Decrypting Argentina Tango and Mathematics Connection

Math, dance, and music are interlinked. According to Pythagoras, ‘’Geometry is visible in the humming of strings and music in the positioning of spheres. So, the relationship between math and musical expression was established decades ago. Every dance form is intertwined with math and tango is not an exception. 

The ballroom tango looks oriented and stylish, so it seems like the dancer’s dance for spectators. On the other hand, Argentina’s tango is different. The couple dance for themselves. The embrace between the leader and follower is close and it depends on improvising instead of rehearsed movements. 

All dance forms have some specific patterns you need to learn, so when you learn Argentina tango it feels like you are breaking tango code. You can continuously explore multiple possibilities.

It is the leader that leads and the follower picks the cue from his subtle moves. It is challenging but intimate, where the couples move like one body with four legs. It is the inspiration of the leader, rhythm, and technical abilities that work together in connecting the couple and the response of the followers to the leads. 

Argentine tango teacher always mentions oscillate, 90-degree angle, L-shape, and out of the circle. It indicates that there are permutations; symmetry, topology, and logic present spontaneously and consciously during tango dancing. Structured steps, improvisation, geometry, and technical ability have appealed to mathematicians, professors, and IT specialists towards decrypting Argentine Tango and mathematics using models like symmetry and solid 2-torus 


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In tango steps, four kinds of symmetries are embodied.

  1. Translation
  2. Reflection
  3. Rotation
  4. Step forward [represents reflection + translation]

The couples stand on a single foot. Changing foot will easily represent reflection. Translations and rotations are easy to understand. A step forward means when you change foot reflection happens simultaneously you move forward thus it is a blend of reflection and translation. This indicates that tango is intrinsically a reflectional dance form. 

Solid 2-torus with two dimensions

The solid 2-torus form is a simplified mathematical form similar to a doughnut. In tango, the arms of the couples are linked as their bodies are embraced. It represents a geometrical model solid 2-torus. In general, torus means a ring with a hole and is a concept very helpful in high dimensional space therefore it resembles the donut confectionary. The couple’s head and legs come in close contact when they dance the tango. This solid 2-torus consistently transforms and becomes solid 3-torus or 3-dimensional form or 3-manifold.

The distance and virtual position between the chest of leader and follower stays consistent throughout the dance segment. When the chest of the leader moves forward and rotates to his left then his foot also moves forward on the left. According to the Argentine tango logarithm, the cue from the chest position & twist of the leader helps the follower to determine their hip position and movement. 

With this understanding, you can consider the leader’s chest position and his twist angle as key parameters. When the couple regains their original position in the end their steps represent a loop on 3-torus called a standard continuous model. In the discrete model, the technical tango rules are stringent. The points corresponding to the music beats are only considered.