Vitruvian Man Baroque Dance Is there a basic mathematical principle underlying Baroque dance and its “movements of the soul” The Ancients, having taken into consid - navel. According to Pythagorean tradi - through space with their body. eration the rigorous construction of the tion, a circle represents the spiritual This interest ultimately led me to human body, elaborated all their works, “Fractal Components in the Gothic and as especially their holy temples, accord - realm and a square represents material ing to these proportions; for they found existence. Thus, according to Leonardo’s in the Baroque Architecture,” an inter - here the two principal figures without formulation, the human body represent - esting article posted on the Internet by which no project is possible: the perfec - ed the perfect marriage of spirit and mat - Nicoletta Sala of the Academy of Archi - tion of the circle, the principle of all ter. Leonardo related man to nature and tecture of Mendrisio at the University of regular bodies, and the equilateral Lugano. It takes a look at the mathemati - square.—from De divina proportione by considered the Vitruvian Man a cosmo - Luca Pacioli, illustrated by Leonardo da grafia del minor mondo (cosmography of cal proportions of Baroque architecture Vinci the microcosm). He believed the work - in relation to fractal principles. ings of the human body to be analogous Sala introduces the concept of self- similarity: SADANCER , I have experienced the to the workings of the universe. Bones art of dance to be a science of were akin to the rocks of earth. Flesh Another example of self-similarity in the A Baroque architecture is in the dome of proportions, angles, and geometry. was like the dirt or earth covering the Church of San Lorenzo (Turin, 1666- Thoughts, emotions and psychological rocky support. Blood moved like the 1680), designed by the Italian architect states are expressed by angles within the water in a river running to the sea; it Guarino Guarini (1624-1683). Guarini body as well as lines in space defined ebbed and flowed in a motion directed used the octagonal star to define the by the moving body or bodies of the by the ocean tides. bearing structure of the dome. The self- dancers. similar components are an octagon and I have found Leonardo’s thoughts an octagonal star which are repeated in When teaching historical dance, I on architecture and the human body different scales. begin by asking the dancers to envision to parallel my thinking on dance chore - Leonardo da Vinci’s Vitruvian Man and Fractal geometry and its complexity can ography, and I often ask dancers to help to introduce the new paradigm in to imitate the figure in the circle and think of architecture as they carve architecture.… This paper introduces then the figure in the square. only an approach to observe the Gothic Leonardo used both image and text to and the Baroque architecture using a express the ideas and theories of Vitru - fractal point of view. The property of the vius, a 1st-century Roman architect and self-similarity present in these two dif - ferent styles has been chosen for an aes - the author of De Architectura . He related thetic sense; in fact the Gothic and the Vitruvius’s writings on architecture to Baroque architects did not know the his own ideas on the structure of the fractal geometry, because it is a recent human body. discovery. Thus, it is possible to refer to the work as an “unintentional” use of If you open your legs so much as to the fractal geometry.” decrease your height 1/14 and spread and raise your arms till your middle fin - The concept of “unintentional” use of gers touch the level of the top of your fractal geometry made me take a deeper head you must know that the centre of look into my own experience in histori - the outspread limbs will be in the navel cal dance. I wondered if Leonardo da and the space between the legs will be an equilateral triangle. The length of a Vinci’s writings may have the seeds of man’s outspread arms is equal to his fractal theory embedded in the idea of height.—from The Notebooks of Leonar - the microcosm/macrocosm. Images of do DaVinci , Vol. 1 (Dover, 1970) nature can remind today’s dancer that, The center of Leonardo’s square is the in the 15th century and well into the pelvis. The center of his circle is the 18th centuries, art and science were 26 Winter 2012 Early Music America and Fractals By Catherine Turocy The dome of the Church of San Lorenzo (Turin, Italy) shows some self-similar components like those of fractal designs. united and both were concerned with The art of dance was an aural as well as the divine as found in nature. Truth and visual representation of the ratios beauty existed in a work of art precisely between numbers and is made clear by Domenico in the theoretical section of because it was rooted in nature. Leonar - his treatise. …one way in which do’s drawing and text for the Vitruvian Domenico represented the cosmos in Man gives dancers a visceral sense of his choreographies was by using the their own bodies as being tied to the uni - Pythagorean ratios to describe the tem - verse and to the divine order of God. poral relationship between the four misura …it was also represented in the However, it is not only the dancing choreographic patterns…. body that reflects the cosmos, but also The choreographic patterns in the the choreography devised by the dancing Renaissance and the Baroque play with master that reflects the carefully meas - symmetry, repetition, and proportion. As ured spatial patterns in proportion to an historical choreographer, I have been the music. fascinated with these patterns. As some - Jennifer Neville, in her book, The Elo - one who has danced many of the period quent Body (Indiana University Press, dance notations published in the 18th 2004) explores the dancing masters of century, I sometimes feel that the larger 15th-century Italy in relation to human - geometry of the paths traced in space by ist ideas of divine proportion. She quotes the dancing body are similar to the ener - Renaissance dancing masters referring to gy paths going through the body as it the cosmografia del minor mundo and moves on a micro level. Am I experienc - talking about the “movements of the ing a fractal model as I dance? soul.” In Giovanni-Andrea Gallini’s Johannes Kepler in his 1611 essay, book, A Treatise on The Art of Dancing On the Six Cornered Snowflake, mentions (London, 1762), I have also found refer - the “Divine Proportion” and the ences to “the movement of the soul,” “Fibonacci Sequence”: which I believe to be a short code for the “I sometimes feel that the larger It is in the likeness of this self-developing concept of cosmografia del minor mondo . geometry of the paths traced series that the faculty of propagation is, The Enlightenment certainly challenged in my opinion, manifest: and so in a in space by the dancing body beliefs of man in relation to the divine, flower the authentic flag of this faculty are similar to the energy paths but were humanist concepts so integrat - is shown, the pentagon. going through the body as it ed into the structure of dance that they Kepler’s idea of the self-developing moves on a micro level. remained covertly? series , together with Gottfried Leibniz’s Am I experiencing a fractal Neville outlines the efforts of Renais - theory of recursions, gave a hint of the model as I dance?” sance dancing masters to align dance future concept of fractals. with music and Pythagorean theory. —Catherine Turocy Domenico da Piacenza ( c. 1400- c. 1470) The essence of fractals “…devotes a substantial part of the theo - For me as a dancer, it is the self-devel - retical section of his treatise to explain - oping series that is the essence of fractals. ing how dance shared the numerical I first imagine the shape of the move - basis of music and the other mathemati - ment in the dance in my mind, and it cal arts of the quadrivium. ” The trivium quickly passes through the nerves and and the quadrivium comprised the seven muscles to manifest itself in the motion liberal arts of the Latin school; the trivi - of my body, reflecting the choreographic um consisting of grammar, logic, and structure of a published historical dance rhetoric and the quadrivium of arith - following the rules of Divine Proportion . metic, geometry, music, and astronomy. Not all choreographies are based on Dance was seen as a sister art to music. such geometrical patterns, but quadrilles For myself, I often see the dancer’s body published in the 18th century certainly as a musical instrument with its own feel tied to geometrical perfection. I hope voice. At times it blends with the rhythm a dancer/mathematician will investigate and harmony and at other times it is a the country dances and quadrilles of the solo line. I experience sound visually, as 17th and 18th centuries from the per - a dancing body that senses proportions, spective of fractals. Other disciplines like colors and textures in physical planes. architecture are already holding confer - Neville goes on to explain: ences on Baroque architecture and 28 Winter 2012 Early Music America fractals. Why not dance? The word “fractal” was first used by Benoit Mandelbrot in 1975 when he published The Fractal Geometry of Nature. His explanation of Chaos Theory is fascinating, and I do not pretend to comprehend all the principles.