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## Mathematical Modelling of Steps in BharathaNatyam

Mathematical modeling involves creating a model of a real-world system using mathematical techniques such as linear programming, differential equations, etc. When the system model has inherent uncertainty, simulation is used in addition to the mathematical model to represent a stationary or dynamic system. (System in motion).

Adavus in BharathaNatyam (South Indian classical dance art form) represent a set of steps that do not involve expression (nrityam). Thus, Adavus can be studied using mathematical models.

Tattu Adavu involves lifting the feet up and down so that the sound of tapping can be heard.

“Sollukattu” (Tamil word translated into English as verbal pronunciation of beats) is performed in different tenses. There are also repeated foot movements on various counts such as 4, 6 and 8.

The four verbal beats can be pronounced as tai,ya, tai,hi. If the four verbal beats occur at T(1), T(2), T(3) and T(4) where T(I) is the ith time instant at which the verbal beat is pronounced by the accompanying artist.

The speed or tempo is given by T(2) – T(1) T(3) – T(2) and T(4) – T(3). Ideally, all these time intervals should be equal. It can be the same if these beats are generated by machine. But when an artist plays these sounds or beats, the intervals will not be uniform and will vary randomly. These variations can be captured by simulation models.

If the whole step of the up and down movement of the feet once takes 30 seconds (for example) at normal speed. It would take 20 seconds and 10 seconds in the second and third times. For example, if tai occurs at instant 0, ya occurs at 13.5 seconds, tai is the wait time for 3 seconds, and hi occurs at 30 seconds, the upward movement of the feet lasts 13.5 seconds and the downward movement takes 13.5 seconds. and the cooldown lasts 3 seconds. A dancer and a vocalist cannot move as precisely as uniformly as the mathematical model shows and there can be variations.

Dancer or artist movement can be modeled by torso position in space or x,y,zi coordinates relative movement of feet, legs, upper hand, lower hand, arms, head, neck and eyes with respect on the torso

For a Tattu Adavu step sequence beginning at time t = 0 and ending at time t = T, the foot equation at an instantaneous time t is given by the position of the dancer’s torso and the relative position of the feet with respect to the Torso.

Since Tattu Adavu involves touching the feet and moving upward, the resulting motion of, say, the toes can be modeled algebraically using the following discrete time equations that result in step functions that describe the motion. Differential equations cannot be used as they would represent a continuous system.

So writing these equations of the Tattu Adavu as y = 0 at = 0 y = hat = T/2 iy = 0 at = T where T is the time period of one beat ih is the maximum height reached. by a foot This can be fixed at 30 cms or can be varied between 25 cms and 50 cms. This is the algebraic model of the 1st Tattu Adavu. If a variational model is to be used, then the algebraic model used should be replaced by a simulation model.

The second tattu adavu or tapping of the feet with twice per beat can be modeled as y =0 at=0 y = hat = T/4; y = 0 at=T/2; y=hat = 3T/4; y= 0 at = T.

If the feet’s locus is plotted for more points along the time interval, the same equation can be described as y = 0 at= 0; y = h/10 it= T/10; y = h/9 at = T/9 etc.

A dancer with natural movement will not be able to replicate the exact mathematical congruence of the height achieved by the moving feet with respect to the divisions within the time frame of the Sollukattu.

If one represents the actual movement of a dancer’s feet while performing the ‘tattu adavu’ (translated into English as tap of the feet), the resulting equation would be h = 0 at = 0, y = 0.6 hat = T/2 ih = 1.1h at = T etc.

These algebraic equations can be used to write computer programs that use graphics to model the movement of a classical dancer’s feet. Therefore, some aspects of mechanical steps or adavus can be generated automatically from the use of suitable models to capture the movement of the feet.

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