30 June 2023, I was happy to facilitate three hours B>Lab “Graph Theory Applied to Dance Making: Dancing through Euler Paths and Circuits” at the 15th BIDE (Barcelona International Dance Exchange) annual B>Meeting in one of the La Caldera Les Corts choreographic studios, in Barcelona accordingly. The B>Lab was accompanied by about 10 min long showing later in the day.
Two practices introduced to the dancers at the BIDE were created for the joint workshop paper “Puppetry, Poetry, Dance, and Sound on the Bridges of Königsberg: Embodied Work with Euler Paths and Circuits” co-written with Susan Gerofsky, S. Brackett Robertson, and Karl Schaffer. It is going to be presented at the Bridges conference on mathematical connections in art, music, architecture, and culture in Halifax, Nova Scotia, Canada on 27-31 July 2023.
To document the B>Lab, it went through the stages:
- Introducing the Seven Bridges of Königsberg math problem to the group: “Is it possible to walk about Königsberg crossing each of its seven bridges exactly once?” Actually, there are two problems here, depending on whether we are asking whether there is a walk in Königsberg that ends where it begins or whether there is a walk that ends in a land region different from the one where it begin. Respectively, we deal with Eulerian circuit or Eulerian path.
Königsberg and this problem can be represented by a multigraph (in the centre of the image below). A graph is a collection of vertices connected by edges. If there is more than one edge connecting two vertices, we deal with a multigraph.
Leonhard Euler (1707-1783) is one of the most important mathematicians. In 1735, Euler presented the first solution of the Königsberg bridge problem, showing that it was impossible. Next year, he published a paper “The Solution to a Problem Relating to the Geometry of Position”. In this paper, Euler showed that every vertex of an Eulerian graph [such graph contains an Eulerian circuit] is even and that there are exactly two odd vertices in a graph containing an Eulerian path. Here, an even/odd vertex means that the vertex has an even/odd degree, i.e. number of edges incident with the vertex is even/odd respectively. For an Eulerian path, it starts at one of the odd vertices and ends at the other one of them.
Any graph can be “Eulerised” by adding or subtracting edges. Subtracting one edge from the graph which represents the Seven Bridges problem, we can obtain an Eulerian path. Adding two edges to this graph, it results in becoming Eulerian, containing an Eulerian circuit.
2. The first dance practice. It came from the participants of the B>Lab that this practice represents traveling through the body. Here, we work with different body parts transitioning between them according to the Eulerian paths created by each of the participants themselves. We did the practice as it was described in the joint paper (link, p. 576).
3. The second dance practice. We started traveling through the original landscape of Königsberg (marked with colourful tape on the dance floor) along our Eulerian circuits (each participant made their own Eulerian circuit) and then, we danced through the ‘natural’ landscape of the room 1 (there were marks on the floor in the room) of the La Caldera Les Corts choreographic centre (see the image below) dancing along the same Eulerian circuits. Participants chose to work with one of four different substances in each area: N – lava, E – sand, S – ice, W – mud. It was not specified either you become the substance or you work with this or you step into this; explore the substance in any way. We spent a lot of time researching on what kind of movement each of us wants to use in each area. During the lab, the importance of transitions arouse, and participants decided to bring contact whenever there is more than one dancer in the area. For the final practice, we worked with the musician Angel Faraldo, who as well traveled with his sound through four rooms having nine transitions in between, he gave us some time for the transitioning.
Videos of the dancing along Eulerian circuits in the replica of the original landscape of Königsberg marked on the dance floor; we divided into two groups, so the other half could watch.
Videos of the dancing through the same Eulerian circuits using another spacing arrangements (see the figure above):
The B>Lab dancers: Alejandra Rivera, Citlali Rojas, Ekaterina (Eka) Zharinova, Esti San, Javiera Gazitua Charnes, Nuria Manzur-Wirth, Simone Cita Kieltyka, Zoë Leigh Gadd; the B>Lab musician: Angel Faraldo.
Eka Zharinova 