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The romance between art and the science of numbers and functions goes beyond western art history, if one considers math to be the aesthetic production found in nature. Plants and thees develop through algorithms, the spiral of the Nautillus shell grows in the golden ratio proportion, honeycombs have a hexagonal shape that allows optimal storing, crystals have a delicate symmetry. There are many examples of beauty and efficiency walking hand in hand.

Indeed, mathematics and mathematicians, art and artists mingle all the time in the most diverse practices. Their territories are constantly crossed through the exploration of concepts such as topology, chaos, beauty, proportion, form, symmetry, space, rhythm, flow, (dis)continuity, among so many others. In the history of painting, for instance, the construction of perspective is among the most important concepts. More recently, it is worth remembering Catalan painter Salvador Dalí (1904-1989) had mathematician   Thomas Banchoff , researcher at Brown University, Providence, USA, as faithful squire.

In Brazil, out of many working artists, architect Tania Fraga and the pair Daniela Kutschat and Rejane Cantoni have distinguished themselves for their interest in the relationship between the body and interfaces (gloves, eyeglasses etc) and environments from other dimensions. Tania Fraga, whose work is internationally acclaimed, has dedicated herself to the expansion and consolidation of computer art in Brazil by creating immersive and virtual environments. She is passionate about mathematics and studies and does the programming for her works herself, often subverting the codes in order to propose new aesthetics and/or objects that are only possible in the virtual world, like the quaternions, complex objects of this science. Among her most recent works are Fragmentos (2207/2008), an artificial intelligence opus-program and stereoscopy (with 3D eyeglasses) and Caracolomóbile, an interactive installation to be carried out in a parabolic flight, as part of the research activities of the group Space Art.

Project OP_ERA, developed since 1999 by artists Daniela Kutschat and Rejane Cantoni, comprises a series of tools and interaction and immersion environments where the body and the machine intertwine in symbiotic experiences that explore space dimensions and cognitive sensations simultaneously. The show OP_ERA was the first implementation of the series and was produced in 2001 during the now extinct Dança Brasil showcase, at Centro Cultural Banco do Brasil, in Rio de Janeiro. It was conceived for a black box theatre, where the interaction platform, four projection screens and the mesh of sensors turned the stage into a cube sensitive only to the interference of the dancer. In the following implementations of the project, the “cube” would be “reassembled” in many different ways, exploring specific relationships. The following installations were different from the first one in one essential aspect, they were open to the participation of anyone willing to venture perceiving worlds from other dimensions, possible only with the help of these wonderful artificial tools.

Dancing numbers, figures and functions

And in dance’s case? What role can a certain mathematic concept have in the composition of a choreography? Which different choreographic organization modes are configured in the presence of abstract concepts? As in art history, the evolution of dance can also be considered in a mathematic perspective. Choreographers use it to create compositions that produce spatial drawings, for example, implementing ideas of harmony and symmetry, present in classic ballet, in choreographies and in the gesture structures of dancers.

In the past, Oskar Schlemmer (1888-1943) dedicated the years between 1916 and 1922 to the development of the Triadic Ballet, a piece in which the geometrization of the body defined the dance. At the stage of Bauhaus, Schlemmer continued working on a model of a human figure determined by geometrical and mathematic formulas. During his workshops, participants made masks and costumes and the studies of movement, mechanics, optics and acoustics were requirements for the scenic work. In aesthetic terms, a close relative of this amazing artist was none other than the amazing Alwin Nicolais.

But the approximations and relationships between dance and mathematics go beyond. We observe them in the works of American choreographer Merce Cunningham, with the use of chance – through random choices and i-ching – as composition operation; in French choreographer Gilles Jobin’s The Moebius Strip (2001), in which the moebius tape is the theme; in the cumulative and metric logic of Trisha Brown’s Accumulation; among so many other examples. In Brazil, group Cena 11 uses formation and recognition of patterns for the elaboration of its latest choreography, Pequenas Frestas de Ficção Sobre Realidade Insistente (2007). Clube Ur=hor (‘U’ of ‘r’ equals ‘H’ ‘zero’ of‘r’), which borrows the name from the formula for the expansion of galaxies, also presents issues related to space, disconcerting it in the piece Prop.Posição #1, necessário a posteriori, directed by Adriana Banana.

Artscience

When we start mapping areas of artistic and scientific research emerged from the relationships between dance and mathematics it is immediately possible to identify three main axis: 1) mathematics teaching through the use of resources from the body and dance; 2) artistic and scientific experiments/projects (choreographic pieces, processes based on motion capture, softwares for creation, artistic projects using video and sensors etc); 3) analogical and digital movement notation projects (development of interfaces and specific softeware).

The first axis is basically composed of research aimed at the development and application of methodology for teaching mathematics through corporal resources. Anne Watson’s article, Dance and mathematics: power of novelty in the teaching of mathematics, shows how powerful the partnership between the two areas is for the teaching-learning process. By combining both fields, it is possible to experience physical sensations of abstract mathematic concepts. “For many people, having a synesthesic experience of an abstract idea is extremely useful for the understanding of what such concept means”, says educator and mathematician Karl Schaffer, from John F. Kennedy Center for the Performing Arts. The link http://www.sciencedaily.com/videos/2008/0503-do_the_math_dance.htm is a record of the experience of how mathematicians and choreographers use dance to create mathdances, mathematic problems to be solved through the creation of small dances. Such field is yet unexplored and almost nonexistent in Brazil.

The second axis encompasses a great amount of processes, experiences, projects and (mainly hybrid) artistic and scientific projects interfacing art and science. In Brazil, there are excellent experiences like the already mentioned group Cena 11, which has been making advances in the relationship between dance and technology by investing in the development of interaction systems between movement and environment modification through physical/digital interfaces. Abroad, researches between artists and scientists proliferate and some names are worth mentioning, like Scott De Lahunta, Emio Greco, William Forsythe and the Siobhan Davies Dance Company, among others.

The third axis is extremely related to the previous one, it focuses on the challenge of creating efficient notations of dances that are produced, since their composition material is fleeting, Although the relationships between dance and record can be dated back to the 1960’s, a period in which the first movement notation softwares were developed, we know the ties between the two areas go back further, as we can observe in analogical records from the 17th and 18th centuries. The computer culture triggers an expansion in this field and, at the same time, new challenges. One of them is the difficult task of creating choreographic “scores” that are able to record and give instructions to decode and recreate dances and/or corporal states at the same time. Besides, as the most recent researches have revealed, it is about favoring the creation of new pieces and new chorographic material.

Movement, space, time

Dulce Aquino’s A dança como tessitura do espaço (1999), is one of the only researches developed in Brazil that focuses in the relationship between dance and space, the latter being considered an element of choreographic composition. The author draws attention to space configuration as scenic representation topoi. This point of view introduces a new perspective on dance, now observed as symbolic space-time organizations. As mathematic concepts are used to compose movement series and sequences, new proposals for occupation of space are created.

We can verify different space/spatiality concepts in dance and space interfaces such as the ones found in classic ballet, modern dance and contemporary dance. Ballet is structured according to the political-philosophical-aesthetic ideas of its time, when it was a novelty. Pieces framed in the theater’s black box were in conformity with a kind of space usage that highlighted some elements, the first dancers (the king and the queen), placing more importance on them. After them, the other elements lose their value as they are placed farther away. Around the scene, some dancers organize themselves as a kind of frame: equal and synchronized, they give an impression of a collective body, without individuality and, in this case, even less important. The whole ensemble composes a hierarchic situation that organizes this dance. The space in which static ballet takes place, according to Isaac Newton’s conception. A space that is absolute, rigid, fixed and independent from time and matter. Which means it does not change with time nor with the different compositions it might have. That is why the relationships that take place there do not change it.

According to Aquino, “the scenic space of the Italian stage is the absolute space where actions are originated in a representation of the natural world under the imperative of tridimensional perspective. This representation was subject to the origins of the renaissance perspective of a linear graphic pattern. Actions develop at the forefront against a background, like a more or less realistic screen. The general composition from the spectator’s perspective corresponds to the composition of renaissance painters.”

Einstein’s relativity theory occupies a highlighted spot in the modern concept of space. In this concept, space is not separated from time, which is a collection of all instants. Spacetime would be equal to possible places and instants. In a parallel with dance, in the 1940’s chorographer Merce Cunningham and composer John Cage respectively started to think about different dances and music and how they could inhabit the same space, without a fixed hierarchy. In this proposal the stage would no longer have a center nor would it have a more important dancer than the other and the fourth wall would no longer be the reference for the front part. Any direction could be the front, which now related directly to the body of the dancer, who builds space upon himself and his dance. Pieces like Beach Birds( 1991), CRWDSPCR (1993), Biped (1999), among dozens of others, reveal this kind of organization and this way of understanding the relationship between body and space. In the case of “Biped”, dancers danced with gigantic holographs, which were the result of processes based on motion caption and computer treatments.

Since the recent scientific evolution and the development and implementation of digital and technologic devices, dance has been building new artistic experiences that draw attention to the construction of other space dimensions.

Mathematics to strangle hearts

After working with Joffrey Ballett anf the Nederlands Dans Theater, William Forsythe directed the Ballett Frankfurt between 1994 and 2004, a period in which he choreographed pieces like Limb’s Theorem” (1991), The loss of small Detail (1991),  Alie/naction (1992), Eidos:Telos (1995), Endless House (1999) and Kammer/Kammer (2000). As The Guardian’s dance critic Judith Mackrell said, that was where perfected the style she defines as “a aesthetic of perfect disorder, which radically breaks the norms of ballet, disorganizes balance axis and shatters space, encouraging dancers to violate all codes of gender, courtesy and romance”. Since 2004, the choreographer started to work on his own company, The Forsythe Company, between Dresden and Frankfurt.

The CD-ROM Improvisation Technologies – A Tool for the Analytical Dance Eye, produced by ZKM (Center for Art and Media Karlsruhe) and launched in 1999, is an important tool to sensitize the eye and cognition to the movement organization logic of this thinker of dance. Here is an example in which memory and creation get mixed up, as the CD records features of this knowledge of dance and favors repertory learning and, beyond that, it collaborates to the understanding of an operating mode that can be used in future improvisations that generate new pieces. The CD contains explanations and video demonstrations about some improvisation methods Forsythe uses to compose, described by him through graphic supports and animations, among other visualization modes.

Helena Katz, dance critic of O Estado de São Paulo, wrote about the excellent piece Eidos:Telos in the article entitled Dança usa a matemática para estrangular corações (“Dance uses mathematicsto strangle hearts”), published in 1996. “Forsythe thinks dance through algorithm (mathematical translation of information) – for that, he is unique. He doesn´t organize movements in space, he builds space with or without movement. A difficult enterprise. For him, his dancer and his audience. We all need to learn that Forsythe choreographs like someone who programs bits and not like someone who uses our bodies made of carbon as model. […] Forsythe’s head reads the world through algorithms. He stops, points to a tree losing its leaves in autumn and says: ‘look how clear it is, this is pure algorithm’. For him, the world that moves evolves because these algorithmic structures reproduce themselves in an evolutive way. And, like almost everything else that moves can be ‘algorithmable’, which means they can be translated mathematically, the mystery for him is unfolding mathematic’s way of working”.

Choreographer Anne Teresa De Keersmaeker, director of Rosas, residing in Brussels, Belgium, distinguishes herself for the way she proposes the relationship between dance and music, permeated by a special logic. Fase, four movements, produced in 1982 for Steve Reich’s music, had an explosive impact in the international dance scene. Since then, the choreographer has been creating a set of choreographies that are becoming an essential reference for contemporary art. In the article Fibonacci Fragments, Anne Teresa de Keersmaeker and music, Jean-Luc Plouvier (2002) tells that in the beginning of 1983, Anne and Thierry De Mey invented a new way to co-relate them (music and dance). The dancer quartet in chairs that can be seen in the 35mm film (commercialized in DVD) Rosas danst Rosas, from 1997, is another example of this relationship: the choreography is associated with music, which originated from a number game. Small gestures set erotic poses in motion, like covering a naked shoulder. According to Plouvier, this initial experience in the career of the young choreographer helped to synchronize a history of fascination between music and dance that her pieces keep exploring and enchanting audiences.

For the piece Achterland, from 1990, the choreographer used the Fibonacci series in the composition of choreographic phrases. Fibonacci sequence was traced by Leonardo de Pisa to describe the growth of a rabbit population: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987… The golden number (1,618…) or divine proportion is related to this series and also represents a growth constant. Between 1942 and 1948, architect Le Corbusier used the divine proportion, Fibonacci’s number and the average dimensions of the human body to build his modulor, a system of measurements used to establish harmony in architectonic projects.

The presence of algorithm concepts and the golden ration in Forsythe’s and Anne Teresa de Keersmaeker’s work, respectively, produce different aesthetics. The application of mathematical concepts in another domain has aesthetic-spatial consequences, which in its turn, demands a construction of another body. It is the presence of mathematical concepts that produces the aesthetics of these choreographers’ work. Thus, it is suitable to question how the transposition of concepts from one domain to the other operates. Which operations and transformations are implied in such transfer? How this translating operation relates to aesthetic choreographic production?

Resources

AQUINO, Dulce. A dança como tessitura do espaço. Tese de Doutorado. Programa de Pós-Graduação em Comunicação e Semiótica, da Pontifícia Universidade Católica de São Paulo, 1999.

KATZ, Helena. Dança usa matemática para estrangular corações. Jornal O Estado de São Paulo, Caderno 2, 1996.

MATOS, Adriana Perrela. Trishapensamento: o espaço como previsão metereológica. Dissertação de Mestrado. Programa de Pós-Graduação em Dança da Universidade Federal da Bahia, 2008.

PLOUVIER, Jean-Luc. Fibonacci fragments, Anne Teresa de Keersmaeker and music. Bélgica, 2002.

SANTOS, Ronaldo Bispo. Processos e estados estéticos em sistemas complexos naturais. Dissertação de mestrado. Programa de Pós-Graduação em Comunicação e Semiótica, da Pontifícia Universidade Católica de São Paulo, 1999.

Maíra Spanghero is post-doctorate candidate at Brunel University, in London.