A version of this article originally appeared in The Oxford Invariants Society magazine The Invariant in Hilary Term 2016.

Everyone knows that music and mathematics go well together. There are myriad examples of this, from Albert Einstein, who loved to play the violin, to John F. Nash, who listened to Bach and Mozart while solving mathematical problems. It is no coincidence that Marcus du Sautoy and Douglas Hofstadter named their brilliant books The Music of the Primes and Gödel, Escher, Bach, respectively. I argue, however, that classical ballet is in fact the truly mathematical art – more so than even mathematicians or dancers might appreciate (1). Ballet incorporates core principles found within theoretical computer science, linear algebra, and geometrical aesthetics. When a ballet performance combines these mathematical fundamentals with graceful expression, evocative music, and sublime visual arts, it brings mathematics to life. Let me explain what I mean.

1. Choreography as automata theory

Every classical ballet dance, such as the Dance of the Sugar Plum Fairy in the Nutcracker, or the Black Swan solo in Swan Lake, is constructed by stringing together individual elements from the alphabet of ballet. These individual elements are called the steps of the dance. The process of inventing new dances is called choreography. When ballet was invented in the Baroque court of King Louis XIV in 1665, in France, it was decreed that every dance step in ballet should begin and end in one of the so-called Positions of the Feet (Fig. 1) (2).

Figure 1. The basic positions of the feet in ballet. In fact, there are two versions of `fourth position’: open (ouverte) and crossed (croise). Generally speaking, a ballet step will begin and end in one of these basic positions (3).

These positions form the basis of a ‘ballet automaton’ as defined by automata theory. In brief, a deterministic finite automaton (DFA) is an abstract machine that processes input strings, either ‘accepting’ them or not. The automaton has a finite number of states, a finite set of input symbols, and a transition function that specifies the next state that the automaton will move to, given its current state and an input symbol. The automaton also has a set of final or accepting states. An automaton therefore reads in a string of input symbols, moving from state to state as it processes each symbol in a manner determined by its transition function. If the automaton finds itself in an accepting state after the final symbol, it ‘accepts’ the string, otherwise it does not. In this way, automata can decide whether a given string is a member of some particular language. As strings can code for logical expressions, graphs, integers, and much more, automata can decide answers to many interesting mathematical problems (4).

We observe that the process of choreographing a ballet dance is equivalent to finding an input string that is accepted by the ballet automaton (5). Each of the positions of the feet are states of the automaton, and are also accepting states, since it is valid for a dance can end in any of these positions. We define an additional ‘impossible’ state, which is not an accepting state, to represent a physically impossible choreographical request. Each ballet step is a symbol in the ballet alphabet. The transition function maps these symbols to a specific movement between the possible states, that is, the foot positions of the dancer. The process of choreographing a dance by sequencing steps together is then, at a first approximation, equivalent to creating an input string that this automaton will accept.

The input string, which represents a sequence of steps, dictates a specific path to take through the states. If the input string is valid, it will move the automaton from state to state, that is, foot position to foot position, ending on one of the final positions. If the input string is invalid, for example by having two consecutive steps where the end position of the first step is not the same as the start position of the next step, then the automaton will move into the ‘impossible’ state, which is not a final state, and the transition function will ensure that the automaton remains in that state to the end of the string. In this case, the string will not be accepted. The set of strings, or language, accepted by the ballet automaton therefore defines the set of all possible valid ballet dances.

Of course, the above automaton is a simplification. It would be too complicated to describe an automaton that models the entirety of classical ballet, at least in the present article. But, we define and consider a small example in the Technical Appendix.

2. ‘En avant et en arrière’: linear transformations and inverse functions

Another founding principle of ballet is the linear transformation of three-dimensional Euclidean vector space R³. First, most ballet steps can be translated in space, that is, they can be performed going forwards (en avant), backwards (en arrière), to the left, and to the right. For example, the step known as a chassé, a sliding step, can be performed in each of these four ways. Secondly, many steps involve rotation, either of the body or of the step itself. A striking example is the pirouette, where the dancer spins on one leg. Thirdly, many steps undergo scaling. For example, the basic jeté (thrown leap) comes in three sizes: petit jeté, a small hop; standard jeté, a medium-sized jump; and grand jeté, a large leap (Fig. 3a). To a mathematical mind, this organisation of ballet steps according to principles of linear transformation is very pleasing.

Many ballet steps are asymmetrical (chiral). Each such step can be reflected along the main vertical axis of the body, that is, can be performed on the left side or on the right side. A wonderful example is the attitude, a held pose that can be performed on either the left leg or the right leg (Fig. 2). Ballet also makes much use of repetition, which can be viewed as an application of the identity function. Most interestingly, many ballet steps have an inverse—the step can literally be danced backwards from its finishing position to its starting position—an almost perfect example of an inverse function. This means that a dance made up only of these invertible steps can technically be performed backwards, equivalent to reversing the equivalent input string and thereby tracing backwards the path through our ballet automaton defined in the previous section!

Figure 3. An attitude pose in ballet, performed here by Yamamoto Mashiko in the ballet Le Corsaire. Attitude can be performed either on the left foot or the right foot, perfectly reflected (6).

In fact, in ballet classes it is not uncommon for the teacher to set a short exercise, and after the students have performed it, to ask the students to perform it backwards, giving them little time to think. This speaks to the somewhat unrecognised mathematical ability of dancers. See the Technical Appendix for a detailed example of such an inversion.

 3. The beauty of lines, angles and symmetry

Many mathematical objects, such as groups, geometric structures or topological spaces, are considered beautiful when we ‘see’ them in our mind’s eye or in an illustration. For example, there is a fundamental, pleasing beauty to a perfect sphere or a tetradedron. Just as in mathematics, fundamental to the core beauty of ballet is the composition of shapes, angles and structures in visual space.

Figure 3. Illustrations of curves, symmetry, angles and extensions in the ballets (a) Don Quixote, (b) Swan Lake and (c) Apollo (7).

Figure 3 illustrates some of these geometrically aesthetic ideas. In the grand jeté (Fig. 3a), the soft ellipse created by the arms stands in sharp contrast to the 180º angle created by the legs. The principle of symmetry, made possible by the fact that every ballet step can be reflected (as discussed in the previous section), is utilised to full effect by the corps de ballet in the First Act of Swan Lake (Fig. 3b). A obsession with repeated angles, in addition to symmetry, is seen in this iconic moment from the ballet Apollo, in which three female dancers create the illusion of multiple legs radiating out from a central origin (Fig.3c). The radiating lines, generated by the concept of extension of the limbs in ballet, seem as though they could go on forever, evoking the impression of infinity. These are just a few examples of the general principle that classical ballet choreography often searches for moments of profound geometric perfection.

4. Conclusion

This paper argued that ballet is based upon mathematical foundations, both at a constructional and aesthetic level. Choreographing and dancing ballet requires fast mathematical problem-solving, consciously or unconsciously. Moreover, recognising the computational and geometrical underpinnings of ballet can increase one’s appreciation of any ballet performance. For these reasons, it would be natural for ballet to become the art of choice for mathematicians!

5. Technical Appendix: Deterministic finite automaton for petit allégro

We present a deterministic finite automaton (DFA) A to exemplify petit allégro exercises, as may be found in a beginner to intermediate ballet class. Petit allégro refers to the part of a ballet lesson or dance that involves small jumps. For simplicity, we limit our exercise to the steps changement,entrechat, and various sautés (jumps) between the foot positions (Table 1).

The DFA A is the five-tuple A = (Q, Σ, δ, q₀, F), where Q is the set of states, Σ is the set of input symbols, δ is the transition function, q₀ is the start state, and F is the set of final accepting states.

For our simplified automaton A, the set of states Q is limited to first position (1st), second position (2nd), fifth position-right-foot-in-front (5thR), fifth position-left-foot-in- front (5thL) (Fig. 1), and the physically impossible state I. That is, Q = {1st, 2nd, 5thR, 5thL, I}. We designate 5thR as the start state q₀ because the majority of ballet class exercises start in this position. The set of input symbols Σ consists of six petit allégro steps, specified in Table 1. Transition function δ , which controls how each step moves the dancer from one state (foot position) to another, is specified in Table 2 (see also Fig. 5). Note that if the choreography asks for an impossible sequence of steps, that is, the end position of one step is not the start position for the following step, the automaton moves to non-accepting state I and remains there for the remainder of the input. We define valid input strings, that is, valid choreographies for a dance exercise based upon these six petit allégro steps, as those strings that would be accepted by automaton A.

An interesting petit allégro choreography to consider is represented by string1 = ch-ch-S2-S1-S2-S2-S5L-ec. It is easy to verify that this string, which starts at the start state 5thR and ends in the state 5thL, will be accepted by automaton A. Since the step S5L is chiral, we can also define the reflection of this exercise as string2 = ch-ch-S2-S1-S2-S2-S5R-ec. By concatenating string1 and string2 to form new string3 = ch-ch-S2-S1-S2-S2-S5L-ec-ch-ch-S2-S1-S2-S2-S5R-ec, we can create a new exercise in which string1 leads straight into string2. We can do this because step ch is valid from the state 5thL, that is, it does not send the automaton into the non-accepting state I. Since the exercise defined by string3 starts and ends on the same position (5thR), the exercise can be repeated consecutively as many times as you like—the only limit being the dancers collapsing from exhaustion!

Moreover, since the exercise encoded by string3 consists of steps that each have an inverse (see Table 1), the entire exercise can be danced backwards by stringing together the inverses of the steps in string3, in inverse order, to form string4 = ec-S2-S2-S1- S2-S5L-ch-ch-ec-S2-S2-S1-S2-S5R-ch-ch. The inverse of a valid string, if it exists, will be valid if the start state is also an accepting state. Both the original exercise (string3) and the inverse exercise (string4) can be traced in the diagram of automaton A in Fig. 5.

Automaton A does not accept every possible string constructed from the set Σ. For example, a string that includes the following consecutive steps …-S2-ec-… will certainly not be valid: from any state, transition function δ sends automaton A to state “2nd” when S2 is read in. But if symbol ec is read in state “2nd”, δ defines that A moves to state I. Since all input symbols read in state I keep automaton A in state I, which is not an accepting state, the string will not be accepted as valid. In the dance, this represents the situation in which the choreography requires the dancer to perform an entrechat after a sauté to second position—which is physically impossible since the entrechat can only be danced by the feet starting in a fifth position.

Figure 4. The entrechat step begins and ends in the same fifth position, in this case, fifth position with the right foot in front (5thR). While in the air, the legs are crossed with the right foot behind, and then crossed again to return the right foot to the front (8).

Figure 5. Diagram of deterministic finite automaton A, showing the five states (foot positions) and the ways the different alphabet symbols (steps) described in Table 1 move the dancer between these states. For clarity, only the transitions used in string3 are depicted (see main text).

symbol	step	description	inverse
ch	changement	Jump up from, and land, in fifth position, having swopped which foot is in front.	changement
ec	entrechat	Jump in fifth position, crossing the twice in the air and landing back in the same fifth position (Fig. 4).	entrechat
S1S2S5RS5L	sauté to firstsauté to secondsauté to 5thRsauté to 5thL	Jump from any foot position to land in the specified foot position.	A sauté back to the foot position from which the original sauté started.

state\step	ch	ec	S1	S2	S5R	S5L
1st	I	I	1st	2nd	5thR	5thL
2nd	I	I	1st	2nd	5thR	5thL
5thR	5thL	5thR	1st	2nd	5thR	5thL
5thL	5thR	5thL	1st	2nd	5thR	5thL
I	I	I	1st	I	5thR	I

~ Footnotes & References ~

(1) But see Wasilewska, K. (2012) Mathematics in the World of Dance. Bridges 2012: Mathematics, Music, Art, Architecure, Culture, 453:456.

(2) As ballet developed, further positions of the feet were used, including a sixth position, a seventh position, and various positions of standing on only one foot. However, the positions shown in Figure 1 remain the core positions.

(3) For more information see the excellent textbook Introduction to Automata Theory, Languages, and Computation by John E. Hopcroft, Rajeev Motwani, and Jeffrey D. Ullman.

(4) Figure adapted from “ballet: five basic positions. Art. Britannica Online for Kids. Web. 12 Dec. 2015. http://kids.britannica.com/comptons/art-167043.”

(5) For other ideas along similar lines, see Schaffer, K. (2011) Mathematics and the Ballet Barre. Bridges 2011: Mathematics, Music, Art, Architecture, Culture, 529:523, and LaViers, A. & Egerstedt, M. (2011) The Ballet Automaton: A Formal Model for Human Motion. Proceedings of the 2011 American Control Conference, 3837:3842.

(6) Image adapted from cc Fanny Schertzer http://creativecommons.org/licenses/by-sa/2.5/

(7) Images adapted from (a) cc Peter Gerstbach, (b) cc Fanny Schertzer http:// creativecommons.org/licenses/by-sa/2.5/, (c) Andrea Mohin/The New York Times.

(8) Image adapted from Ortia at Wikimedia Commons.