Hot swing: How to keep your venue cool, the smart way

It's Friday night. You've had a long week and now, finally, you get to STFO. You rock up at your favourite weekly venue, swap a smiling hi with your buddy on the door, and push through into a sweaty blast furnace of a venue that melts that smile half way down your chin. And you only brought two spare shirts. Happy dancing, chump.  

It's been a while but last weekend inspired me to post here again. There's more to say yet about dance dynamics but this post will steer a little wide of the beaten track, into a different branch of physics: thermodynamics. We're taking this detour because recently, while sweating up a storm at a dance, I realised that I've been quietly complaining to myself about irrational ventilation and cooling set-ups in dance venues for a long time. And it occurred to me that the world might be a happier place if I stopped being lame and tried to actually do something about it.

So, here goes: A nerd's guide to reducing the spare-shirt requirements of your venue, with only the equipment you already have.

Dancers get hot. Just in case, you know, you haven't noticed. How hot? Well, while running, a human typically outputs 700-1400W of power (Smil, 2008). I figure that dancing at medium-to-high tempos is about as tiring as running, but slow tempos considerably less so. So, let's take 500W as a typical power output for a reasonably energetic dancer. That power is quickly converted to heat because dancers don't go anywhere except round and round and up and down a bit, before ending where they started.

How much is 500W of heat? Well, you've seen one of these things before, right?

A small, radiant heater usually outputs about 2000W on its highest setting. So, get 2 couples swinging out in a room, and you might as well have switched on your bathroom heater. Get 50 couples and you've got 25 heaters all running at the same time. Now, that might be just the ticket if you're dancing in in a Siberian winter with the windows open. But otherwise, you'll know it's usually not long before even the people sitting by the bar are sweating.

When it comes to keeping people comfortable, how much heat they produce is less interesting than where it goes. The first place it goes is into the air around them. For practical purposes, there are really only two ways to then get the heat out of the room. One option is to use a heat pump (air conditioner) to extract the heat to the air outside the room, while leaving the inside air where it is (switch on the A/C and keep the windows closed). The other option is to remove the hot air itself and replace it with cooler air from outside (assuming it's actually cooler outside) by opening the windows and using fans. There is a third option too: Temporarily remove the heater from the room (e.g. to a balcony) and let him/her radiate into the outside air for a while before going back inside. But that's not really cooling the room, it's just not heating it up in the first place. 

With the right set-up these different methods can all be effective at cooling the room but there's something else to add here, in the interests of sustainability. An air conditioning unit uses a lot more energy than a fan does. Like, about 10 times more. A hall that takes 3 A/C units to cool effectively usually won't need anywhere near 30 fans if the A/C is left off! So, the first option to consider for cooling a hot room is always smart ventilation, assuming there are enough windows/doors for adequate air flow, and assuming it's actually cooler outside than inside. 

But what do I mean by 'smart' ventilation? 

The picture below shows what is, in my experience, the usual set-up for trying to keep people cool at a dance if there's no A/C (and often, even if there is):

Unfortunately, many venues were designed without good ventilation in mind. Often, there are windows only on some walls and they don't always open as they should. Anyway, let's assume that our windows, on two walls in this example, open ok. There's also a door in another wall, and one corner of the room with no natural ventilation at all. There are three fans, all aimed at the dance floor, and two of them have hot, sweaty dancers standing in front of them. 

Is this the best way to keep people cool?

Remember, to cool this room through ventilation, one-way airflow is needed, to and from the outside world. That is, hot air leaving and staying out, and cool air entering and staying in, until it warms up and finds its way out again through another window. Fans can help with this but in the set-up here, they're doing something very different. 

Presumably, the idea is that it's refreshing for any given dancer who manages to stand/dance in front of a fan. The trouble is that all the heat blown off that dancer is simply distributed to the rest of the room, making it even hotter for everyone else. The fan and dancer together are functioning precisely as a fan-forced heater. That's, well, not cool.

The fact that all the fans are pointing in different directions makes matters worse because it prevents consistent air flow, in through one set of windows and out through another ('cross-ventilation'). Instead, flow is complex and turbulent, preventing quick heat extraction to the outside world. 

Now, let's consider what might be a better way to use the same equipment in the same venue:

Here, the fans are not being used with individuals in mind but rather, the whole room. Two of them are now aimed out the windows. This isn't as silly as it seems. The point is to blow hot air out of the room on one side so that cool air is sucked in on the other side, rather than just recycling hot air. Note that this is usually more effective than trying to suck cool air in, by standing a fan next to a window, facing inwards.

The third fan is still aimed at the dance floor, but this is done in such a way as to promote cross-ventilation from the 'intake' windows, across the hot dance floor, to the 'exhaust' windows (indicated by the long, grey arrow). Individual dancers who want to be cooled down by the fans can stand behind them  (shown), rather than in front, or they could possibly squeeze between the fans and the windows they're aimed at. Either way, the point is to immediately extract their heat from the room, rather than blowing it all over everyone else.

Finally, let's take a look at what an 'ideally' ventilated venue might look like (Well, indoor venue, anyway; ultimately, there's no substitute for having no walls at all!). 

Firstly, this venue has more windows and, importantly, some of them are in opposing walls, allowing for direct cross-ventilation. Two of the fans are directed out of the windows in one wall, close to the dance floor, sucking cool air in through the other windows, over the dancers and out the exhaust windows. The third fan is placed to assist this process. Conveniently, the fan-equipped windows open onto a balcony, where sweaty dancers can stand in the airflow as it's ejected from the room. Voila!

Of course, this description and the others above are over-simplifications. Air flow in these rooms would in reality be complicated by many factors, depending on the venue design, the number of people, the heights and sizes of the windows, etc. 

Nonetheless, I hope the basic ideas sketched here: choosing smart ventilation over A/C where possible, promoting cross-ventilation and extracting hot air rather than recycling it, might be useful to venue operators and dancers everywhere!

Smil, V. Energy in Nature and Society: General energetics of complex systems. MIT Press, 2008. 


From complex to simple 2: 3 thousand moves or 3 degrees of freedom?

Early in 2006, after about five years of regular dancing, classes, workshops, etc, I was feeling frustrated. I had a reasonable repertoire of moves, slides, dips and aerials, which were a lot of fun on the social dance floor and had served me well in performances and contests. But I knew that I was still lacking something major. When I watched the A-list internationals out on the floor, I could see that they had something that was still out of my reach: Whereas my social dancing was just a move salad, they had an element of genuine originality in every dance, creating things in real-time, right before my eyes. I had moves that I could lead on a partner, they had movement that they could share with a partner. Though we weren't sure how to bridge the gap, my dance partner at the time and I decided it was time to focus on getting beyond moves. But how? Like the geeks we were (and resolutely still are), we decided to fight moves with science.

One of the most useful tools in the Swiss army knife of theoretical physics is something called vector decomposition. Forget this jargony name though; the process itself is simple, and I've found it to be a very useful way to think about dancing, and about teaching dancing. The basic idea is this: Any complicated-looking movement can be broken down into a small number of simple movements, which can be learned, led and followed simply, then just added together to re-create the complicated-looking movement. This might not sound very surprising in the context of dance teaching, since all teachers worth their salt will 'break down' moves for their students. But this isn't the kind of breaking down I'm talking about. Whereas it is usual for moves to be broken down in sequence, I'm talking about breaking things down in parallel. That is, rather than telling students to do such-and-such on beat 1, whatever on beat 2, etc, I'm talking about components of a movement, which are happening at the same time. I'm not so interested in what happens on particular beats. I'm more interested in how people are moving through the beats and it's this flowing, ongoing movement which can be broken down simply. Let me explain.

Imagine for a moment, a three-dimensional space which is empty except for a solid ball (or whatever shape you like - check out the figure below, which shows a box rather than a ball but demonstrates the same principles), which is free to bounce around the space any which way. It can move in straight lines, curves, and it can spin. It turns out that there are six fundamental ways in which the ball can move, and all of its possible movements, without exception, can be broken down (decomposed) into those six ways. The six ways are called degrees of freedom (DOF) and for our ball they are: straight line movement up and down, side-to-side and back and forth, and rotation around a vertical axis and around each of the two perpendicular horizontal axes (ie. the up-and-down axis, the side-to-side axis and the back-and-forth axis). So, concisely, we can say that the dynamics of the ball are spanned by six DOF. That is, the movement within those degrees of freedom can together 'reach out and fill' all of the possible movements of the ball. Now imagine that the ball is restricted such that it can no longer move up and down. That is, imagine that it is stuck in a two-dimensional plane. This robs the ball of one of its degrees of freedom, so its movements are now spanned by only 5 DOF, 2 translational ('translation' is the technical term for straight-line movement) and 3 rotational. Now, let's add another restriction and say that the ball can rotate around only the vertical axis. This removes two more DOF, so we are left with just three in total: One rotational DOF and two translational DOF. This system - a mass restricted to translations in a 2D plane and rotations only around the vertical axis is a good analogy for a dancer on a social dance floor.

Yeah, ok, strictly speaking, a dancer has 6 DOF like the unrestricted ball does but not much of consequence happens in 3 of those 6 D's of F, so we can do away with them to simplify how we think about things. Dancers don't move up and down much. Sure, they bounce, jump and dip, but those movements are generally small compared with the horizontal movements in the dance. And, moreover, the up-and-down movements always average to zero quickly because what goes up must come down and vice versa, if the dancers start and finish each move standing on their feet. As for the two rotational DOF we're ignoring, they are only important if the dancers are doing aerials or other tricks which involve rolling movements like backsaults, frontsaults or cartwheels. Those don't happen much. What does happen a lot are translations across the floor, back and forth and side-to-side, as well as rotations around the dancer's vertical axis ('spins').Excepting a few details, pretty much every move done on the social dance floor can be broken down into sideways & back/forth translations, and spins.

Alright, so what? Well, let's see how this kind of thinking simplifies things for both dancers and teachers. Consider a basic Lindy swingout. The conventional way of thinking about this 'move' is as a pre-defined sequence of steps and movements on particular beats within eight counts (a 'dancer's bar') of the music; a recreation of something done historically. This is how I used to think about a swingout: As the leader, I 'rock-step' on 1&2, and while doing this, lead my partner towards me. She steps or swivels in. I get out of her way and we make an effort to keep facing each other ('spotlight' each other) as we pass each other by. I 'collect' her momentum as I catch her and we reach beat 4 facing each other with a nice stretch in our 'frames', ready to be released to create the second half of the move. On '4-and' I lead her back in the direction she came from and either 'lead her out backwards' or 'forwards', depending on my stylisic preference. By the time we get to 8, we have a nice stretch built up again and are ready to repeat the move if we want to.

For years, I travelled long distances and emptied my poor student's pockets repeatedly to hear lots of different teachers tell me slight variations of the above. Each couple had their own version of the basic, with its mysterious subtleties. Such-and-such started 'leading in' slightly earlier or later than someone else. One awesome leader  had his right foot forward on 4 while this other awesome guy had his left foot forward. This couple put lots of rotation in their basic while this other couple kept it very linear. I constantly had the feeling that if only I could somehow absorb this sublety or that one, I would be able to reach some previously unobtainable level of awesomeness in my dancing. Eventually, I came to realise that this kind of thinking wasn't making me a better dancer, and I was just getting ever more confused about what I should teach. There had to be a simpler way to think about things!

How about this:

In a 'Lindy basic', both partners move back and forth once, roughly equal distances, in a (more or less) straight line, passing by each other, and also rotating through 360 degrees, and this is done with continous connection/communication. That's it. End of story. The movement is within only 2 DOF, one translational, one rotational (excepting a small deviation into the second translational DOF, to get out of each other's way, as you pass by).  For reasons described at length in earlier posts, what, exactly the partners do with their feet along the way is not particularly important, provided it facilitates movement that is not jerky, allowing for mutual trackability. No one cares if you do a rock-step or a kick-ball-change or a hold step, or whatever on this or that beat; that's a personal stylistic choice but it means almost nothing to the partnership. What does matter for the partnership is that the leading and following is done simply, elastically, as described earlier. That's it. Controlled movement and elastic connection. All the rest is details.

At anything but the beginner level, there is just no need for conceptual fluff like 'rock-stepping', 'spotlighting', 'leading in on 2', 'over-rotation', 'leading out backwards/forwards', 'dancing more American style than Swede style', or whatever. Instead, each partner simply moves and connects in accordance with the simple rules we've seen in earlier posts, within 2 degrees of freedom (for a swingout). That is the dance. The dancers are able to choose to move and connect through a shape which happens to be known as a 'Lindy swingout' for historical reasons, and they can add whatever special 'variations' they like but there is nothing special about this 'move'; it's just a shared movement like any other in the ongoing flow of a genuinely connected, co-creative dance. On one of the occassions when I had the pleasure to meet Frankie Manning and hear him speak and answer questions from young dancers, he was asked "How many different styles were danced at the Savoy?" He answered, in that inimitable Frankie way, "Ahhh, we didn't really worry about styles! We just danced!" And, how did they do that? In the most natural ways their bodies knew how to, the ways that allowed for the greatest cooperative expression with the fewest rules, the ways just felt good. I'm yet to hear a world-class dancer get worked up about a 'rock step'.

But enough with the rant, already. I was trying to explain how thinking in terms of degrees of freedom simplifies complicated movement! Having looked at the example of a 'swingout', perhaps the best way I can summarise the merit of thinking in terms of DOF is that, in an important way, it makes every 'move' the same. When a student asks me, "How are you leading that?", my immediate impulse (which I squash because I don't want to seem rude) is to say, "The same way that I (try to) lead everything else." I believe that it really is that simple. Once one learns how to move with smooth control and connect elastically, the answer to every how-do-you-do-that question is just a slight variation on, "Move with smooth control and connect elastically." Then, the dance just happens and endless avenues for expression open up. What's the slight variation? It's simply the number of degrees of freedom you're moving and connecting in! How do you lead a sugar-push? Move smoothly and connect elastically within one degree of freedom - back-and-forth. How do you lead a swingout? Move smoothly and connect elastically within two degrees of freedom - back-and-forth and a vertical-axis rotation. How do you lead a Lindy circle? Move smoothly and connect elastically through three degrees of freedom - back-and-forth, side-to-side and a vertical-axis rotation. How do you create a brand new move, musically, in real time? Move smoothly and connect elastically through as many degrees of freedom as you like, making shapes and a movement trajectory that reflects what you hear in the music. You get the point. This way of thinking can be extended to every single 'move' that is ever done on the dance floor, including aerials, dips, etc, if one adds in the necessary extra degrees of freedom.

The next post will deal with rotational connection. We have discussed straight-line ('translational') connection at length in earlier posts and now this post has introduced the notion that rotational and translational components play equal, complementary parts in any complex dance movement. So, to complete the picture we must grasp the fundamentals of rotational connection. At that point, we will finally have a more-or-less complete picture of on-balance dancing. Hooray! That's not yet the whole story though; a discussion of counterbalance is yet to come!


From complex to simple 1: Exploring elasticity

In the last post, it was proposed that when two advanced dancers dance together, they control their muscles in such a way that their centres of mass (COMs) behave as though they were simply connected by a spring. We have called this system elastic connection. This post will start at the other end and meet the previous post in the middle; here we shall begin by discussing elasticity in a general sense and then describe dancing as a specific example.

The whole point: Learning how to think simply about a complex thing
Something to keep in mind as we proceed: the whole point of attempting a scientific analysis of good dancing is to make it simpler to think about and easier to learn. When we first come to dancing, either as observers or participants, it's an impressive, complex-looking thing. We sit and stare, awestruck, at world-class dancers and think, "What is that magic that they've got, that exquisite quality of movement? How can I possibly take that magic and make it my own?" Trying to learn to do something that's complex is difficult. We fumble around enthusiastically, learning lots of 'moves' and 'variations' and feeling spiritual about a few vague concepts of connection. We imitate the eye-catching shapes and stylings of our favourite dancers, and try to hold a thousand little snippets of information together in our minds as we dance, hoping that it will also somehow click together and voila, we too will make the magic happen. My experience with trying to learn dancing this way has often been frustrating. The whole process can be made easier and less dramatic, I believe, if we are able to dig deep enough down into the fundamentals of what's going on in good dancing, to show that it's actually not that complicated after all, but rather is just a small collection of simple things added together. Those simple things can then be learned one at a time and put together piece by piece, to build solid dancing from the ground up.

Luckily, this approach is applied all the time in physics, with great success. The process for doing so was described earlier in this blog, in the post on simple models of the human body and the magic of its centre. There is an important point to add here: physics typically proceeds when it is recognised that a newly-studied system in nature, which isn't yet understood, looks like some other system which has been studied in the past and is understood. One can then take the model which was developed for the old system and see if it works for the new system too. Sometimes it works brilliantly, right off the mark; sometimes a bit of tweaking is required. In many cases, the new thing quickly becomes just as well understood as the old thing; suddenly, it becomes possible to think about a complex, mysterious looking thing in simple ways.

I have spent some time trying to figure out what well-understood model might be applied to a dance partnership in order to allow us to think about dancing more simply. In the last post we introduced a key feature of that model: elastic connection. In this post, we will explore more deeply, the consequences of having dance partners connected elastically. We will begin by talking about some abstract things but see if you can spot the connections to dancing along the way. I promise that if you stick with me through the abstract ideas and get a feel for them, new vistas of simplicity will open up in how we are able to talk about dancing down the line. Here we go.

Something well understood: A mass on a spring

Let's consider the above picture, which shows a mass M connected to a solid wall by a spring. We assume that the mass is sitting on a frictionless surface. In the top frame, the mass is sitting such that the spring is at its rest length. ie. its natural length when it's not being stretched or compressed - all springs have one of these. At this length, there is no elastic potential energy stored in the spring. If we also assume that the mass is sitting at rest, there is no kinetic energy in the system either. We will call this the zero energy system. In the lower frame, the mass has been moved by a distance x, such that the spring is now stretched. Now there is some elastic potential energy stored in the spring. If we assume that the mass is again at rest in this new position, there is still no kinetic energy in the system. We will call this a pure potential system.What will happen if the mass is released from this position?

The answer, of course, is: The mass will be pulled towards by the wall by the spring as it shortens back to its rest length. As this happens, the potential energy stored in the spring is gradually converted into the mass' kinetic energy as it speeds up. By the time the mass has returned to the position shown in the top frame (the spring is again 'relaxed' at its rest length), the system has become a pure kinetic system, with all of the spring's original potential energy now having been converted into the mass' kinetic energy. The mass does not stop here. It keeps moving and the spring is gradually compressed until all of the mass' kinetic energy has been returned to potential energy in the spring (at which point, the mass momentarily stops again), which now stores it as a compression, rather than a stretch. And so on, the mass will oscillate back and forth as energy is traded between potential and kinetic states. If the system is genuinely frictionless and no energy is either given to or lost from the system by any other mechanism, the mass will bounce back and forth forever. The technical term for this kind of system, which shows up everywhere in physics, is a simple harmonic oscillator (SHO). Readers who would really like to get a feel for how SHOs behave are encouraged to check out the wikipedia page on SHOs, which is excellent and includes some great animations, which help to make things more intuitive.

There is an important property of this system, which must be mentioned because it is essential to understanding the elastic model of dance connection. Let's introduce it with a question: How quickly does the mass oscillate? Are the vibrations quick like a plucked guitar string or slow like a kid on a swing? At this stage, we can't say either way; the model as we've introduced it is entirely general so we need to specify a couple of things in order to answer this question. There are two properties of the mass-spring system, which determine how quickly it oscillates: The size of the mass M and the elastic constant of the spring k (ie. whether it's a stiff spring like in car suspension or a soft spring like a slinky). If those two quantities are known, we can calcuate exactly, the resonant frequency of the system. Every mechanical system has a resonant frequency. It's the system's natural frequency of vibration - how fast it will vibrate  (the number of times that its mass moves back and forth per second) if it's struck or plucked. Two familiar examples are a crystal wine glass and a guitar string, both of which will vibrate fast enough to make a musical note.

A quick summary, which is important to remember: When a mass on a spring is given some energy (ie. the mass is moved by some external force - like a hand pulling it - and the spring is stretched/compressed accordingly, then the external force is removed and the system is allowed to freely move), it will naturally oscillate at a particular frequency, which is known as its resonant frequency. Or, stated in reverse, if a mass of certain size, attached to a spring, is required to resonate at a particular frequency, the property of the system that must be carefully chosen is the spring constant (or elastic constant) k.

Losing energy: Damping
In real mechanical systems, there is always some loss of energy through friction, so there is no such thing as a mechanical SHO that oscillates forever. The vibrations of a SHO under friction will gradually get smaller and smaller (though they will keep the same frequency) until the mass stops moving altogether (the SHO has returned to a zero energy system). But unwanted friction isn't the only thing that can steal energy from a SHO; the general name for a force that does this is a damping force. Sometimes mechanical systems are deliberately designed to include significant damping forces because ongoing vibrations are not desired. The classic example is a suspension system in a car, which is composed of a spring and a shock absorber. Here, 'shock absorber' is just a fancy way of saying 'damper'. When a car hits a bump, the suspension springs compress and energy is stored in them. Without shock absorbers, the car will bounce up and down at its resonant frequency until its vibrations are gradually damped by friction with the environment (air resistance, etc). This would make for an awful ride. So, shock absorbers are added and the oscillations of the car are very quickly damped, making the ride more comfortable.

Adding energy: Driving
No, not that kind of driving. We're not talking about cars now. A driving force is the technical name given to any force which gives energy to the oscillations of a SHO. The classic example for this is a parent pushing his/her child on a swing. Without repeated pushing, the swing's oscillations will be damped by air resistance and the ride will soon be over. So, a periodic driving force is required in order to keep the swing going. If the swing is to be kept going at the same height, the driving force and damping force must cancel each other exactly, and the swing will move simply as if there were no driving or damping at all.

Muscles: drivers, dampers
Muscles work in groups to achieve complex tasks. Sometimes they contract, sometimes they stretch. Sometimes they give energy to limbs, sometimes they take energy away from limbs. How a given muscle behaves is dictated by the role it is required to play in making an overall body movement happen. Consider the act of throwing a ball in the usual way (like pitching a baseball). Early in the movement, the muscles of the chest and shoulder contract quickly, giving the arm great speed and energy (ie. acting as drivers of the arm's movement). But not all of that energy can be given to the ball, so after its release, there will be an arm left behind, moving at dangerous speeds. In order to prevent the injury that might result from this speeding arm crashing into the body after the ball is released, the shoulder and chest muscles switch into 'active stretch' (as we called it in the last post) and damp the arm's motion, slowing it down to a safe stop.

Stripping the dancers
Ok, great. We've talked some physics and some physiology, and now we can get down to talking about dancing. Let's begin here, by describing the step-by-step process of reducing the complex interactions of a dance partnership to a simple, elastic connection model. Consider the following figure:

In frame A, we see a connected dance partnership in all their silhouetted glory. If we could see them moving, it might look quite complex and at this stage, we think of them with all the complexity with which we might usually imagine two connected human bodies, each with the potential to move in so many amazing ways.

A few essential mechanical features have been added in frame B. We see both partners' centres of mass (COMs) as blue blobs around the heights of their belly buttons. The basic structure of the skeleton is shown by the white 'wireframe' and overlaid on it are grey springs and some red and green lines, which are all together intended represent the major muscle groups. Green represents the potential to apply a driving force, which adds energy, and red the potential to apply a damping force, which subtracts energy. The presence of the springs represents the potential of each muscle to execute elastic, active stretch. In this frame, all the muscles are both red and green because all muscles, strictly speaking, have the potential for both driving and damping. So, we haven't made any simplifying assumptions yet but rather just introduced the mechanical characters of our story.

In frame C, things are starting to get a bit simpler. Frame D is the same as C, with the silhouettes removed for ease of viewing. We have now removed all the green lines from the 'muscle chain' - that is, the sequence of muscles which connects the two partners' COMs. So, we have now removed the ability of these muscles to add energy through contractions. Rather, they can now only do two different things - elastic, active stretch, which transmits energy between partners, and damping, which removes energy from the partnership, as when a stop is led. In this frame, the leader's leg muscles can both add and subtract energy from the partnership; they are the ultimate drivers of the dynamics. At this point, the follower's legs retain the ability to add energy but this is intended only in the limited sense that she must continually add some energy to replace the energy naturally lost to friction as she moves. She's not trying to add energy to the partnership but rather only making sure that no energy is lost overall. It's a bit like driving a car down the highway - if the road were frictionless, the motor could be switched off and the car would cruise forever. But this is not the case and the motor must continually compensate for energy lost to friction. Indeed, I have heard several teachers encourage followers to 'keep motoring' once led; that is, conserve energy and don't slow down until a slow down is led.

In frame E, the legs remain the same but the detailed structure of the upper body is done away with and replaced by two springs, connecting their COMs in the middle (where the hands would be). As stated in the last post, it can be shown mathematically that a series of connected springs behaves like one big spring, so we can think of all the muscles of each dancer's upper body behaving simply as a single spring, connected to a single spring provided by the other partner.The capacity for damping remains in the connection here, though it should be emphasised that damping does not act constantly, but rather only when one or both partners wants to subtract some energy from their connection.

We extend this reasoning in frame F, representing the two partners' springs as a single spring (plus damper) connecting their COMs. The legs are again unchanged.

In frame G, we arrive at our final simplified model. Here, the legs have been replaced by wheels for further simplicity. The leader has one green and one red wheel, indicating that he can both add energy and subtract energy from the partnership's motion. The follower simply has two black/uncoloured wheels, indicating that her role is to simply conserve energy, neither adding nor subtracting it. The two partners are identified with their COMs and are connected to one another by a spring and a (voluntarily operable) damper. If we were to describe this model in technical language, we might do so like this: A pair of point masses connected by a damped spring with time-dependent spring constant and damping function. That is, both masses are acted upon by an elastic force and also by a damping force. In addition, one of the masses (the leader) is also subject to a time-dependent driving force and an additional time-dependent damping force.

Compared to the original dance partnership, in all its complex glory, this model might seem a bit yawn-inspiring. However, despite being a lot simpler than what we started with, describing it mathematically (which is what we must do in order to explore it completely and decide conclusively whether it's a good representation of a dance partnership) will take some doing. And I must confess that I haven't yet tried. When I do, I'll post some results. Moreover, this model only describes linear motion and says nothing about rotations. We will have to build up a complementary model for rotations down the line, and we will. But let's get started here first, without getting mathematical. We don't really need to go that far in order to use the model intuitively to talk through some familiar moves, which is what we will now do.

One move to describe them all: the sugar-push, in all its glory
Let's start with (arguably) the simplest move out there, the sugar-push. I love this move because it's so simple it's difficult. You can't fake it. If you're not on top of your movement and connection, there just ain't no sugar in your sugar-push, no matter how nice your lines or fancy your footwork. We will talk through the move in terms of our simple model - rather than talking about bodies, arms and steps, we will focus on masses, springs and forces. If at any point you find yourself unsure of how this translates back to bodies, take a look back at the above figure and discussion. There is something to keep in mind as we go along - the process of leading a sugar-push from start to finish contains absolutely everything that any linear move ever contains. This is it - understand this well and understand everything, do this well, and you can do anything. So, really, this is the only move we need to describe in detail. Everything else is just a variation on this.

Before we launch into it, it might be worth checking out the cool little interactive animation found at this site. It lets you watch the dynamics of two masses connected by a spring. I must stress that this model is not strictly the same as the one we're proposing for a dance partnership (because both springs here are also connected to two walls by two other springs and our dancers aren't doing that) but under certain conditions, it's close. The animation takes a while to get going but if you wait for about 5 oscillations, it starts to look remarkably like a dance couple doing some nice sugar-pushes. Well, their centres of mass, anyway. Think of the red mass on the right as the leader, and the blue one as the follower. You can play with the spring constants (the K values) if you like. Notice how things change if you do. Once you've got a visual feel for things, the below discussion might seem more intuitive. 

We assume that prior to the beginning of the move, both masses are stationary so there is no kinetic energy. The spring between them is at its rest length so it contains no potential energy. Overall, the partnership is a 'zero energy system'. To get things started, the leader adds some energy by applying a driving force with his legs, forcing his COM backwards, away from his follower. When this happens, the spring connecting the two COMs stretches immediately and immediately applies a force on the follower's COM. This means that her COM begins to move immediately. Let's address this right away. It is common for teachers to tell followers that they have to wait when they're being led, to force a delay between the leader's movement and their movement. Strictly, this is not true. Watch a couple of world-class dancers and you will see that it's not true. Good followers don't force anything; they follow. When the leader moves his centre and transmits energy through the connection to the follower's centre, it moves immediately (ok, well, strictly it takes a tiny fraction of a second for the energy to reach her since it travels through the connection, roughly at the speed of sound in water: about 1.4km/s). What is true for the follower is that she must not take the leader's giving of energy as a signal for her to add energy to her own movement. The energy of her centre will build gradually as it is transmitted to her from the leader. She will begin to move immediately but only because she is being moved, not moving herself. Remember, in our model, neither her legs nor her muscle chain can add energy. Ok, back to the sugar-push....

So, the leader's legs have moved his centre, causing the spring between the partners to stretch. The force which this stretch applies on the follower's centre is only small to begin with because the stretch is only small (remember F = - k.x from the last post). So, her centre begins to accelerate immediately (because a force causes acceleration by Newton's second law, F = m.a, where m is the mass) but only a little bit while the stretch is small. Bus as the leader's driving force moves his centre further and further away from hers, increasing the stretch length of the spring, the force on her centre increases and she accelerates ever more quickly. We see from this that the delay between partners is a natural consequence of elastic connection and has nothing to do with a deliberate action on behalf of the follower; she speeds up somewhat after he does because it takes time for the elastic force to build as a result of the stretch in the spring, which is caused by his movement. So, he moves, stretching the spring, which gradually moves her. Voila, a delay.

When the leader sees and feels that the follower is moving nearly as fast as he wants her to move, he shuts off the driving force from his legs, meaning that he stops accelerating is own centre. This means that he is no longer adding stretch to the spring, or even maintaining the stretch that's already there, so the spring then naturally returns to its rest length. ie. The follower catches up to the leader - the distance between them returns to what it was before the move started. The difference now is that the partnership is moving together; they have kinetic energy. If he were not in the midst of leading a sugar-push, the leader could simply choose to cruise across the floor with his partner. The spring would remain at its rest length, nice and relaxed, until he chose to reactivate the driving force to add energy in a new way. BUT, he is leading a sugar-push, so something different happens. He instead applies a new driving force equal and opposite to the one he applied originally, forcing his COM to move forward, towards his partner, which begins a compression in the spring (which is still mediated by active stretch in muscles, don't forget - just different muscles to those which were stretching during the spring stretch at the beginning of the move). Again, the follower's centre begins to slow immediately but the slow-down builds gradually as the spring compression builds. Eventually, the leader will have slowed himself to a stop and the follower will come to a stop shortly thereafter. However, the stop is only instantaneous and both partners move straight through it because the leader's driving force continues unchanged. It is only when both partners have achieved the desired speed in the opposite direction to before that the leader shuts off his driving force and begins a new driving force backwards (for him), to initiate a stretch and begin the whole process over again. In reality, the transitions between the forward and backward driving forces are not sharp and clunky like this but rather, gradual, ramping up and down like a wave.

In fact, everything about this whole process is like a wave, with different parts waving at different times, in a sequence. The diving force wave gets started first, the wave of the leader's speed follows shortly afterwards, which drives the wave of the elastic force applied to the follower. Coming slightly after that is the wave of the follower's acceleration and after that again, is the wave of her speed. So, we see the natural delay inherent in the process when everything is mediated by an elastic connection. To get a better feel for this sequence of waves, take a look back at this wiki page and take a close look at the third animation from the top, which has the arrows that grow and shrink over time. These arrows are like the waves for the quantities we've just discussed, but just for a single mass.

Right, there we have it! The sugar-push, cut to its bare bones! No magic, just some simple physics, which lets two bodies work together effectively. I promised it would be simple, didn't it? What's that?,"Then why is the above discussion so long and why does it require so much mind-bending visualisation?!" :-) Well, firstly, I guess I should say that simple doesn't necessarily mean easy! But that's not the whole story. Now that we've thought through everything in terms of forces, accelerations, speeds, etc, let's clear our heads for a second and then think of it in a new way, which will, hopefully, will be even simpler and tie this whole post together.

Tempo, frequency and spring constant: Resonance between music and dancers
Recall this, from the top of this post:

'if a mass of certain size, attached to a spring, is required to resonate at a particular frequency, the property of the system that must be carefully chosen is the spring constant (or elastic constant) k.'

Hold that thought as we discuss the following question: What is the most fundamental property of the music we dance to? This might seem like a controversial question to begin with but I don't think it is. I would say the answer (provided we restrict ourselves to a single genre of music, hot jazz, say) is tempo. Tempo is even more fundamental than rhythm. After genre, tempo is the only property of music which divides up events: contests have slow and fast divisions, exchanges have slow rooms and fast rooms. Tempo sets the energetic baseline of the dance.

Now, consider this: For all but a few exceptional dances, all dance movement is periodic - it is made up of waves. Everything that goes in one direction also must come back the other way in equal measure. You step out onto the floor with a partner and you move in all kinds of complicated ways but you (almost) always end up where you started, at (roughly) the same spot on the floor. You might move back a bit, but you always come forward again. You might move left but you return right. You bounce up and you drop down again. You end where you started. This means that, just like rhythms in the music you dance to, your movement is periodic; it goes in cycles and cycles within those cycles, and so on. Cycles always have a frequency - how fast they cycle/vibrate. When we dance, there are fast cycles (like bouncing with the beat), medium cycles (like the speed of repeated sugar-pushes or swingouts) and slow cycles (like drifting across the floor and back again). Most of those cycles - at least the fast and medium ones - are founded on the tempo. Let's tie this back in to our elastic model of connection.

Tying it all together
If a couple dancers repeat sugar-pushes or swingouts, say, at a particular tempo, their bodies are required to move back and forth - a kind of vibration - in sync with the music. In physics, when vibrations are in sync with each other, we say that they are on resonance. Now, remember that sentence from before again:
'if a mass of certain size, attached to a spring, is required to resonate at a particular frequency, the property of the system that must be carefully chosen is the spring constant (or elastic constant) k.'

Aha! If we assume that dancing is all about resonance with the music, and dance partnerships are held together by elastic connection, then it becomes clear that the tuning of the partnership's spring constant is of utmost importance. With this in mind, here's a genuinely simple recipe for good sugar-pushes, and indeed, any periodic, linear connected movement:

1. Listen to the tempo and use it to decide how quickly the dance partnership will have to resonate
2. Set your spring constant accordingly. Fast speeds need high constants (stiff springs), slow speeds, low constants (loose springs).
3. Apply the leader's leg driving force in time with the resonant frequency of the partnership, as determined by the spring constant. That is, like a father pushing his kid on the swing, the leader needs to add energy to the partnership's movement in time with its natural resonance to the music. All the rest of the nice, flowing dynamics is taken care of by the beautiful simplicity of the elastic connection; no fancy movements, no pushes or pulls, need to be added by either of the dance partners.

Of course, just repeating the same movement over and over again gets boring but there are lots of different movements that might be chosen and dancing is really just a matter of switching between them musically. Sometimes, not even one oscillation is completed before switching to a new movement but even a fraction of an elastic oscillation still follows the same rules:

1. Pick the speed of (part)oscillation required; how quickly does the leader want to follower to get to where he's going to send her?
2. Once he knows, he sets his spring constant and applies the driving force with his legs. A good follower will quickly detect her leader's spring constant and set her own to match it - this is the 'frame matching' that teachers sometimes talk about.
3. Simply hold an elastic relationship, obeying the rules of movement we've discussed earlier.
4. Change movement in interesting ways and repeat.
5. When a reduction in energy or a complete stop is desired, one or both partners apply damping, deliberately wasting their kinetic and potential energy so that they have less speed and/or less stretch, or stop completely, returning to a zero energy system.

Phew! Ok, that's it for this post. Now that we have thoroughly explored both movement and connection, at least for linear movements, in the next post we will discuss how even the most complex dance movements can be broken down in terms of just three directions of movement. If a pair of dance partners can move and connect in just those three directions, they can do literally anything, even the most complex 'moves'.


Connection 2: Elastic muscle magic

Alright, it's business time. If you've ever wanted to understand dance connection - I mean really get to the bottom of it - this might just be the post for you. Recall the following definition from the last post: Connection is achieved when the motion of one partner's centre of mass is influenced by - and influences - the motion of the other partner's centre of mass, in a predictable way. A stronger way to state this is that when two dance partners achieve continuous connection, any observer (including the dancers themselves) who has knowledge of the motion of just one partner's centre of mass (COM) can in principle predict the motion of the other partner's centre of mass. Given all the possible complexities in the motion of two human bodies, this is quite a feat. It is natural ask, how is this possible?! In this post, we will introduce a system of muscle control, which can be followed by each dancer in a partnership in order to achieve this predictable mutual influence simply. In a word, this system is elasticity. But that familiar word means many things to many different people. We will make it our business here to come to a clear understanding of its fundamental meaning and how that applies practically, in dancing.

Haptic communication

First though, we will return to the idea introduced earlier, that the physics obeyed by each partner's body is a slave to the need for cooperation between partners. We don't use a system of elastic connection just because; we use it because it allows each partner to feel how the other's COM is moving, and to influence that in a controlled way which is predictable to both partners. It's pretty magic, really. Ok, consider this: Experienced dancers can dance well with their eyes shut; both follower and leader (ignoring the obvious problems associated with crowded dance floors). I love leading with my eyes shut but it only works with experienced (though not necessarily familiar) followers with whom I am able to move and connect in such a way that my brain receives enough clear information just from the mechanical, tactile connection between us that I am able to predict my follower's motion and interactively control it. The inverse is also true - a follower can only rely on the signals from touch alone if her leader is able to move and connect in such a way that she gets enough information to feel and predict how he is moving (Note: As discussed earlier, there is nothing wrong with a follower predicting how her leaders is moving - if fact, it is useful to do so - so long as she's still truly following and not 'correcting' her motion based on her assumptions about what he might be trying to lead. There are no choices about what to do while following purely; there are only choices about how to be - how to hold oneself and control the elasticity of one's muscles - as we shall see). The technical term for this kind of touch-based communication and control is haptic. Connection allows dancers to achieve haptic communication and shared control.

Listening within yourself to hear your partner

Now, for an interesting question: How can it be that one dancer can predict the motion of the other dancer's COM from haptic signals alone when (s)he is receiving no sensory cues directly from that place? There are no nerves which connect a follower's COM to her leader's brain. The only part of the follower's body with which the leader's nerves are in direct contact is the small area of skin or clothing that (s)he happens to be touching at any given moment - hand-to-hand or hand-on-back contact, say. And the information available from this point of contact (texture, temperature, pressure) is limited in its usefulness for telling much about the follower's motion. So, how does the leader ('he'/'his' from now on, for simplicity) do it? He might not know the details of what his follower's ('she'/'her', from now on) body is doing, but he is privy to a wealth of sensory information coming from within his own body. So, what the leader is doing is monitoring what's going on inside his own body and on the basis of that information is able to reliably infer what his partner is doing. Or, put another way, a leader is able to tell what his follower's centre is doing just from all the information that his brain receives about his own muscles and limbs. And vice versa for a follower. It's amazing!

How is this possible? We will eventually see that it's because both leader and follower have learned to hold a particular kind of relationship - and elastic relationship - between the shapes in their 'frames' and how hard they are pushing or pulling on their partner. We will have to build up the story gradually before this is obvious, however, so let's get started. Every skeletal muscle in your body has little sensory organs attached to it, which continuously sense essential information about it, and tell your brain what it's doing. One organ, called the 'muscle spindle', tells your brain how long/stretched your muscle is and how quickly that stretch is growing or shrinking. If someone grabs your hand unexpectedly and pulls hard on it, the muscle spindles in your arm muscles tell your brain that those muscles are getting longer, quickly. At the same time the 'golgi tendon organ', which is integrated into the tendon that connects each muscle to a bone, measures the tension force in the tendon (and therefore, the muscle). So, in the example, if your arm is being pulled on, if you have a limp arm and don't create any tension by resisting, the golgi organ will tell your brain there's no tension in the tendon/muscle, even if the muscle spindle is saying that the muscle is stretching and stretching quickly. It turns out that together, all of your body's muscle spindles and tendon organs provide your brain with all the information it needs to put together a complete picture of how all the parts of your body are moving, and make predictions about how to control that by activating various muscles (Ok, so in reality there is more information required from other organs as well for a truly complete picture, but for the purposes of our discussion, we can make do with the above). So, if you did decide to firm up your arm to resist the person pulling on it, your brain would quickly figure out that the pull might put you off balance and it might need to make one of your legs take a step in order to prevent you from falling over. This, of course, is exactly what happens in a follower's brain when she is led by a pull on the arm.

Ok, so we now have an idea of how a person's brain is able to monitor the dynamics within his/her own body. But how does this allow him/her to infer reliably, where a connected dance partner's centre is? Answering this question is a bit more involved and we will have to build up to the answer, one step at a time.

Chain, not frame

To begin, I'd like to introduce the concept of what I call the 'muscle chain'. By this, I mean the sequence of muscles which spans the distance from one dancer's centre/COM to his/her partner's centre. It now becomes convenient to identify the centre with the hips (as is often done by teachers, and it's a reasonable approximation), since the centre itself is only an abstract thing and has no muscles attached to it. So, we can think of the muscle chain as beginning with the 'core' muscles attached to the leader's hips and running up his torso. Next come the leader's chest, upper back and shoulder muscles, and then his upper arm and forearm muscles. Progressing to the follower's half of the muscle chain, we begin with her forearm muscles and proceed all the way to her hips through the same sequence of muscles as for the leader, but in reverse. The muscle chain is a mechanical communication device between partners. Each half of the muscle chain is commonly referred to as each partner's 'frame'. I prefer not to use this term because I think it places the focus on shape, and shapes by themselves are of little consequence to good connection (If I had a dollar for every time I've seen a world-class dancer 'break frame' by letting their elbow drift away from/behind their hip, as I was told as a beginner never to do, I would be a rich man. It has been a long time since I've felt that 'frame vs. breaking frame' is a concept useful to anyone but beginners.) What is important, as we shall see, are relationships between shapes and forces. So, we will stick with the term 'muscle chain' and avoid 'frame'. In order for the leader's centre to lead the follower's centre, energy must be transported through the muscle chain. There are many, many ways in which this can happen and most of them are not conducive to good (by our definition) dancing . In order to understand why not, and more importantly, which ways are conducive to good dancing, we must take a step back for a moment and take a look at how muscles work.

Subtle muscles

One can think of a muscle as being made up of lots of little fibers which interlace like the fingers of two hands pointing in opposite directions. And, like those fingers, the fibers can slide over each other as their far ends move further away or closer together (as happens if the arms attached to the hands are pulled apart or pushed together). When a muscle contracts, chemical energy from food and oxygen is used by molecules at the surfaces of the muscle fibers to force those surfaces to slide over each other such that the overall muscle gets shorter. This is the usual process we associate with a muscle doing work. But there is another way the muscle can do work, and that's by extending. Imagine you're standing up, holding onto a shopping bag with one hand, and you've lifted the bag quite high off the ground (by contracting muscles). Now, you want to gently lower the bag to the ground to protect the contents from the damage they would suffer if you simply dropped the bag. So, instead of dropping it, you slowly lower your hand, lengthening your arm - and its muscles - in the process. Your muscles are doing work - giving energy - by preventing the bag from falling suddenly but they are extending while they're doing this work, burning up food and oxygen as they go, in the same way they do for a contraction. We will refer to this kind of behaviour in a muscle as active stretch (The technical term for it is 'eccentric contraction' but this is just a little too confusing, I think!). It is important to note here, that you can choose the rate at which the bag lowers to the ground by choosing the elastic strength in your arm muscles as you lower it. One can think of this as the level of muscle 'flex' or 'tone' which is sustained throughout the movement. If you choose a high level of elasticity/tone, the bag will lower more slowly. If you lower the elasticity/tone - relax the arm more - you will let the bag fall faster. If you relax completely, the bag will fall at the same rate as it would if you just dropped it. We can think of your arm like suspension (as in the springs+shock absorpers attached to car wheels) for the bag's fall.

Active stretch: the key to advanced connection

Ok, so far so good, I hope. Now, we need to push - no, I should probably say stretch - this concept of active stretch a little further. This is important, so stick with me here! An arm actively stretching can be used not only to slow something down, but also to speed something up. Let's take the above bag example in reverse. Imagine now that you want to pick up a heavy bag off the floor and lift it onto a shelf. There are many ways to do this. The most intuitive/familiar way is to grab the handle and pull it upwards by contracting your arm/shoulder muscles. Once the bag has reached the desired height, you 'lock' your muscles, keeping them at a constant length, and walk to/lean over the shelf before switching your muscles into active stretch and lowering the bag as described above. Now, I'd like you to consider a less intuitive way to lift the bag onto the shelf. This way is not really practical/necessary for lifting bags but it essential for good dance connection, so humour me. In this way of lifting the bag, you hold all your upper body muscles - your muscle chain between your and the bag's COM - at a comfortable, relaxed length (instead of straightening your arm to reach down for the bag) . In order to get your hand down the height required to grab the handle, you bend your legs as far as necessary, keeping you muscle chain at a relaxed length. Now, once you've grabbed the handle, you take a look at the path of motion required to get the bag from where it is to where you want it to go and you estimate how much energy you will have to give it in order to get it there in one smooth motion (without any muscle 'locking'). This estimate might not be easy to make the first time but with practice it becomes easier. The key parameter to estimate is how much elastic strength you will need to hold in your muscle chain. Having made your estimate, you contract your leg muscles, pushing your hips upwards and towards the shelf, sending a pulse of energy up through your centre and through the muscle chain, to the bag's COM. All the while, the muscle chain is only stretching. That is, the muscles between your centre and the bag's centre are only getting longer throughout the entire process of moving/giving energy to the bag. Eventually, after your legs have pushed and your muscle chain has stretched enough, the bag will be moving fast enough to get where you want it to go, all by itself. It will cruise through the air and land on the shelf. Of course, the landing will be hard if you don't move yourself along with it and use some more active stretch at the end to slow its landing. Even in this lifting-through-active-stretch case. we can again think of your arm as acting like suspension, buffering the acceleration of the bag off the floor from the rapid acceleration of your hips as they are pushed by your legs. You muscle chain passively conveys energy by actively stretching. This contrasts with actively adding energy with your arm muscles by contracting them.

Just before we leave behind this gory exploration of active stretch, I'd like for us to step out of the leader's shoes and into the follower's. Here's another non-dance example that illustrates how active stretch works, but for a follower. Imagine you are a particularly acrobatic follower who is partial to climbing trees. While you are climbing one day, you decide you will drop down off one branch and catch yourself by grabbing onto a lower branch with your hands as you are falling past that branch. How will you hold your arms as your first grip the branch, and how will you use your muscles to buffer what might otherwise be a clunking halt? Anyone who has ever done something like this knows that you don't grab onto the branch with dead straight arms. Instead, you have your arms at a comforable bend, leaving plenty of room for them to actively stretch and slow your stop. How fast you are falling when your hands first grip the branch will determine how much elastic strength you will need to carry in your muscles in order to slow yourself to a stop over the course of their stretch. What you are most certainly not doing is trying to contract your muscles at any point between gripping the branch and stopping at a comfortable hang; you will have a hard enough time just letting them stretch in a safe, controlled way! The branch here is playing the role of the leader, changing the direction of the follower's motion (from downwards to stationary; this requires an upward force). What the follower is doing (inadvertently in this case) is continuing to move in the direction opposite to the force being applied by the leader, while she uses her muscle chain to actively buffer that force so that it gradually changes the motion of her centre in as controlled a way as possible. 'Bad following' or 'anticipating' in this example, would be like trying to contract your arm muscles to actively pull yourself upwards as soon as you have a grip on the branch. Shoulder dislocation, anyone?

Alright, hopefully we've seen enough about active stretch to have a feel for the concept. Now, let us consider the muscle chain in a dance partnership again. Every one of the muscles in the chain is capable of giving energy by either contracting or stretching. Moreover, it is possible to contract some of the muscles in the chain while letting others stretch at the same time. If more muscles are allowed to stretch and fewer are contracted, then the overall muscle chain will stretch. However, it's not clear exactly how it will stretch. With all the complex ways in which the individual muscles might stretch or contract, it's difficult to predict how the entire muscle chain will behave overall. It is common among inexperienced dancers for there to be inconsistency between the various muscles in the muscle chain. For example, a beginner leader might have a relaxed (stretching) core, a strongly contracting upper arm/shoulder ('arm leading'), and his follower might have a very relaxed (stretching, 'noodle-like') arm and very tense core. The various muscle dynamics combine in a complex way and make it difficult for each partner to get a feel for how the other partner's centre is moving based on touch alone. This is one of the reasons that partner dancing is hard for beginners and why teachers place an early emphasis on the (generally vague) notion of 'frame'. For beginners, perhaps the only way for them to get any kind of idea about how centre-to-centre connection feels, is to have them hold their limbs in the contrived shapes that we might call frame. Anyway...

Elasticity: The whole is the sum of the parts

At this point, a question naturally arises: How should each individual muscle in the muscle chain behave if the whole thing together is to behave in a way that is predictable to both partners? This is where things really get interesting. The answer to the question is 'elastically'. Let's see what this means and why it is the answer.

Elastic is not just the stuff that keeps your pants from falling down. Strictly speaking, 'elastic' describes a relationship between a shape and a force. Imagine you are holding a rubber/elastic band, with one end held in each hand (if you have one nearby, grab it and try this for real). Now, hold one end still while pulling the other end away from it. At first, the band is completely slack. You quickly reach a point known as the 'rest length' of the band, where it pulls taught but is not yet stretching. From there, as the band starts to stretch, the further you pull, the harder the rubber band pulls back on your hand. We can write the relationship between the length of stretch and the force with which the band pulls on your hand, as a simple equation. If you don't like math, don't freak out; it's simple, I promise. Here it is:

Force = - (spring strength) x (length of stretch)


F = -k x

for short.

The force is simply how hard the rubber band is pulling on your hands. We can also call this the tension in the band. We can think of the spring strength (called this instead of 'band strength' because the same law applies to springs and is usually presented in that context) as simply the thickness of the rubber band. A thick rubber band is stronger than a thin one, and for the same length of stretch will pull harder on your hands. The length of stretch is how much you've increased the length of the band from its rest length (defined above). The minus sign is there because the force pulls in the opposite direction to the direction of the stretch; as you pull to lengthen the band, it pulls back on you, trying to get shorter again. What we have here is a simple relationship between a shape (length - a 'straight line shape') and a force. Anything which can accurately be described as 'elastic' must obey this relationship.

To see what's so special about muscles that stretch like elastic, we consider just two muscles connected together in a chain and ask the question, 'What kind of force-shape relationship must each muscle display in order for both muscles together to exhibit the same force-shape relationship as each muscle individually?' This question must be answered mathematically. After crunching some algebra, it turns out that there are only two kinds of relationship which will allow this to happen, and one is useless for dance connection for a reason that we need not go into. The remaining answer is, if the pair of muscles is to stretch like elastic, each of the individual muscles must also stretch like elastic. If just one of them stretches elastically and the other does something different, the overall pair will not stretch elastically, but rather in a more complex way that is harder to predict. This reasoning is easily generalised to the entire muscle chain, with its long sequence of many muscles. If the whole muscle chain is to stretch elastically, each muscle within it must stretch elastically.

But why do we want the whole muscle chain to stretch elastically? The key reason once again is that the whole acts like each of the parts. Recall our discussion in the last post, about the need for a simple grammar of dance connection. What each dancer needs is a short list of assumptions, which will allow him/her to step out on the dance floor with a complete stranger and know that "As long as I uphold my half of the bargain and (s)he does the same, this should all work out ok." We can state this with more specific reasoning based on the above discussion. What each partner needs is to be able to monitor and predict the behaviour of the other partner's centre of mass, based only on what he/she can feel to be the case for his/her own muscles. Since this is true for both partners, what is needed is a system of force-shape relationship which will apply to the whole partnership if each partner can make it apply to him/herself individually. As we have seen, the only system which will achieve this is elastic active stretch of each muscle in the muscle chain.

Compressions are stretches

At this point, you might be wondering something: Sure, there's all this stuff about stretch and that's all very well, but what about compression? Good teachers talk about compression all the time, so how can we account for this in our elastic model of connection. The answer to this question has two parts. The first part is simply the acknowledgement that springs don't just stretch, they compress too, and when they do, they obey the same law for the relationship between length and force as they do when stretching. Imagine holding a strudy spring between the palms of your hands. Initial, at its rest length, it does not push back on your hands. As soon as you start to compress the spring - make it shorter - it pushes back on you. And, the shorter you make it (the more compressed it is), the harder it pushes back. Compression in dance connection behaves in the same way. BUT, this leaves one important question open: How can this possibly work when an individual muscle does not - indeed, cannot - resist compression? Muscles can only give energy under tension; under compression, they are just limp pieces of meat. The resolution to this problem is the way that muscles work in pairs with each other, also in cooperation with a stiff skeleton. If you lean against a wall, it is true that you can apply a compression force on that wall because at least some of the musles in your body are under tension and that tension is converted into a compression force by the structure of your skeleton. This is a complicated way of saying that even when you're pushing on something, you're still doing it by contracting a muscle somewhere. When you do a push-up, during the push phase, your tricep contracts; during the lowering phase, your tricep actively stretches. In both cases, it is under a tension force. Compression in dance connection works in the same way. If you want your connection to feel elastic under compression, you still have to learn how to control the stretch of each muscle in your muscle chain so that it behaves elastically.

Tuning, not relaxation

One final point before proceeding to a summary of this post. It is common for teachers to tell their students that in order to achieve good connection, even at fast speeds, they have to relax. This always seemed strange to me, right from my beginner days, and now I realise why; because, if one takes 'relax' literally, it's simply not true. It seems to me that what people are really trying to say when they say 'relax', is to use an elastic connection with a spring constant no larger than is necessary to pass the required amount of energy from one partner to the other while using a comfortable range of stretch lengths in the muscle chain, in order to achieve the desired shared movement. Admittedly, stating it like that is not exactly a pithy gem to have in one's teaching repertoir. But conciseness is, in my opinion, not a worthy trade for truth, and the key point here is that simply relaxing one's frame does not make for good dancing. A 'relaxed frame' - in the literal sense - is a limp, noodle-like frame; exactly what we tell beginners not to have. If I tried to dance at any tempo above maybe 50 bpm with a genuinely relaxed frame, I would risk injuring myself, my follow, and having her tell all her friends afterwards that my dancing is 'flacid'. Point made, I hope. A relaxed frame is not a functional goal. What really separates amazing connection from good connection, I think, is each dancer's ability to fine tune their spring constants to match the energy of the movement being communicated. Tension is definitely required, and it's required in proportion to the energy being passed between partners. A functional goal for dancers aspiring to good connection is to never hold more or less elasticity than is required in order to work smoothly and easily with one's partner, ramping up and down quickly and efficiently when the energy level changes.


So, time to wrap it all up for this post. What does the above small novel of theoretical mumbo-jumbo mean in practical terms? Here's a summary for practical application. In order to achieve good connection, each partner should make sure that:

- The muscles in his/her half of the muscle chain are only stretching, not contracting, at almost all times (almost because in reality, even the best dancers use contractions in their muscle chains sometimes; we will talk more about this later) no matter what movement is being danced or what pace the music is. This is counterintuitive and needs practice but to do otherwise amounts to the 'arm leading' that teachers warn against, which does not allow for efficient communication/cooperation between partners.

- Energy given by the leader to the follower originates in his leg muscles, which contract to move his centre. The motion of his centre then influences the motion of her centre in a simple, predictable way as energy flows through the elastic muscle chain from one to the other. Both partners must resist the temption to add energy by contracting some of the muscles in the muscle chain. Following purely requires no addition or subtraction of energy by the follower herself. She conserves her momentum and uses her half of the muscle chain simply to buffer and smooth out energetic transitions which find their way to her through the leader's half of the muscle chain. The whole process might not be easy but it is profoundly simple.

- The most important thing to learn how to finely control is the spring constant that you hold in your half of the muscle chain. Fast, powerful communication through the chain (like that used when dancing Lindy hop quickly) requires a high (but no higher than necessary) spring constant. Slow, relaxed communication (like in slow blues) uses lower constants. What really separates amazing connection from good connection is the ability of each partner to dynamically adjust his/her spring constant to match that of the other partner.

The next post will further explore the application of elastic connection in practical dancing. We will see that many aspects of connected dancing, which might initially seem complex and mysterious, become much simpler to understand when they are discussed in light of an elastic model of connection.


Connection 1: Body language

We have finally arrived at what I think is possibly the most interesting - and most confusing - issue discussed by partner dancers: Connection. We have already flirted with questions of connection in earlier posts but now we will dive into them head first. The discussion will begin by introducing connection as a kind of tactile language; a means of communicating through forces and shapes experienced/observed in two dance partners' bodies. We will then consider an argument that a particular system of such language is more effective than others that are commonly used in facilitating musical, co-creative dancing. It will be argued that this system is more-or-less universal between the various styles of improvisational partnered jazz dancing, being different only in fine-tuning between Lindy, Blues and Bal (and other styles as well, even outside the jazz genre). Finally, this system will be explained in depth. In order for the explanation to be as clear as possible, a simple physical model of connection will be developed and explored. Please don't worry if you've never studied physics; I will attempt to explain things in a way that is accessible to everyone. For those who are interested in the quantitative details of the model, there will be small 'technical supplement' (read as 'shameless geekfest') sections along the way.

It is useful here to state a very brief summary of much of what has been discussed in earlier posts:

  • There are many ways to define 'good swing dancing'; we have chosen to define it as musically co-creative connected movement which is innovative within a historical frame.
  • Both musical auditory processing and sensory-motor control of your body are evolved human capacities and carry the legacy of that evolution.
  • Gestural and (later) verbal language provided the survival advantage of efficient communication in cooperative social groups. Current theories of the neural processing of music suggest that it evolved in parallel with language processing. We have speculated that musicality in dance results from the music-verbal language-gestural language links.
  • Fast, efficient sensory-motor control of a single human body (ie. the one which houses the brain that controls it) is biologically evolved and therefore intuitive. Control of two connected bodies, shared between two brains, has evolved not biologically but culturally, through trial and error, by many dancers over many years. These systems involve self-imposed restrictions on the movement of each individual partner, which facilitate shared control.
  • Particular systems of shared body control have emerged as more effective than others. These are the systems used by the most respected dancers. Good teaching consists in distilling these systems into basic principles and communicating those principles accessibly, in entertaining ways, to students.
  • Individual partners move in a way conducive to good shared control when they move with maximum predictability over time scales short compared with a whole dance (ie. less than a second to a few seconds). Predictability over longer time scales (eg. Dancing the same move or sequence of moves over and over) can make a dance uninteresting, so good dancing combines predictability over short times with originality over longer times. We have speculated that maximum predictability over short time scales is achieved when individual dance partners move in such a way as to minimise the jerk of the motion of their individual centres of mass. This makes the dancers' motion smooth, preventing sudden, difficult-to-work-with changes, each partner helping the other to have the experience, "Ah, I can feel and see what you're doing and where you're going. I can work with that!".
  • For the purposes of good shared control, the primary function of steps and footwork is to carry movement; where you put a foot is much less important than how you control the motion of your centre of mass through the space above it.
  • The following steps are required for shared control:
  1. Partition the control into roles of lead and follow. The roles can be passed back and forth but in general, only one person is leading and the other is following at any given time.
  2. Restrict individual movement in the ways described for leader and followers in an earlier post.
  3. Restrict the connection system to obey a particular set of rules, which is known to both partners, or can be learned through 'tuning in' to each other on the social dance floor.
  4. Cooperate to create shared movement, which reflects the music.
Now, for a very simple functional definition of connection: Connection is achieved when the motion of one partner's centre of mass is influenced by - and influences - the motion of the other partner's centre of mass, in a predictable way. Let's keep this in mind as we proceed. First, however, we will take a step back and consider connection as a kind of language-based conversation, which will provide a familar analogy against which to frame the later technical discussions.

The various partnered jazz dance styles are primarily social dances, meaning that the primary goal within each is for a dancer to be able to visit a place where (s)he has never danced before, dance with someone (s)he has never met before, to music (s)he has never heard before and still have fun. This extreme flexibility requirement places heavy constraints on the complexity of the movement/connection system which dancers should learn. Strict systems with long lists of specific rules to learn are not likely to facilitate this kind of flexibility. So, if dancers seek to learn lots of specific moves as their end goal, they may find that unfamiliar partners don't know any of the same moves, making it difficult to find common ground on which to share a fun dance. We can make a linguistic analogy here, imagining that two strangers - one English speaker and one Spanish speaker, say - might try to prepare to have a conversation in a new language (eg. Japanese) which is unfamiliar to them both. If they each prepare by learning randomly selected phrases from a phrase book, it is unlikely that their conversation will achieve any depth, if it manages to get off the ground at all. What's required then, are some general rules - preferably as few of them as possible - allowing people to communicate flexibly. In language, these are the rules of grammar, which allow for the building of meaningful, original sentences. Two mutually unfamiliar dancers can have a creative, original dance then, if they just know the same rules of grammar, which is a significally smaller set of things to learn than a long list of pre-cooked sentences.

I think it's interesting to spend a moment discussing the language spoken between regular dance partners, particularly in the case of rehearsing/performing choreography. Frequently, such partnerships are able to achieve things in their dancing, which neither partner can achieve while dancing socially with someone else: particularly impressive tricks and aerials; quick, subtle changes of direction, etc. At the same time, the dancers in the partnership might not have as much fun while dancing with each other as they do when dancing with unfamiliar others, because all too often, with each other they are having the same conversation over and over. We might think of this as analogous to a couple of actors playing out the same scene, learned from a script. They have each got their own lines down, complete with emotional nuances, and can predict each other so well that the scene flows smoothly enough to convince an audience. But the internal experience of the actors themselves is not the same as it would be if they were having a real, original conversation. Yes, if they are good actors they are still able to bring some genuine emotion to the script and 'play off' each other but they both know that there is only one way the conversation will go.

Ok, so, point taken, I hope. What does this have to do with connection - the physical mechanism by which dancers achieve movement together? The key point, I believe, is the amount of information which must be pre-learned by the dancers. Choreography still uses connection but it uses very complicated connection. It does this because it can afford to - there's lot of time to learn the rules and rehearse. Choreographed performances might involved sections where two dance partners are able to precisely coordinate their movements without being in physical contact at all. They might use subtle visual signals in order to communicate when it's time to change and do something differently. And as a result of all this complexity, they are able to achieve things which are rarely achieved on the social dance floor. It would be *great* if every dance we ever had looked as good as the best choreographed performances. Indeed, when most of us first come to partner dancing, we dedicate ourselves to learning cool new moves and tricks and love dancing them on the social floor with familiar partners who know the same stuff. But we soon realise that the prospect of having the experience of executing precise choreography with everyone we ever dance with is going to require a godlike memory and a preparatory discussion before every dance. "Hi! Would you like to dance? Great! Do you know such a such a move? Do you know how to get into it from this move? What about this sequence? Ok, well, let's forget the sequence but we can do that move. Great! Let's do it!" Imagine having to go through all that every time you want to have a dance! It might be fun and impressive to pre-learn a script and act out an amusing 'conversation' for an audience but trying to take the system by which that is achieved and imposing it on social dancing doesn't result in much fun. So, what's the alternative? A simple, powerful system of tactile connection; a mechanical grammar which allows two unfamiliar dance partners to achieve amazing things on the social floor without any shared preparation whatsoever. Dancers the world over pay good money to learn from instructors who have mastered this, and students with a passion for excellence spend months and years trying to put the teachings into practice.

I don't think I'm describing anything new here. I think dancers know that they need to learn general rules of movement and connection. I skeptical, however, that there exists a single, well-formulated grammar of dance connection, which is agreed upon by everyone. Different teachers teach different grammars and when their students try to dance together, it is analogous to two people who speak different languages trying to have a conversation. The languages aren't usually all that different; it's not like Mandarin and French coming together. I think it's usually something more like different dialects of the same language. Nonetheless, this can make things difficult. Sometimes the two partners can manage to figure each other's dialects out and have a great dance. Other times though, there is an ongoing clash. One might argue that there simply isn't a singluar 'best' grammar and there's no point trying to find one. I think this is too strong a statement. I think that there are good reasons to argue for a single, optimal system of connection - a single, most effective grammar - which, if learned by everyone, would facilitate 'better' (according to our definition) dancing. Sure, there will always be dialects but perhaps the goal should be to leave them behind, or least be aware of them so that one can choose one's particular, quirky dialect consciously, rather than because it's the only think one knows how to do. I am not arguing here against individual styles; far from it! I'm arguing for a clear system of communication. Two people who speak exactly the same language can certainly have fun, meaningful, creative, exciting conversations!

In what follows, I will attempt to describe what I believe might be the optimal dance connection grammar, which is used by the world's best dancers. I should make a small qualifying point here that Lindy, blues, Bal, etc, are definitely different dialects. But they are NOT different languages from a biophysical standpoint. I cannot emphasise this strongly enough. The systems of muscle control used by Blues dancers are the same as those used by Bal dancers. The two dances differ only in so far as those systems are subject to historical constraints. Bal dancers only 'talk' (ie. have tactile discussions on the dance floor) about the topics which are accepted by the Bal community, and vice versa for blues dancers. But the language they speak when having those topical discussions is the same. The (relevant) nerves attached to your muscles do not know whether you are dancing Bal, Lindy, Blues or Polka. All they are able to tell your brain is how long/stretched each muscle is, and how much load/force it is bearing. Your brain accepts signals of this type from every skeletal muscle in your body constantly, synthesises them, interprets them in the context of whichever dance you are choosing to do, decides what to do next, and then activates the relevant muscles when required. This is not to say that there are not different systems of shared body control out there. There are. But as far as I can tell, the systems used in all of the various improvisational partnered jazz dances are fundamentally the same. The differences between them are less than fundamental.

In the next post we will get down to business and begin describing the simple, fundamental rules of connection, the rules which relate the shapes in dancers' bodies to the forces they are applying on their partners.


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