term that describes a family of similar maneuvers.

For example, the Navaho "Many Stars" figure begins with a transverse exchange, after Opening A:

1. 1 moves over 2 loop and picks up 5n.
2. 3 picks up 1f. Release 1.
3. 1 moves through 2 loop from above, under 3 and 5 loops, and returns with 5f. Release 5.
4. 5 removes 3 loop from above.
5. ...then perform the rest of Many Stars!

If you analyze the movements and the final string configuration, you'll see that the result of the above steps is to move the near transverse thumb string to the far side of the little finger, and to move the far transverse little finger string to the near side of the thumb. It's a transverse exchange.

Another form of the transverse exchange occurs in the Tikopian maneuver called "Tao-sokotosi, Ta-sokotosi" (which means "hold one, manipulate one"). Beginning with any configuration where 1, 2, and 5 have a loop on them (e.g. Opening A):

1. 1 moves over 2n and holds it down to the palm.
2. 2 and 3 move under 2f, pinch 1n, and by rotating 2 away and up, place 1n on 2, releasing 1.
3. 1 moves through lower 2 loop from above, under 5 loop, and returns with 5f. Release 5.
4. 5 removes upper 2 loop from above.

A third published form of the transverse exchange is in the Nauruan "Eongatubabo". (This has been described in some places as the Banaban ending, I believe, though I could be wrong about that!) Again, assuming an initial configuration like Opening A:

1. 1 moves over 2n and holds it down to the palm.
2. 2 enters the 5 loop from above, hooks back 5n and 2f, and hooks up 1n (releasing 5n and 2f).
3. 1 moves though the 2 loop from below, picks up 5f and returns.
4. Release 5.
5. 5 removes upper 2 loop from above.

In all three of these examples, the final result is to have the former TV 1n at 5f, and the former TV 5f at 1n. In fact, there are a great number of ways to do a transverse exchange, but they all consist of 4 distinct phases, and each of those phases have some very distinctive characteristics.

Structure of the Transverse Exchange

In "Phase A", there are loops on (at least) 1 and 5, and 1n and 5f are both transverse. There must also exist 1f and 5n (so First Position alone is not sufficient). Note that there is no requirement that loops exist on any other fingers, although it is most common to see a transverse exchange from an "opening A" configuration.

In "Phase B", the 1 loop is held "in storage", so to speak, making room for the 5 loop to come to roost on 1 during Phase C. Generally speaking, Phase B involves using the thumb to pull some string (or strings) toward you, exposing 1f. Then, either 2 or 3 (or, conceivably, 4, though I've not seen it done anywhere) pick up that exposed 1f, and the thumb is released from all loops. In some cases, though (as in Eongatubabo, or "Tao-sokotosi, Ta-sokotosi"), the 1 loop is removed via 1n, instead of 1f. Regardless, though, the original 1 loop is kept "aside", on finger 2 or 3 (or 4), until Phase D where it will eventually land on 5.

In "Phase C", the 5 loop rotated a half turn and transferred to the now-empty 1 finger. This is done by weaving the thumb through the existing loops and returning it with 5f, releasing 5 in the process. The final result of this phase is to have the former transverse 5f string on the thumb, as 1n.

Lastly, in "Phase D" we complete the exchange. The former 1 loop, held in "storage" on 2, 3, or 4, is given a half turn rotation and transferred to 5, such that the transverse string becomes 5f.

"Transverse Exchange" Variations

Note, though, that in none of these phases is there a strict guide for how the strings need to be moved. The transverse exchange is not a description of a specific maneuver! This means there is a great deal of possibility for exploration with this class of figure.

For example, in phase A, you can start with Opening A, Nauru Opening 1, 2 or 3, or any other opening that gives you transverse 1n and 5f strings. Given the requirements of TV 1n and 5f, and the existence of 1f and 5n, the simplest configuration that you can apply a transverse exchange to would be the "X Open" (opening A, release 1, and transfer 2 to 1), but there is no "upper bound" on the complexity of your opening.

For phase B, there are even more possibilities. Consider just a few of the ways in which you might expose 1f to the 2, 3 or 4 finger:

• 1 picks up 2n
• 1 moves over (or under) 2n and returns with 2f
• 1 moves over (or under) 2 and returns with 5n
• 1 moves over (or under) 2, over (or under) 5n, and returns with 5f
• 1 moves under 2 and returns with both 2f and 2n
• 1 moves under (or over) 2 and under 5 returns with both 5f and 5n
• 1 moves through 2 from above (or below) and returns with 5n
• 1 moves through 2 from above (or below), over (or under) 5n, and returns with 5f
• 1 moves through 2 from above (or below), under 5, and returns with both 5f and 5n
• etc, etc, etc.

Using an opening configuration in which there are loops on more fingers (like Nauru Opening 1 or 2) gives you even more possibilities.

Then, consider phase C. The thumb is typically moved through the (lower) 2 loop from above, and returns with the 5f string, but there is no reason it must be that way. The Navaho "Two Coyotes" figure, for instance, moves the thumb through the lower 2 loop from below. You might try returning with 5f by moving entirely over all the strings, or entirely under. Or you might try some more complex weaving of the thumb, especially if there are loops on more of the fingers.

Lastly, even phase D offers opportunities for variation. There's no reason the little finger needs to directly remove the "storage" loop. Try moving that loop to 5 by bringing it entirely around the near side of the figure, or by weaving it through other loops (if there are any available). For example, if the storage loop is the upper 2 loop, you might try inserting 5 from below into the lower 2 loop and hooking down the upper 2f, pulling it through the lower loop and removing it from 2. Then, rotate 5 down and away from you to bring the loop to rest on 5, with a transverse 5f string. As long you end up with a transverse 5f, it's all fair!

And, interestingly, note that the final configuration is exactly what is needed for the transverse exchange in the first place; this means you can chain transverse exchanges together, one onto another, for even more variation!

Next?

In the end, though, a transverse exchange is only really interesting insofar as it is results in an interesting presentation. I'll cover possible ways to present a transverse exchange figure in my next post.

In the meantime, experiment! See what kinds of interesting exchanges you can find. I'm not exaggerating when I say that this family of figures has kept me occupied for many, many hours. There is a lot of room for variation.

More on Mataka

I've been playing with the Mataka series some more, and I've refined it a bit. I've been able to answer a few of my questions since my last post, too.

Firstly, here's a general method for iteratively building any even-numbered Mataka figure (e.g., Mataka-0, 2, 4, 6, etc.). It uses a maneuver for which I've never seen a name, but which I've seen used in several other figures. It is closely related to the Kwakiutl Butterfly Opening, and is functionally equivalent to a Tikopian move called Tao-sokotosi, Ta-sokotosi ("hold one, manipulate one", see "Tikopia Web-Weaving Techniques" in BIFSA Vol. 4). I'll call this the "Net Opening" here, but if anyone knows a previously published name for it, let me know!

Assuming you have a loop on each 1, 2, and 5 (e.g. Opening A or similar):

1. 1 picks up 2f.
2. 3 picks up 1f.
3. Release 1.
4. 1 moves through 2 from above and picks up 5f from below.
5. Release 5.
6. 5 removes 3 from above.

So, to create any even-numbered Mataka figure:

1. Opening A.
2. Net Opening (zero or more times in succession).
3. 1 moves through 2 loop from above and removes 5.
4. Gilbertese Extension.

For every time you perform the Net Opening, you'll get 2 crossings on the transverse strings. In other words, skipping the Net Opening altogether will give you Mataka-0, but doing the Net Opening 3 times in a row will give you Mataka-6. Yay!

So, that's the general case. There are short-cuts for many of these figures, though. I wrote about three of them in the previous post about the Mataka series, and here are a few more I've found while experimenting.

Mataka-6:

1. Opening A.
2. Rotate 2 loop a half-turn away from you.
3. Rotate 5 a full turn away from you.
4. Gilbertese Extension.

Mataka-8:

1. Opening A.
2. 1 move under 2 loop and remove 5.
3. Rotate 1 away from you twice.
4. Gilbertese Extension.

Lastly, I finally made the connection between the Gilbertese Extension and the Nauru Amwangiyo maneuver. Essentially, Amwangiyo is the Gilbertese Movement, plus a thumb rotation, plus the Gilbertese Extension.

Though it's been documented elsewhere, I'll describe the Gilbertese Movement here for convenience. It assumes an initial configuration like Opening A, generally, with loops on 1, 2 and 5:

1. 2 removes 1.
2. 1 moves through lower 2 loop from above and removes 5.
3. 1 removes upper 2.
4. 2345 move over upper 1 loop and pick up lower 1n, removing lower 1 loop.
5. 1 removes 2345 loop.
6. Repeat steps 4 and 5.

The Nauru Amwangiyo maneuver is, then:

1. Gilbertese Movement.
2. Rotate 1 a full turn toward you.
3. Gilbertese Extension.

Making this mental connection, I then wondered if you could somehow mix the Mataka figures with Amwangiyo? Playing around a bit, I found this pretty figure:

1. Opening A.
2. Net Opening.
3. Rotate 1 a full turn toward you. Rotate 2 and 5 a full turn away from you.
4. Amwangiyo.

It's basically the Nauru "Amwangiyo" figure, but with two central suns. Very pretty! For every time you do the Net Opening in step 2, you'll get another central sun (but it can get really hard to extend with more than two suns).

Note the similarities to the Mataka series, though. There is a single loop that circles the palmar strings, and an even number of crossings on the transverse strings. Also, the central suns can easily form a mesh if their transverse crossings are arranged manually.

Mataka Series

After coming to string figures a few months ago, I started reading through the archives of the ISFA mailing list. My efforts were well rewarded when I stumbled upon this post, by Wil Wirt in April 2001, in which he describes the instructions for a Fijian figure called "Mataka".

1. Opening A.
2. 1 move under all loops and enter 5 loop from below.
3. Rotate 1 away from you, catching 5f on its back, continuing down, toward you, and up.
4. Release 5.
5. 345 move under 2 loop, enter 1 loop from above and hold 1f to the palm.
6. 1 shares 2 loop.
7. Navajo lower 1 loops over upper loop.
8. Touch tips of 1 and 2 and transfer 2 loop to 1.
9. 2 enters 345 loop proximally and picks up double strings across back of 345 fingers.
10. Release 1 and slowly extend with fingers pointing away.

I fell immediately in love with it. (Mostly with the process, though the presentation is lovely, too.) I've since played with it quite a bit, trying variations and experimenting. Recently I realized that the extension it uses (steps 5-10) is identical to the "Gilbertese Extension", the difference being that this Fijian version is done entirely on the hands. (The Gilbertese Extension itself uses slightly different techniques to achieve the same ends, employing the teeth to pull 2n through the 1 loop, instead of the Navajo maneuver that this figure uses.)

While experimenting, I discovered that I could make a "sparse" version of Mataka, with only two loops suspended from the transverse strings, instead of four:

1. Opening A.
2. Rotate 1 a full turn towards you.
3. Rotate 2 a half turn away from you.
4. Proceed with steps 2-10 of Mataka, above.

This made me wonder, what would a version of Mataka with zero loops suspended from the transverse strings look like?

1. Opening A.
2. Rotate 5 a full turn toward you.
3. 1 moves over 2 loops and removes 5 from below.
4. Gilbertese Extension (steps 5-10 of Mataka, above).

Knowing, then, how to make these net patterns with 0, 2, and 4 crossings across the transverse strings, I wondered next how to move from one to the other. Starting with the 0-crossing version, could you go to the 2-crossing version?

Indeed, you can. Starting with "Mataka 0":

1. 1 removes 5 loop proximally.
2. 5 removes 2 loop proximally.
3. Each palmar string has a loop encircling it. 2 picks up the distal string of that loop.
4. Release 1.
5. 1 moves to the far side of 5f and removes 5 from above.
6. Rotate 2 a half-turn away from you.
7. Gilbertese Extension.

The result is "Mataka 2". And if you apply those same movements to "Mataka 2", you get "Mataka 4"! So, what happens when you apply it "Mataka 4"?


The process may be continued indefinitely, although at 6 or more crossings the net pattern begins to require more and more manual arranging. I must admit to feeling more than a bit of pride in having puzzled this all out! (I wouldn't be surprised at all to learn that someone else worked this all out before me; but that wouldn't reduce my satisfaction in having solved it myself.)

So, where next? There are two questions in particular that I'd like to research further regarding this series:

1. Can you make a net with an odd number of crossings on the transverse strings? Is there a Mataka-3, for instance?
2. How do you make Mataka-6 or Mataka-8 directly, without starting from Mataka-4?

This is what I love about string figures: there's always something more to explore!

Navajo "Golden Eagle"

Today I was reading "String Games of the Navajo", from Volume 7 of the Bulletin of the International String Figure Association, and I fell in love with the "Golden Eagle" figure. Compared to some figures, it isn't particularly striking to look at:

But it's not so much the presentation I fell in love with, as it was the process. It has a really novel weaving step (novel, at least, as far as my own experience with string figures goes). Even the opening is novel, with the index reaching all the way around and under the figure to pick up the near thumb string. Here are the full instructions (rephrased in my own words):

1. Opening A.
2. Move 2 away from you, past 5f, then under 5f and all other strings. Hook down 1n and return, releasing 1. (I love that opening; I'm definitely going to experiment with that to see what else can be done from it.)
   
3. 1 moves through the lower 2 loop from above, under all strings, picks up 5f and returns.
4. (Here we begin the novel weaving step. It's harder to describe than it is to perform, so pay attention!) 1 hooks down upper 2n, letting the original 1 loop slip off.
5. 1 moves under 2 loops, and over 5n, hooking 5n down and through the 1 loop. Move 1 below the old 1 loop (originally was upper 2n).
6. 1 picks up, on back, the old 1 loop (originally was upper 2n), releasing the 5n loop from 1, and returns.
7. Release 2 and extend with fingers pointing away from you.

After that last step, the figure forms (as if by magic) from what previously looked like a tangle of string. Love it! I also really want to analyze that last weave step (steps 4-6, above) to see if I can understand what's going on. It's too novel to let pass with a simple "ooh, pretty!"

Lastly, the article in the Bulletin mentions that another name for this figure is "Road Going into the Distance Between Two Mountains." I really love that interpretation. It opens all kinds of possibilities for this figure, both from a story-telling perspective, as well as simply "wow, that's deep".