Abstract
Can we use mathematics, and in particular the abstract branch of category theory, to describe some basics of dance, and to highlight structural similarities between music and dance? We first summarize recent studies between mathematics and dance, and between music and categories. Then, we extend this formalism and diagrammatic thinking style to dance.
M. Mannone is an alumna of the University of Minnesota, USA.
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Notes
- 1.
An Archimedean solid.
- 2.
Families of circles where every circle of a family intersects every circle of the other family orthogonally.
- 3.
We can define 2-categories and n-categories.
- 4.
More precisely, an abstract (and oriented) diagram is mapped into points and paths in a topological space.
- 5.
Conversely, the violent bow’s movements of strings in the movie Psycho by Hitchcock well evoke the knife hitting.
- 6.
According to [9], “in the dance studio, conscious use of breathing patterns can enhance the phrasing and expressivity in movement.”
- 7.
The category of gestures has gestures as objects and gestures of gestures (hypergestures) as the morphisms between them. The composition of hypergestures (paths) is associative up to a path of paths [16].
- 8.
From [5]: “The physical signal of dance consists of a change in position of the dancer body in space with respect to time.” We can see positions as vertices and changes of positions as arrows. Also: “movement is physically created by an infinite sequence of continuous positions in space unfolding in time.” We can represent the transition from a position to another one via an arrow representing a morphing.
- 9.
In fact, a transformation between movements requires a homotopy, and homotopy is not associative (it requires a reparametrization). However, homotopy classes are associative.
- 10.
In the case of total improvisation (music and dance), the diagram would be commutative.
- 11.
As defined in [14], a 2-functor between two categories A and B is a triple of functions that map objects, arrows, and 2-cells of A into objects, arrows, and 2-cells of B respectively, preserving “all the categorical structures.”
- 12.
Dancers can move without any (external) music if they are able to think of their pulse and to communicate with each other via touch and non-verbal indications (and the leader should be particularly clear in such indications), especially for couple dancing; but this is an unstable situation. Permutations of roles should be clearly signaled via, for example, a variation in touch or visual communication.
- 13.
For tango music, this is more complicated than a simple ‘pulse extraction.’ We can instead think of categories enriched with maps from a beat to another beat containing inner maps, that is, inner pulses. In salsa, there often are recognizable patterns with clave rhythms between strong beats. The pulse can depend on the specific music style, and it can be the object of future research in itself.
- 14.
Limits and colimits are generalizations of products and coproducts, respectively; they are obtained the ones from the others via reversing arrows [12].
- 15.
An ellipse is used for the center of attention in [23]. The ellipsoid of Fig. 2 is a diagram connecting different elements. If the listener/audience recovers the pulse contained in conducting gestures, the two extremities of the ellipsoid can be joined, transforming it into a torus with a section collapsed into a point.
- 16.
A connected category J is a category where, for each couple of objects \(j, k\in J\), there is a finite sequence of objects \(j_0,j_1,...,j_{2n}\) connecting them, that is: \(j=j_0\rightarrow j_1 \leftarrow j_2 \rightarrow ... \rightarrow j_{2n-1} \leftarrow j_{2n}=k\), where both directions are allowed [14]. Here, the morphisms are the arrows \(f_i:j_i\rightarrow j_{i+1}\) or \(f_i:j_{i+1}\rightarrow j_{i}\).
- 17.
The definition of beauty is beyond the scope of this paper, and it would start a philosophical debate. We can just say that a mixture of symmetry, balance, proportion, and smoothness of movements can be overall thought of and mathematically investigated as ‘beauty’ in dance.
- 18.
- 19.
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Acknowledgments
The authors are grateful to the mathematician, musician, and tango dancer Emmanuel Amiot for his helpful suggestions.
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Mannone, M., Turchet, L. (2019). Shall We (Math and) Dance?. In: Montiel, M., Gomez-Martin, F., Agustín-Aquino, O.A. (eds) Mathematics and Computation in Music. MCM 2019. Lecture Notes in Computer Science(), vol 11502. Springer, Cham. https://doi.org/10.1007/978-3-030-21392-3_7
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