3x3, 4x4, etc.

The official definition of 3x3 and 4x4 (and NxN for higher values of N) says something like this:

Find pairings of people in the original ("2x2") version of the call, that start facing in the same direction, go through the same turning motions, and end facing in the same direction. For the 3x3 call, expand the setup so that each such pair has a person "interpolating" the two people in that pair, that is, halfway between them and facing in the same direction. (That person is commonly called the "cheese", in analogy with a cheese sandwich.) The 3x3 call is done by having the people in each pair doing the normal call but with the extra space, and the interpolating person going through the same motions and staying halfway between them.

Consider Bend the line:

\phfour{bAn,bCn,bXs,bZs}	\phBB{bAe,bCe,bXw,bZw}

before Bend the Line	after

A and C form a pair, as do X and Z. Interpolate B between A and C, and Y between X and Z.

\phsix{bAn,bBn,bCn,bXs,bYs,bZs}	\pvBBB{bXw,bAe,bYw,bBe,bZw,bCe}

before 3x3 Bend the Line	after

This case is easy, because the pairs are trivial to identify, are clearly related in the call definition ("as couples 1/4 in"), and are adjacent both before and after the call. That won't always be the case.

For 4x4, interpolate two people between the people in each pair, all equally spaced, maintaining the correct order at all times.

\phfour{bAn,bDn,bWs,bZs}	\phBB{bAe,bDe,bWw,bZw}

before Bend the Line	after

\pheight{bAn,bBn,bCn,bDn,bWs,bXs,bYs,bZs}	\pvBBBB{bWw,bAe,bXw,bBe,bYw,bCe,bZw,bDe}

before 4x4 Bend the Line	after

And similarly for higher numbers.

Things quickly become more complicated. The paired people might not be adjacent at the start of the call:

\phfour{bAn,bXs,bCn,bZs}	\phfour{bXn,bZn,bAs,bCs}

before Switch the Wave	after

\phsix{bAn,bXs,bBn,bYs,bCn,bZs}	\phsix{bXn,bYn,bZn,bAs,bBs,bCs}

before 3x3 Switch the Wave	after

Or they might not be adjacent at the end of the call:

\phfour{bAn,bCn,bXs,bZs}	\phfour{bXn,bAs,bZn,bCs}

before Cross Roll	after

\phsix{bAn,bBn,bCn,bXs,bYs,bZs}	\phsix{bXn,bAs,bYn,bBs,bZn,bCs}

before 3x3 Cross Roll	after

Or they might be along different axes before and after the call:

\phfour{bAn,bXs,bCn,bZs}	\phBB{bXn,bZn,bAs,bCs}

before Ah So	after

\phsix{bAn,bXs,bBn,bYs,bCn,bZs}	\pvBBB{bAs,bXn,bBs,bYn,bCs,bZn}

before 3x3 Ah So	after

or the pairs might be made of people who just happen to go through similar motions but aren't logically related by the definition. 3x3 Split Transfer is an example of this.

Furthermore, there might be multiple ambiguous pairings. In an 8-person call, there are usually 4 people facing in the same direction. How do we decide on the pairings?

\pvBBBB{b.s,b.s,bDn,bCn,b.s,b.s,bBn,bAn}	\pvBBBB{bDs,b.n,bCs,b.n,bAs,b.n,bBs,b.n}

before Beaus Advance to a Column	after

Let's try pairing A with C and B with D.

\sdform{\danc b.s,\danc b.s\cr\danc bCn,\danc bDn\cr\danc b.s,\danc b.s\cr\danc b?n,\danc b?n\cr\danc b.s,\danc b.s\cr\danc bAn,\danc bBn\cr}	\sdform{\danc b.n,\danc bDs\cr\danc b.n,\danc bCs\cr\danc b.n,\danc b?s\cr\danc b.n,\danc bAs\cr\danc b.n,\danc bBs\cr}

before 3x3 Beaus Advance to a Column	after

The interpolated people for the two pairs would be on the same spot. So this can't work. If we pair A with D and B with C, we get a similar paradox. So we try A with B and C with D.

\sdform{\danc b.s,\danc b.s,\danc b.s\cr\danc bCn,\danc b?n,\danc bDn\cr\danc b.s,\danc b.s,\danc b.s\cr\danc bAn,\danc b?n,\danc bBn\cr}	\sdform{\danc b.n,\danc bDs\cr\danc b.n,\danc b?s\cr\danc b.n,\danc bCs\cr\danc b.n,\danc bAs\cr\danc b.n,\danc b?s\cr\danc b.n,\danc bBs\cr}

before 3x3 Beaus Advance to a Column	after

So that's the correct way to do the call. (In practice, the caller needs to say "12 Matrix Beaus Advance to a Column" or "3x4 Matrix Beaus Advance to a Column" in order to legitimize the phantoms.)

Now try another case.

\pvBBBB{b.s,bDn,b.s,bCn,b.s,bBn,b.s,bAn}	\pvBBBB{b..,b..,b..,b..,bDn,bCn,bBn,bAn}

before Scoot and Fancy	after

Try pairing A with C and B with D.

\sdform{\danc bDn,\danc b.s\cr\danc bCn,\danc b.s\cr\danc b?n,\danc b.s\cr\danc b?n,\danc b.s\cr\danc bBn,\danc b.s\cr\danc bAn,\danc b.s\cr}	\sdform{\danc b.s,\danc b.s\cr\danc b.s,\danc b.s\cr\danc b.s,\danc b.s\cr\danc bCn,\danc bDn\cr\danc b?n,\danc b?n\cr\danc bAn,\danc bBn\cr}

before 3x3 Scoot and Fancy	after

Or we could pair A with B and C with D.

\sdform{\danc bDn,\danc b.s\cr\danc b?n,\danc b.s\cr\danc bCn,\danc b.s\cr\danc bBn,\danc b.s\cr\danc b?n,\danc b.s\cr\danc bAn,\danc b.s\cr}	\sdform{\danc b.s,\danc b.s,\danc b.s\cr\danc b.s,\danc b.s,\danc b.s\cr\danc bCn,\danc b?n,\danc bDn\cr\danc bAn,\danc b?n,\danc bBn\cr}

before 3x3 Scoot and Fancy	after

Both would be legal, so this call would appear to be ambiguous.

The definition says that, in cases like this, the ambiguity is broken by pairing the people that are physically closer. So the second of the above examples is the correct one.

Another call that shows the ambiguity, and the need to resolve it this way, is 3x3 Vertical 1/2 Tag. We will discuss that below.

The reader should have noticed a fundamental truth: Knowing the official definition of the NxN concept, and being able to execute the concept in real time, are very different things. The official definition is suitable for careful analysis and for theoretical discussions, but it isn't enough:

1. It requires dancers to undergo mental execution of multiple parts of a call, rather than the usual method that is based on people doing just one part of a call.
2. It requires proposing alternative theories about the pairings.
3. It requires mentally testing those theories with "thought experiments", noting whether they are unsound for various reasons, and picking the correct theory.

To do the concept in practice, one needs a lot of experience, and one needs to know a lot of tricks.

Overview of the Tricks

The first thing we notice is that, for many calls, the paired people are obvious, and work together in an obvious way:

\phsix{bAn,bBn,bCn,bXs,bYs,bZs}	\pvBBB{bXw,bAe,bYw,bBe,bZw,bCe}

before 3x3 Bend the Line	after

\phsix{bAn,bBn,bCn,bXs,bYs,bZs}	\phBBB{bCs,bZn,bBs,bYn,bAs,bXn}

before 3x3 Wheel and Deal	after

Some are not quite so trivial, but one can quickly develop a sense for how to do them:

\pvBBB{bXs,bAn,bYs,bBn,bZs,bCn}	\phsix{bAs,bBs,bCs,bXn,bYn,bZn}

before 3x3 Peel Off	after

\pvBBB{bXs,bAn,bYs,bBn,bZs,bCn}	\phBBB{bCe,bZw,bBe,bYw,bAe,bXw}

before 3x3 Box Counter Rotate	after

A few are more complex:

\pvBBB{bXs,bAn,bYs,bBn,bZs,bCn}	\phBBB{bCn,bZs,bBn,bYs,bAn,bXs}

before 3x3 Walk and Dodge	after

(Note that one's intuition about shape-changers and breathing needs to be re-learned.)

\phBBB{bAs,bXn,bBs,bYn,bCs,bZn}	\pvBBB{bCs,bZn,bBs,bYn,bAs,bXn}

before 3x3 Vertical 1/2 Tag	after

(And similarly for 3x3 Vertical Tag Zero and 3x3 Vertical Tag Full. Only the facing couples version of Vertical Tag can be used with 3x3.)

We mentioned earlier that this call could be ambiguous. This occurs when the starting setup is thought of as facing lines of 6. If a Triple Box Vertical 1/2 Tag is done from that setup, one can see that it satisfies the pairing requirement for 3x3. This shows once again the correctness of the "closest possible pairing" rule, and the need to analyze the call in the smallest possible setup, facing lines of 3 rather than lines of 6.

And a few are serendipitous and surprising:

\pvthree{bAn,bBn,bCn}	\phthree{bAn,bBn,bCn}

before 3x3 Mesh	after

(Only the all-facing-same version of this call works. Note that it is a 3-person call, not 6, and not 12.)

\pvBBB{bZs,bCn,bYs,bBn,bXs,bAn}	\phBBB{bAs,bXn,bBs,bYn,bCs,bZn}

before 3x3 Couple Up	after

\phBBB{bAn,bXs,bBn,bYs,bCn,bZs}	\pvBBB{bAs,bXn,bBs,bYn,bCs,bZn}

before 3x3 Leads Wheel the Ocean	after

\phBBB{bAn,bXs,bBn,bYs,bCn,bZs}	\pvBBB{bXn,bAs,bYn,bBs,bZn,bCs}

before 3x3 Leads Wheel the Sea	after

\phBBB{bAs,bXn,bBs,bYn,bCs,bZn}	\phBBB{bZe,bCw,bYe,bBw,bXe,bAw}

before 3x3 Circle to a Wave	after

The Tricks

1. If a call has people obviously working together in groups of 2, work in the equivalent groups of 3 or 4 or whatever. This covers things like Bend the Line, Wheel and Deal, Turn and Deal, and Ferris Wheel from a 2x6 matrix.

\sxhBBBstwelve{g4s,b4s,g5s,b5s,g6s,b6s}	\syhBBBstwelve{g3s,b3s,g2s,b2s,g1s,b1s}

before 3x3 Ferris Wheel	after

(In an actual situation, there would be 4 phantoms present, of course.)
(But beware: 3x3 Ferris Wheel can also be done from triple 2-faced lines (3x4 matrix.) We will revisit this.)
Notice that we have lost some of the generality of some calls. We can't do a 3x3 Turn and Deal from a wave of 6. This loss of generality occurs for many calls and works in the dancers' favor.

2. If a call is defined, or could have been defined, in terms of couples or tandems, turn them into couples or tandems of 3. Do this even if the definition isn't actually in terms of couples or tandems. Do it even if this "cheat" loses some of the generality of the call — the fully general version can't be done 3x3 anyway.

This gets Bend the Line and Wheel and Deal again. It also gets Drift Apart. We know that Drift Apart isn't defined as Tandem Cross Roll, and has more generality than that. But 3x3 Drift Apart loses that generality. It's just tandems of 3 Cross Roll.

This also gets 3x3 Counter Rotate from columns of 6. Box Counter Rotate is just Tandem Hinge. It also gets 3x3 Stack the Line.

\pvBBB{bZn,bCn,bYn,bBn,bXn,bAn}	\phBBB{bAe,bZw,bBe,bYw,bCe,bXw}

before 3x3 Stack the Line	after

In order to do 3x3 well, one needs to know a lot of "cheats".

3. If a call can be defined in terms of Couples Twosomes or Tandem Twosomes, turn them into Threesomes. This gets Turn and Deal and Turn to a Line. It also gets By Golly, which is a Couples Twosome (natural) Touch 1/4. It also gets suitably restricted Tag the Line and Follow Thru.

This also shows that there is another another version of Ferris Wheel. Ferris Wheel could be defined as "As Couples, Tandem Twosome, Single Wheel". We have already seen the version in which the Couples were turned into couples of 3. When we turn the Tandem Twosomes into Tandem Threesomes, we get this:

\sxvBBBstwelve{b4s,g4s,b5s,g5s,b6s,g6s}	\syvBBBstwelve{b3s,g3s,b2s,g2s,b1s,g1s}

before 3x3 Ferris Wheel	after

4. If a call can be defined in terms of Fractional Twosome, or Twosome Fractional Solid, do the same thing. This gets Peel Off, Trail Off, and a suitably restricted version of Crossfire.

One needs to be really good at seeing things in terms of geometrical patterns of this sort.

5. Learn things like Beau Walk / Belle Dodge, or With the Flow, or Vertical, or Finish Wheel the Ocean / Sea. They don't actually fit into the above Couples / Tandem / [Fractional] Twosome categories, but they are geometrically very similar. 6. If a call (or restricted version of same) can be formulated in terms of a sequence of other calls that can be done 3x3, do so.

Know that Ah So is Hinge and Counter Rotate, Split Recycle is Counter Rotate and Hinge, Wheel the Ocean / Sea is Wheel Around and Finish Wheel the Ocean / Sea, Vertical 1/2 Tag is Vertical and Tandem Touch, and similarly for Circle to a Wave, Circle the [1/2] Tag, etc. (Note that 3x3 Split Recycle is an extremely nonintuitive call.)

7. 3x3 Arm Turn any amount (i.e. miniwave Hinge / Trade / Cast) is just Arm Turns in the individual miniwaves. This is why 3x3 Ah So and Split Recycle work. Touch, Pass Thru, and Step Thru are similar.

The same is true for Quarter Left or Right, or Roll, or doing anything Stable. For example, Walk and Dodge is Stable Counter Rotate, and Counter Rotate is Tandem Hinge, so one can deduce how to do a 3x3 Walk and Dodge. Similarly, Couple Up can be reformulated as Counter rotate and Roll. Shakedown is Quarter Right, Counter Rotate, and Roll.

8. Learn how to do 3x3 Slither. From a wave of 6, go past everyone who is facing opposite you, until everyone facing in the same direction is in one adjacent group. To go the other way, from couples of 3, spread yourselves all the way out until everyone is in a miniwave.

\phsix{bAn,gCs,bBn,gBs,bCn,gAs}	\phsix{bAn,bBn,bCn,gCs,gBs,gAs}

before or after 3x3 Slither	after or before 3x3 Slither

This gets 3x3 Switch the Wave and 3x3 Cross Roll, because they can be formulated in terms of Trade and Slither.
Also, Scatter Circulate is Couples Twosome Circulate and Slither, so 3x3 Scatter Circulate is Couples Threesome Circulate and 3x3 Slither. Go First Class is Slither and Couples Twosome Circulate, so 3x3 Go First Class is 3x3 Slither and Couples Threesome Circulate.

\sxhBBBstwelve{g4s,b4s,g5s,b5s,g6s,b6s}	\sxhBBBstwelve{b4s,g3s,g4n,b3n,b5s,g2s}

before 3x3 Scatter Circulate	after

\sxhBBBstwelve{b4s,g3s,g4n,b3n,b5s,g2s}	\sxhBBBstwelve{g3s,b3s,g2s,b2s,g1s,b1s}

before 3x3 Go First Class	after

Also, Cross and Divide, from a wave, is Slither and Retreat the Line, so 3x3 Cross and Divide, from a wave of 6, is 3x3 Slither and 3x3 Retreat the Line (couples of 3 1/4 Out.) (Cross and Divide can also be done from a 2-faced line, but the turning motions are such that it can't be done 3x3 or 4x4.)

These calls don't arise with 12 people in real dancing, but the 1x3 versions do arise with 8 people, so it is useful to be aware of the 3x3 Slither.

9. There is a kind of "vertical Slither" that shows up as a building block in some calls.

\phfour{bAe,bBe,gBw,gAw}	\phfour{bAe,gBw,bBe,gAw}

before or after Vertical Slither	after or before Vertical Slither

\phsix{bAe,bBe,bCe,gCw,gBw,gAw}	\phsix{bAe,gCw,bBe,gBw,bCe,gAw}

before or after 3x3 Vertical Slither	after or before 3x3 Vertical Slither

For example, Dixie Style to a Wave can be formulated as Vertical, Vertical Slither, and Left Touch 1/4. So 3x3 Dixie Style to a Wave can be formulated as 3x3 Vertical, 3x3 Vertical Slither, and Left Touch 1/4. That is, do a 3x3 Vertical, everyone except the very ends walk forward until everyone is facing someone (but no farther!), and all Left Touch 1/4. This is rather difficult.

\syhlines{g3s,b3s,g2s,b2s}	\syvlines{g1w,b2e,b1w,g2e}

before 4x4 Dixie Style to a Wave	after

This also shows up in the starting-DPT variety of Grand Chain 8. This can be formulated as a centers Pass Thru (Vertical Slither), Quarter Out, and Courtesy Turn.

\sxhBBBstwelve{b1w,g1w,b2w,g2w,b3w,g3w}	\sxhBBBstwelve{g6n,b6s,b1n,g1s,g5n,b5s}

before 3x3 Grand Chain 8	after

This is another difficult call.

10. There are a few calls that could be formulated in terms of a nonexistent "fractional anti-twosome" concept, in which the people in each group orbit around each other opposite the direction they would if working solid. (Such a concept was proposed once, but, fortunately, no one could figure out how to express it.)

Scoot Apart could be thought of as a sort of Tandem 1/4-Anti-Twosome Trade. So 3x3 Scoot Apart is a Tandem 1/4-Anti-Threesome Trade.

\pvBBB{b4s,b3n,g4s,g2n,b1s,b2n}	\phsix{b4n,g4n,b1n,b3s,g2s,b2s}

before 3x3 Scoot Apart	after

3x3 Split Swap is analyzed similarly.

This covers almost all situations that will arise. The rest can usually be covered by redefining [a restricted version of] the call in terms of other calls.

Here is a hard example. We know that Countershake is not properly defined as Quarter Right, Counter Rotate, Roll, and Solid Box of 4 Touch — The real definition can be done from other setups, and fractionalizes differently. But this cheat shows how to do a 3x3 Countershake. In fact, the cheat shows that there are two different 3x3 versions of the call. The groups can be paired laterally or front-to-back. It's too complicated to figure out the pairing with the definition, especially with front-to-back pairing. Instead, just know the cheat, and do the call in whatever shaped groups you have.

\syvBBBs{.+.,g2n,.+.,b2n,b3n,g1n}	\syhBBBs{.+.,g4e,.+.,b4e,b1e,g3e}

before 3x3 Countershake	after

\syhBBBs{.+.,g4n,.+.,b4n,b1n,g3n}	\syvBBBs{g1e,b3e,b2e,.+.,g2e,.+.}

before 3x3 Countershake	after

Here is another one that does violence to the proper definition. From a completed DPT, Roll Out to a Column could be described as Couples 1/4 Twosome, Tandem Twosome, Roll Right to a Wave. So 3x3 Roll Out to a Column is either

Couples 1/4 Threesome, Tandem Twosome, Roll Right to a Wave

\syhBBBs{.+.,g4n,.+.,b4n,b1n,g3n}	\syvBBBs{.+.,g1n,.+.,b2n,b1s,g2n}

before 3x3 Roll Out to a Column	after

or Couples 1/4 Twosome, Tandem Threesome, Roll Right to a Wave

\syvBBBs{.+.,g2n,.+.,b2n,b3n,g1n}	\syvBBBs{g2s,b1n,.+.,g3n,b2s,.+.}

before 3x3 Roll Out to a Column	after

Here is one last one, which seems to be very common. Run the Wheel in general is a space-invader. But Leads Run the Wheel can be considered to be equivalent to Box Counter Rotate 3/4. Hence 3x3 Leads Run the Wheel is just a Counter Rotate 3/4 in columns of 6.

How Hard Can it Be?

We have pointed out that these concepts are really not always suitable for dancing "cold" except in the simple cases, and that one needs to learn "tricks" to get through the hard cases without having the dance degenerate into a mathematics symposium. Perhaps the most notorious case of this, as of this writing, is 3x3 and 4x4 Load the Boat. Load the Boat has the property that the person who is undergoing the same turning motion as you, whom you have to quickly locate, starts once removed from you, finishes once removed from you, never interacts with you, and is following a definition completely different from yours. But the 3x3 and 4x4 concepts are indifferent to all of this, so 4x4 Load the Boat is legal.

\sxhBBBBs{.+.,b3s,g2n,.+.,.+.,g3s,b2n,.+.}

before 4x4 Load the Boat

\sxhBBBBs{b1w,.+.,.+.,g4e,g1w,.+.,.+.,b4e}

after

So what "trick definition" of Load the Boat does one use to get this result? A little checker pushing will show that Load the Boat is equivalent to a big line-of-4 Wheel Around followed by a 1/4 In to your immediate partner. Those "trick" components are suitable building blocks for doing the call 3x3 or 4x4. For a 4x4 Load the Boat, from facing lines of 8, do a huge line-of-8 Wheel Around, preserving phantom spots of course, followed by a 1/4 In. 3x3 Load the Boat is similar, with a line-of-6 Wheel Around.

If one really wanted to do this in a proper, stylish, elegant, and aesthetically pleasing manner, one would actually Pass Thru, Own the Belles for a big line-of-8 Wheel Around by big line-of-8 Reverse Wheel Around, Pass Thru, and then 1/4 In.

Supplemental Material:

The 3 By 3 Concept (Sue Curtis)