Any problem in the world can be solved by dancing.   --James Brown

Topology and tango! This sounds like an interesting mix-up of topics, perhaps a mistake in subject matter? I would think that Topology is a word as familiar to most dancers as dancing is to most topologists! So, let me take a paragraph or two to explain…….

Topology is loosely defined as the mathematical study of shapes. The field of topology was begun in 1736 when the brilliant mathematician Leonhard Euler wrote a paper about seven bridges that connected various islands in the Pregel River in the town of Königsberg, Prussia. Euler proved that one cannot travel in a route that crosses each bridge only once. This result didn’t depend on the length of the bridges or the sizes of the islands, but only on how the islands and bridges were connected to each other. In fact, the shapes of the islands and bridges could be changed in any way, so long as the connection points were the same, and the same result would be found.

Of course there are other famous and useful problems of topology that apply to many fields of study. For our purposes, we can think of topology as a study of the possible movements of the dancers, given the shape of the human body and set of rules that define the dance. We will see that the laws of topology limit the possibilities of where, from a given position, each partner can step. Two partners, holding each other in an embrace, each have limited stepping possibilities. These limitations are a result of the shapes of their bodies (two arms, two legs, chest, and hips) and how they are holding each other. Different dancers may be larger or smaller, taller or shorter, but the topological limitations are the same. A dance that allows partners to let go of each other may have different possibilities than a dance where a one-handed lead is used, or from a dance like tango that uses a full embrace. The step possibilities in a dance between two Centaurs (a mythical half-horse, half-human beast) would surely prove different from those in a dance between normal bipeds. Where, for instance, would the back sacadas go!

Why is it important to study the ‘topological possibilities’ of a dance? For one, many of the problems that dancers face are rooted in a lack of knowledge, or at least a lack of a body awareness, of the options and limitations imposed by the laws of topology. It is all too common for a leader to lead a follower into one position, and them he himself attempt to obtain another position that is topologically ‘awkward’. For example, imagine a couple dancing in an open style embrace. To be specific, let’s say that a measurement of the distance between the couples’ gives four inches chest to chest. If each partner is standing on balance with one foot weighted and the other foot free, then the distance between the balls of their weighted feet would be about the same 4 inches. Now each partner takes a step with his or her free leg. If the new step ends up with the weighted feet greater than four inches apart, then one or both of the dancers will necessarily be off balance, or the embrace will have to be adjusted to match the new distance.  Either of these results (off balance partners or fluctuating embrace) might be felt and seen as lack of elegance, smoothness and style.

On the other hand, a dancer who has internalized these limitations will seem to magically maintain poise, elegance and balance as he takes each step in the correct place within the topological bounds of the dance. Equally important to our study of dance topology is that it can be a powerful way to open up a world of new possibilities. All too often, we learn a few figures that seem to work and stick with them dance after dance; Out of habit, we may be overlooking other new and interesting possibilities at each step. Knowing the complete set of possibilities available to us from any given dance position allows us to create new variations on a given figure and completely new movements in the dance.

Although ‘elegance’, ‘ grace’, ‘style’ and ‘beauty’ may be subjective, we would propose that these descriptions of mastery of dance are closely connected with the degree to which the dancers, either consciously or through body awareness learned during countless hours of trial and error, adhere to the physical rules and limitations of the dance.  This book is about only one small part of tango. It is not about technique, musicality, posture, etiquette, or any number of important things that can best be learned from competent teachers in classes. Rather, it is about underlying physical principles that are often not well understood by dancers and rarely discussed in class. We do not claim that you will learn dancing from this book and, in fact, we hope that readers have some dance experience or are at least learning concurrently with their reading. However, if you follow the chapters carefully, we are confident that you will learn something deep and fundamental about dancing tango.