Square dancing is usually danced by 4 couples who start each sequence in a square formation. This paper describes how 6 couples can start each sequence in a hexagon formation, and dance the same choreography with the same timing as the usual square. Certain dancers will enjoy this twist on square dancing.
Hexagon squares are the next in a series of Square Games. Each is a global twist on normal square dancing which can be danced in real time while the caller is calling to the rest of the floor.
In March 2002, I created a new square game, Hexagon dancing, which requires 6 couples. A history of Hexagon Squares and similar variations can be found in the Appendix.
Arrange 6 couples in a perfect hexagon. There should be two couples lined up on walls (the couple with their backs to the caller and the couple they are facing). The other four couples are not lined up on walls.
Starting with the couple with their backs to the caller and going around the set in promenade direction, identify each couple as heads, sides, heads, sides, heads, and sides. Have the heads raise their hands. Have the sides raise their hands. Point out that the couple directly across the hex from them is not the same as them (i.e., opposite every head couple is a side couple!).
Heads Go Forward And Back. Notice how the person they approach when going forward is not directly facing them, but slightly around the corner from them. This kind of slightly around the corner will happen a lot in Hex dancing.
Heads Go Forward again and Square Thru 4. It should feel like a wrong way grand in a circle of 6. Don't worry about the angles and turns. With any luck, all heads will end facing the sides. At this point the entire set looks like a "Y". This will be very common.
All Do Sa Do To A Wave. Whenever we do 4-dancer calls on the outside (which is very common in singing calls), nothing special happens in the Hex square. We just work in our own group of 4. To see this, dance the following: Swing Thru, Boys Run, Bend The Line, Right And Left Thru, Dixie Style To A Wave, Trade The Wave, Centers Run, Centers Hinge, Diamond Circulate, Flip The Diamond, Centers Trade, Swing Thru.
Now we should still be in waves with boys on the end. Notice how the whole set looks like a Triangle (or a delta). Only one of the 3 sides can be lined up on a wall. The ends of each wave are boys. They must be careful to not touch or get too close to the adjacent wave. It would be easy to hold hands from one wave to the next, but this would be very wrong.
Let's talk about each wave. The dancers who are looking out are just like the dancers looking out of any wave. They should know where to go on a Circulate. Consider the end looking in. On a Circulate, he will move to the next wave, following a back, moving around a slight angle, and finish as an end looking out. Have all the Boys Circulate. Have all the Boys Circulate again. Now the girls turn. The girl looking in is also following the back of another girl in the next wave. Have the Girls Circulate, and then Circulate again. Do a Swing Thru to move everyone into new positions and do some more Circulates and Acey Deuceys. Have the Centers Run and do two Couples Circulate.
At this point, you should check out some animated illustrations created by Justin Legakis. The "Square/Hexagon Animation Gallery" will show you several of formations, and "Hexagon Formation Animations" will show you a variety of calls. If your computer has Macromedia's Flash Player you should also check out these animations by Yoichi Shinohara.
Time for a Ferris Wheel. The lead couples just do a normal Wheel And Deal. The trailing couples (all 3 of them) will head into the middle as usual and wheel and deal but they will under achieve. They will go a little less than they feel they should. Do it, Ferris Wheel. If I end in the center, I double check my alignment in two directions. First, I should be exactly in front of a couple. Second, the center 6 (my couple and the other 2 couples in the middle) form a perfect triangle. A common mistake is to have two of the center couples try to face each other since that is what happens in a normal Ferris Wheel.
Now we are ready for Double Pass Thru. Think of the center of the set as having a giant magnet which repels you away as you try to approach it. The square looks like a "Y". The men are going to go down the left arm of the "Y" and the women are going to go down the right arm. You will lose your current partner! Everyone ready? Go, Double Pass Thru.
Now is not the time to attempt Track 2. Instead have the Leads Trade and all Pass The Ocean. We have a Tidal Wave. In the very center we should have a left hand star of 3 dancers. Do a Grand Swing Thru, turn half by the right and half by the left. The center star only turns 1/3 (remember, under achieve in the center). Another way to think of this is that the very centers arm turn to the next dancer.
Do another Grand Swing Thru. Each wave Recycle. Right And Left Thru. Everyone is happy and we are back to our 4-dancer calls. Pass Thru. What are we thinking? At all times we can be thinking about 2 different 4-dancer formations. In a normal square, these correspond to the box you are in and the line you are in. We just did a Pass Thru. Our focus is on our box of 4 (i.e., Couples Back-To-Back). An easy next call would be Chase Right (don't do it).
What if the next call were Tag The Line? Do you even see a line? Probably not. However, if you find another couple facing close to your direction, and form a line with them, and straighten up the set a little, we should be able to make the set look like a Triangle again with 3 lines all looking out. At this point, Tag The Line would be easy (don't do it).
Equivalent Formations 
Let's undo that adjustment and get the set back to right after the Pass Thru. The overall shape is a "Y". Now make the slight adjustment to lines back-to-back. The overall shape is a Triangle. Here is a really important point. In Hex dancing, when the square is in the equivalent to a 2x4 (lines, waves, columns, 8 chain, etc.) we physically stand on either "Y" spots or Triangle spots (whichever feels good after the last call), but are always ready to adjust to the "other" spots if the next call requires it. A less satisfying way of saying the same thing is that we never know which are the correct spots and must dance as if we are on both. Some dancers (the precise ones who must always know where they (and everyone else) are) will have problems with this. The flow and intuition dancers will do just fine.
Having covered that point, let's finally do the Tag The Line, Face In. All Go Forward And Back. While we usually don't think of forward and back as much of a call, in Hex dancing I use it to establish who I am facing and who is my partner. Do it again -- Forward And Back. Pass Thru. Tag The Line, Face In. Forward And Back. Right And Left Thru.
Let's look at the same issue from the point of view of waves. Pass The Ocean. We have our nice waves. Circulate. Acey Deucey. Swing Thru. Circulate. Everything is fine. Don't do it, but think about Scoot Back. Where is your box of 4? Who do you scoot with? OK, Scoot Back. Do another Scoot Back. What shape is the set? "Y" or Triangle? Probably a "Y" because the Scoot Backs focused attention on the box of 4. So, the next call is Centers Trade. Where is your wave? Now, Centers Trade. In the heat of battle, dancers will make a mistake and have some number of dancers near the center of the set make a star and do some sort of trade. This is totally wrong.
Swing Thru. Split Circulate. Centers Trade. Scoot Back. Centers Trade. Swing Thru. Walk And Dodge. Tag The Line. Partner Tag.
Square Your Set. The next set of information is useful for singing calls. When doing Chains, Promenades, and Single File Promenades from a squared set, think of them as having to take the same amount of time as the normal squares and dance them by counting positions.
For example, Head Ladies Chain has the head ladies make a 3-dancer right-hand star. They skip one man and get Courtesy Turned by the next. Do it, Heads Ladies Chain. Another one, Head Ladies Chain. Side Ladies Chain. Side Ladies Chain. Now, 4 Ladies Chain. This should be a 6-hand star, skip one man and get turned by the next.
4 Ladies Chain 3/4. 6-hand star, skip one man, skip the next man, and get turned by the next man. Every head man should be turning a side lady. Go again, 4 Ladies Chain 3/4.
To make the timing work out, a Right And Left Grand is just 4 hands and promenade the next dancer. The same goes for Weave The Ring. Weave by 4 and promenade the fifth.
Now, say the call is "4 ladies promenade inside, get back home and swing". You will have 6 ladies promenade inside, skip one man, skip the next man, skip the third man, and swing the fourth man. This will time out the same as in the real squares. Every head man will be swinging a head lady (same as in the real squares). You will promenade for the same amount of time as the real squares and every head man will stop in a head position (but not necessarily his original head position).
Square Your Set. When we dance Hex to a caller who calls and resolves normal squares, we will not necessarily get our partners back, nor will we always be in sequence. However, in the patter tip every head man will always get a head lady. If we are heads, we will promenade to nearest head position, or in a singing call, until the music says to stop. In all cases, the men should retain their identity and stop in an appropriate position (head men stop in head positions, side men in side positions). Heads Star Thru, Square Thru 3, Left Allemande, Right And Left Grand, Promenade Home.
If the call is Heads Promenade 1/2 Way, again count positions. Promenade on the outside of the set past one side couple and face into the next gap (where a head couple used to be).
If you are dancing Mainstream, what I have presented so far will be enough to try some actual dancing. You won't get 100% success, but you will improve and it will be fun. Singing calls are easier than patter, but patter is often very accessible.
From a squared set, Heads Pass The Ocean. This is hard for several reasons. First, the adjustment from a squared set to having the heads be facing couples in the middle is so automatic that we get lost doing it in a Hex square. Having the Heads Go Forward and Back helps identify which dancers are facing. Next, we usually dance Pass The Ocean by feel. In this case, the Hex effect removes that ability. We must do it by definition. Heads Up To The Middle and Pass Thru. Find a new partner. Face That Partner. Step To a Wave With That dancer. The center 6 should have 3 mini-waves, and the head women have a 3-dancer left hand star in the center.
Have the Heads Swing Thru (right hand half, left hand star 1/3). Have The Heads Swing Thru again. We are in a 1/4 Tag. Two good calls from here are Extend and Ping Pong Circulate. Everyone Ping Pong Circulate. Centers Swing Thru, Double. Extend. We should now have normal waves.
Relay The Deucey is really cool. Look at where you are in case we break down and have to go back. Look at who you are holding onto. Just like a real Relay The Deucey, you will end up with this dancer. Go by parts. Turn 1/2 Right. Men left 3/4, ladies walk a little around the outside. Girl On. Stop.
We should have the following. 3 waves of 3 joined in the center in a right-hand star of 3 boys. 3 loose women on the outside. These women must not get too close to the star of boys or they will get sucked in. Instead of the women thinking that they walk down the "Y" and up an arm of the "Y", they should focus instead on walking around the outside of the set. Their next act will be to get "on" the tidal wave. They should be looking for where that is.
OK, turn 1/2 by the Right (1/3 in the very center). Boy off and girl on as the others turn 1/2 by the Left. 1/2 Right (1/3 in the very center). Boy off, as the girls turn 3/4 by the left. The boy who is walking around the outside at the end and will become the lead end of the wave has a tendency to go too far (which is also true in normal squares). Make sure he stops sooner. Do the call several times from the same and the other position.
Spin Chain The Gears is the easiest of the "big" calls. Start from normal waves. All turn 1/2. Boys 3/4 while the girls turn back. Center 3 boys find a right-hand star and turn it 1/3. Each 4-dancer star turn left 3/4. 3 girls star 1/3 in the very center. Girls left 3/4 while the boys turn back. This is usually successful but it is important to get practice looking for the various parts. Some dancers are confused by the stars on each side, the star in the center, who they are working with, etc.
Spin Chain And Exchange The Gears is also easy. Everything is the same until the exchange point. The lady who leads her star just has to head in an approximate location and make the wave. The whole set will gel into a Triangle of 3 waves.
Track 2 is tough. From a Squared Set, Heads Star Thru. Double Pass Thru. Focus on this: Every boy is going to get a girl with his right hand. Somewhere close by, we should have right hand waves with girls in the center. If these two conditions are met, we are close enough. Now, start the Track 2. Boys go around the girls and down the side and get some new girls. Every boy get a girl. Which girls have left hands with each other? No one is in the very center of the set. We should have 3 normal ocean waves.
We recently got tricked by Triple Scoot. The #3 dancers in the columns will star with each other. Each dancer skips the first clump of dancers they encounter and joins the next clump. Coordinate has the same center effect and was also took some practice.
From a squared set, Sides Face, Grand Square. I normally don't walk this -- it just works! Encourage everyone to take one step for each beat of music. Since the Hex square is a little larger than a normal square, there is less a tendency to rush. This looks and feels really cool.
Teacup Chain. This one is a real problem. It is hard to do and has few rewards for doing it correctly. To make the timing work, we need to think of it as skipping dancers instead of progressing around the square in a nice order. Here is how it goes. Side men always fetch women from the center and send them to the next head men. Head men always receive women from the side men and send them into the center. When the head women go into the center with a right arm, they star to the third man. When they go in with a left arm, they star to the fifth man. This will cause a weird progression, but it times out properly.
Fast calling, especially when the centers are activated, can be tough. Heads Pass The Ocean, Swing Thru, Extend The Tag and we are dead. Same goes for Pass To The Center, Pass Thru. If you are dancing patter for the first time, expect to break down and have a plan. The plan is to square your set and make lines by having the heads turn slightly to the right and sides to the left (as couples) and go forward and back. The caller will get to lines to pick up some other broken square soon enough.
Memorized sequences with clever choreography can be difficult. From normal lines facing, Touch 1/4, Circulate 1 1/2, Center 6 Trade and Roll, Center 6 Slide Thru, Left Allemande. The center 3 dancers working around the flagpole center for any call (e.g., Slide Thru, Pass Thru, Dosado) is unusually difficult.
I usually start with a few Right And Left Thrus, a Ladies Chain (to bring up the star 1/3 issue) a few Flutter Wheels (girls get a new man and bring him to a place neither of you have been to), and a Square Thru. Then do some box calls (Box Circulate, Walk And Dodge). Don't forget diamonds (from waves, centers run, new centers hinge (dancers will usually turn too far on a Hex Hinge -- you may need to back them up slightly to get an exact diamond), check a diamond). Diamonds look really funny, and not at all like a diamond.
Really hard calls are Fan The Top, Scoot Back, and Wheel Thru. Also keeping track of which dancers "own the wall" (i.e., should be lined up on a wall) is really hard. I find it useful to keep track of who "owns the wall" with 3 couples, but not for 6. With 6 couples, everything just seems to work.
3-couple dancing is also useful when you want to show the Hex stuff to someone but don't want to find 12 interested people. I have never danced the 3-couple stuff live with music. Non-symmetric material does not work so don't get trapped with it.
I try to choose dancers who I believe will enjoy the experience. Usually this means people who have good dance skills, who have been dancing a while, perhaps who know Advanced or Challenge, who have good recovery skills, and who have shown a past interest in doing different things with their dancing.
No matter how we form the set, there are usually one or two dancers who clearly aren't getting it, or whose dance skills were already weak and this has pushed them over the edge. Since these dancers have already spent the time getting trained, I just keep them in the square. Sometimes they catch on, and usually they don't cause a lot of problems. After several tips, it is good to give those that would like a break a chance to escape. Usually there is someone on the sidelines who has been watching and would love the opportunity to join the square. If they have been watching, they don't need any instruction.
While I would prefer to spend 10 minutes or so explaining the ground rules and walking people through the various calls, in one case we had no time for that. Three experienced Hex dancers squared up (as heads) with 3 couples who had never done it before and had no idea what was going to happen. The music was already going and the callers were calling with no breaks at an after party. I believe I squared us up, identified heads and sides, said that we were going to dance what he was calling, that we go 4 hands on a right and left grand, and we wouldn't necessarily promenade to our original home, and to watch us and go with the flow, and not to worry.
With basically no instruction, we did ok at first and got better. Each time we broke down on something we would square up and if I had time, I would quickly explain it or walk it. Soon we switched one head and one side couple so when the sides were asked to do something, at least one couple had done it before. In a few tips we were doing fine and having fun. Soon one experienced couple had to leave and was replaced by a new couple who had been watching on the side. We still did fine.
One way to learn Hex dancing is to watch a square doing it, and then jump in. Another way is to put 1 or 2 new couples in an experienced square have have them go with the flow.
No matter how the Hexagon is formed and trained, it helps to have a leader. The leader signals when promenades stop. The leader decides when the hex is broken down and asks it to "square up". Also, after squaring up, the leader watches the other squares and decides when to "make lines". By making lines, the hex will get a lot more practice dancing. Finally, when there is down time (say they are changing callers), the leader should walk the last few calls that broke the hex down (so it won't happen again).
In a normal square every dancer has a diagonal opposite. In Hex squares every dancer has two opposites. Instead of being across the square, each is down one arm of a "Y". The dancer who is down to his right will always be down to his right. While "following your opposite" can be a useful tool in normal dancing, I have found it to be useless in Hex dancing. The only time I notice my opposites in on the promenade when I am watching the square to see when we will stop. The opposite to my right is always 2 couples ahead of me on the promenade.
In normal squares there are two sequences of couples -- in and out. Promenade home only works when you are in sequence. In Hex squares there are 3 sequences of couple, only one of which is truly "in sequence". This doesn't matter to us because we just promenade to a home position of the correct type (head or side) instead of our original home. If your goal was to call only to Hex squares and to resolve, then the three sequences would matter and you would have to get it right.
From a Double Pass Thru formation, you can think of a normal square as having your half the square and the other half the square on the other side. In the Hex situation you still have your half, but someone put a beam splitter, mirror, or prism in the center of the square which has duplicated the far half and moved one copy to the left and one to the right.
There are 6 sex arrangements for each formation in normal squares. The same holds for Hex squares. Every symmetric formation in normal squares has a corresponding formation in Hex squares.
If the normal square resolves, the hex square will get the heads together and sides together, but sequence and partners may be wrong.
Consider a call from facing couples in which the left side dancers right-pull by with each other. Consider the same call, but the left-side dancers use the other hand and left-pull by with each other. In normal square dancing both version of this call would have the same effect. However, in Hex dancing the two versions would have different effects! In Hex dancing the traffic pattern matters. This means that all choreography programs which know definitions of calls but not traffic patterns (which I believe is all of them) will not be able to dance Hex with their existing database, even if their display software is updated for the odd angles and extra couples.
In a conversation with Justin Legakis we discussed that one could write a computer program to automatically convert animations of 4 couple squares to Hex annimations. He has done just that and you can see the results here. Click on the "2-Couple, Square, Hexagon, and 8-Couple Generalizations".
One dancer commented to me: If you think of square dance calls executed in a normal square in terms of fractions with a denominator of 4, for example, Right And Left Thru would move you 2/4 the way across the square, that can be transformed into hexagon-ese by changing the denominator to 6, but keeping the same numerator. So, Right And Left Thru moves you 2/6 (or 1/3) of the way around the hexagon. But of course, this transformation only applies to operations that either move through the center of the formation or affect the entire formation with respect to the center (does that make sense?). It probably complicates things too much to be explained simply, but it does quantify what you were referring to as "under-achieving". In a sense, it's more or less a different way of saying "you skip the same number of people (or positions) you would skip in a normal square".
In e-mail to me, Justin observed:
You mention several times that the equivalent of a 2x4 looks either like a Y or a triangle, and then when dancing you must be ready to adjust between these shapes. I prefer to think if it as one formation, with curved waves (or columns) so you see the three 1x4s, and the three 2x2s at the same time. Of course, this is how I've thought about Hexagon Squares on paper -- I don't have your experience with dancing it.
With care, one can write choreography which works (i.e., resolves to correct partners, and in sequence) in both normal and Hex squares at the same time. While interesting, I don't think it is worth the effort.
Walls: While dancing Hex, one normally doesn't think of facing "Head Walls" or "Side Walls" even though these terms definitely apply in normal squares. Noriko Takahashi clued me into thinking about this. When the part of the hexagon which is lined up with the walls in the room is close to the caller, everyone is facing head walls. When the part of the hexagon which is lined up with the walls in the room is far away from the caller, everyone is facing side walls. No one should ever face the real side walls in the room.
Sight calling to such a square is easy. That is, picking calls that are legal and flow well is easy since you just have to call your usual combinations. Sight resolution is another matter.
Developing a method of sight resolution that works, is not too hard to learn, that can be used without any long pauses, and that doesn't take too many calls is hard. For those interested in such things, I think trying to develop a Hex resolution system is a great exercise. For many of us, sight resolution systems came to us fully developed. They are presented as an algorithm that we have to memorize and follow and later we will be able to expand on the system. In Hex you have to start from scratch. I recommend thinking about each letter in FASR and what it means in Hex. Consider also that from a Beginning Double Pass Thru with the ends paired and the centers unpaired we would normally call either Pass Thru or Square Thru 3. In Hex we would have Pass Thru, Square Thru 3, and Square Thru 5. Also, in Hex, the partners don't stay with you when the centers Pass Thru.
If you come up with something, let me know. If I try my method and it works, I might post it here.
P.S. I have recently learned that Bill Eyler has been using Hexagons for about 16 years and sight calling them to regular dancers for about the last 10 years.
Hexagons are not for everyone. If you present this gimmick to everyone, a little bit goes a long way. Probably many people would just as soon not dance Hexagons at all. For those that do enjoy them, there are many hours ahead of new-found fun with our existing calls and choreography.
I welcome feedback, corrections, and suggestions on this paper. Also, if you teach or dance Hexagons I would be interested in hearing your experiences. Please write. If you have any questions on how certain calls should work, just ask. I always have an opinion.
My experience with Hexagons is relatively new. While I have been dancing since 1974 and they were invented in 1968, I hadn't encountered them in any depth before talking with Clay in 2002. If you have other history or credit which I should know about (e.g., "Back home, Joe has been doing that stuff for years. Except, he does it slightly differently than what you say..."), please tell me.
Clark Baker
email: cmbaker@tiac.net
phone: (617) 484-0175
In The Handbook of Modern Square Dancing, Jay King describes Lee Boswell's Triangle Squares by saying that they start in an actual hexagon with the couples alternating heads and sides and that each sequence starts with heads or sides leading to the right and circling to a line. This gives 3 lines (each side of a triangle). If you bend the line, the set is in a "Y" shape. While I have no other information on how Lee Boswell used his Triangle Squares, they are exactly what I am talking about.
Bill Eyler wrote an article in the May 1993 issue of Mikeside Management (a caller note service) entitled "APPLE PIE" DANCING; Hexagons - A Little Slice of Square Dance Heaven. Bill says that the July 1986 issue of American Squaredance Magazine featured an article on the concept of Triangle Squares by Ross Crispino, with credit to Harriett Miles. Bill has been using Hexagons in performance exhibitions for 16 years and as an entertainment tool at dances and festivals for the last 10 years.
I have e-mail from Dan Koft from May 2000 on the subject of gimmicks. Rutgers Promenaders has one "Unusual Mainstream" tip each evening and one of the tips was "True Hexagons (3 even couples, 3 odd couples, Square thru 6 to get to corner)". I don't know where Dan first saw this formation or what choreography he used. In talking about gimmicks he did say that True Hexagons were not worth the prep time. I am not sure if he is saying that it took a lot of effort to prepare material that works, or if it takes too long to teach dancers how to dance in a hexagon.
At Callerlab in March 2002, Clay Goss introduced me to the Hexagon Squares he uses with his exhibition group, the Clay Figures. They dance in a true Hexagon (heads, sides, heads, sides, heads, sides) and make use of several of the 8-dancer Plus calls (e.g., Relay The Deucey, Load The Boat). I was full of questions and he explained how Double Pass Thru, Grand Swing Thru and the various Ladies Chains worked. While someone had tried to show me choreography from a Hexagon in the past, it wasn't until I talked with Clay that something started to click.
Soon after I left Callerlab I realized two things:
One evening, I was able to grab 6 couples of Tech Squares dancers and we spent an hour walking various calls from a Hexagon. I didn't have a precise explanation of what the rules were, and several in the group were resistant to just "go along with the flow" and see what happens.
At the New England Convention in April, we were able to get a Hex square and actually try dancing to a live caller. We started with some trepidation, and we had a few breakdowns, but we danced two tips with great success. Everyone was very enthusiastic and thought it was neat how the various calls like "Heads Square Thru" just worked out even though the angles are all weird. Also, the "look" and "feel" of certain calls really changes, which added to the magic.
I have shown Hex dancing to challenge groups in Chicago, Cherry Ridge, L.A., and Ploen, Germany, and have danced it at the National Convention in St. Paul and the Singles convention in Richmond. We have recently danced it for the first time at a regular Tech Squares dance.
Clark asks if there is some way to know when to change 180 degree turns into 120 degree turns. In general, how do you know where to go in Hex dancing, and if every call "works"?
What you would like to be able to claim is the following:
Every call in symmetric 4-couple square dancing has a well-defined analogous call in Hexagon dancing.
I say that we can as least claim this:
Every motion (path, traffic pattern) of symmetric 4-couple dancing has a well-defined analogous motion in Hexagon dancing.
If you can say that all calls in square dancing have a well-defined traffic pattern, then you can say that all calls have a well-defined way to dance them in Hexagons. So, the question is: do any calls have undefined traffic patterns?
I seem to remember learning one or two calls with pull-bys that could be either with the inside or outside hand. The rule was "don't hurt anybody." Cross and Turn, maybe? Or anything and Cross?
The ambiguity only makes a difference of course if it happens in the center of the square, and gives people a choice as to which direction around the center of the square they travel. So, doing an Ends Cross Cast Back in the east coast or west coast style doesn't matter.
The claim above (every 4-couple motion has a well-defined 6-couple motion) can be backed up solidly by giving the mapping from one motion to the other. This is done most naturally in polar coordinates. If a dancer in a 4-couple square moves from
(a0, r0, d0) to (a1, r1, d1)
where a0 and a1 are the polar angles around the center of the square, r0 and r1 are the distances from the center of the square, and d0 and d1 are the dancer's starting and final facing directions, then the equivalent 6-couple dancer's motion would change by the following:
( (a1-a0)*2/3, r1-r0, (d1-d0)-(a1-a0)*1/3 )
In English:
All rotations around the center point are reduced to 2/3 their amount. This is the "under achieve" rule that you discuss in your paper.All distances from the center map directly.
As you rotate around the center of the square, you adjust your facing direction by 1/3 the angle in the opposite direction.
It is important to not take the angles mod 360! If a1 > a0, this implies counter-clockwise motion, and a1 < a0 implies clockwise motion.
This is just a mathematical mapping, and doesn't address the fact that the square will breath, and tend to line up in a Y or triangle. This will map a 2x4 to curved waves (or columns), between a Y and a triangle. But these are all equivalent -- the important thing is that this mapping gives you an "ideal" spot to stand on and direction to face, which provide for a well-defined Hexagon interpretation of any 4-couple motion.
You can also use this mapping to construct a picture of the equivalent Hexagon formation for any 4-couple formation. Start with a picture of the 4-couple formation, centered at the origin. Cut it in half, at any angle through the center. Take the resulting half picture, and squeeze it from a 180-degree angle about the origin to a 120-degree angle. Replicate this distorted image twice, once rotated at 120 degrees and once at -120 degrees.
Instead of extending to 12 people, you can in theory also contract to 4 people. This is not two-couple dancing, but honestly dancing any normal 8-person material with 4 people.
One way to do this is to draw a line on the floor through the center of the set. All four people stay on one side of this line. Whenever the call takes you over the line, you run across the square to the position of your diagonal opposite, and continue his part.
To eliminate the running around, draw the half plane that you are dancing on on a sheet of rubber, and stretch it so that the two sides of the boundary line join each other. Mathematically, if the line is the x-axis, just double the value of theta.
Dancing this way in practice would be possible, but quite hard until you got used to it. Having a copy of your floor distorted appropriately to dance on would make it much easier.
To understand resolving, imagine that 4-person dancing was normal, and 8-person dancing was a gimmick that someone came up with, like hexagon dancing is now. 4-person sequences always end with the heads together and sides together, of course. But when danced with 8 people, sometimes 1 man is with 3 woman, sometimes 2 man is with 4 woman, and sometimes neither of these happen, but the couples are out of sequence. If you want to explain to a 4-dancer caller (who has never seen 8 dancers in a square, doesn't quite believe it works, and doesn't want to think about it) how to write material that will resolve for 8 dancers, you just tell him the following rule:
For each of the 4 dancers, on their path from "bow to your partner" to "right and left grand", keep track of how many times (net) they circle the center of the set; if this number is odd for all dancers or even for all dancers, the sequence works for 8 dancers.
If the number is even for the head man and odd for the head woman, the heads won't get their partners. If it is even for the heads and odd for the sides, it works for everyone, but they will be out of sequence.
The same deal works for hexagon dancing, but instead of looking at the parity of the winding number, you look at the winding number mod 3. If these numbers match for the head man and the head woman, the heads get their partners, and similarly for the sides. If the heads have the same number, say 0, and the sides have the same number, then you will all get your partner, but you will be out of sequence if the sides number is 1 or 2; that's why there 3 sequences, as you observe.
So if you want to write sequences that resolve for both squares and hexes, you just need to make the winding numbers all differ by a multiple of 6. Or if you normally choreograph for 8 dancers, just make sure the winding numbers all differ by a multiple of 3; the multiples of 2 already work by virtue of the fact that the sequence works for 8 dancers.
(If you object that these winding numbers aren't integers, just add something on to make them all integers: they will all have the same fractional part, because the sequence resolves as a 4-dancer sequence, so the head man and head woman are together, meaning they have the same value of theta (mod 2pi). Alternatively, just add "promenade whatever the right fraction is" to the end of the sequence, and this will make all the winding numbers integers.)
All this works because no-one ever touches the center of the set, so square dancing is really done on the punctured plane. Dancing with 8, 12, or 4N dancers is lifting the motion of the dancers to a covering space. These are the only covering spaces of the punctured plane, so there's no way to extend this hack further. If only there was a second spot that no dancer ever touched, the fundamental group would be much bigger, so there would be many more covering spaces, and many more strange ways to dance.
It takes 12 Interlocked Diamond Circulates to get back to where you started! You can't "Check your Interlocked Diamond" because there is no 4-dancer diamond, just one knotty 12-dancer one. The same goes for Blocks (or any 4-dancer formation whose distortion includes the flagpole center).
Consider 3 head men in the center of the set in a Right-Hand Mini-Wave. Counter Rotate 3/2 will get them back to where they started. Stable Counter Rotate 3/2 will get them back to where they started, but they will be turned around even though they danced it stable! This is because the overriding "Hexagon" concept is more powerful than Stable, and one can think of the wall moving as you move around the flagpole center. Dancing Stable Counter Rotate 3/2 is much like driving a 3 point turn. Think about it.
Consider the heads in the center of the set in a normal Right-Hand Two-Faced Line. Couples Twosome Trade three times will get everyone back to where they started, but the couples will be half sashayed even though they danced it twosome! Again, the overriding "Hexagon" concept trumps Twosome.
These are not just special cases -- they actually makes perfect sense once you realize that the "Hexagon" concept redefines words and phrases like "north", "south", and "same wall". Every time you make 1 revolution about the flagpole center of the square, north and south are swapped, and so are east and west. So, after a Stable Counter Rotate 3/2, you are still facing the same direction, it's just that that direction has moved to the other side of the room! Similarly, for Twosome Trade three times, if you started east of your partner, even though you end on the same spots but half sashayed, you are still east of your partner. This is because east is now in the opposite direction.
Another concept with interesting Hexagon ramifications is Once Removed. Consider a 2x4. (Refer to the "Equivalent Formations" figure above.) Can you color with a different color the separate groups of people who work once removed from each other? (Try it!) If the dancers work in their spots, then the answer is no. As with both Interlocked Diamonds and Blocks, dancing Once Removed Circulates repeatedly would take you to all 12 positions. But if the dancers remove the distortion (as they often do), perform the call, and then replace the distortion, clearly there are 3 distinct groups of 4, and every dancer can only end on one of the 4 spots in their group.
This is an example of Andy's observation that if there are two different traffic patterns that take you on different sides of the flagpole center, they will end on different spots in Hexagon dancing. So, to successfully dance Once Removed in Hexagons, there must be a rule that says you never (or always) remove the distortion before doing the call.
Revised: $Date: 2003/07/29 18:58:52 $