Geometrical transformations in Bharata Natyam

At least once in a lifetime, a mathematics teacher would have faced one question from many students, what is the use of learning these concepts in my life? It’s true, all you learn inside the four walls of a school or college might not be useful in real life. But a few concepts will be definitely helpful and that’s more than enough. What if I tell you the subject from an engineering course CAED (computer added engineering drawing) is a modern version of Bharata Natyam. Our ancestors were ingenious, they created a beautiful art form Bharata Natyam with hidden mathematics and science behind it. Let's explore it more.

Ancient scientists and mathematicians saw geometry as a branch of art. So do the Bharata Natyam and the human body, both are the best work of art. The human body is an entity of geometry, it exists in a 3- dimensional plane. It can travel either parallel to the plane or perpendicular or at any specific angle. That's the birth of geometry too. Geometry exists because something is moving, imagine a world where nothing moves, there is no work of geometry there. Dance is a world of movements, so geometry is a part of the dance. Geometry exists in dance forever.

“Design and development” it’s the major keyword of all the thesis papers of technical works nowadays. Design is the first stage of the creation of new technology and product. The entire world is running based on designs of the brilliant mind. Be it the design f car or the design of the cloth. Without design, there is no product. CAED was developed with that vision in an engineering field. Hence it is introduced to the students at the very first stage of learning and technology. 

The geometrical transformations are the major part of CAED, which involves the two & three dimensional, transformations of geometrical figures.

The major transformations are:

1.   1. Reflection

2.    2. Rotation

3.    3. Translation

4.    4. Dilation

Apart from these, there are other operations are that are widely used. The aim of performing transformations of geometrical figures is to be able to view the design in all the possible angles and planes. The dance is watched by audiences from all angles, so there exists a transformation in dance naturally.

“Geometry is the archetype of the beauty of the world,” The very first chapter of Bharata Natyam classes is Adavu, which means a step. There are many groups of adavu. The first adavu is Tattu adavu which translates to, Hitting or stamping steps. The body is pushed down by applying force. The position of the human body while doing this adavu is called, Aramandi or half-sitting position.

 To represent a design on paper a designer considers two planes. Horizontal plane and vertical plane, are popularly known as HP and VP among today's engineers. These planes help in understanding the morphology of design in top view and front view and side view and sometimes a bottom view too.

Reflection is a geometrical concept implemented to perform the mirror images of the shapes and structures in order to maintain the symmetrical nature. In laymen terms, it's just a reflection of an image by 180 degrees. The reflection of the polygons can be performed either in VP or HP, but with respect to the dancer, all the transformations can be observed only horizontally. Here the image or the shape gets flipped across, and hence the transformation happens always in Y-axis.

 

An example of horizontal transformation.

 The concept of right and left exists, extensively in dance. Often, students of dance would have got corrected in classes as they flip the direction of movement, the hands are not symmetrical. This means, the length and the angle to which the body and hand are stretched and bent, is not exactly equal on the other side. Bharat Natyam expects the students to perform the symmetrical reflection whenever there is a change, in the direction of movement(legs or hands).

 

Tattu adavu: A schematic representation of reflection of legs and body

Rotation is a geometrical transformation that turns an object or body about a fixed point called the centre of rotation. A body under rotation should maintain the same size and shape, but can be turned in different directions. Rotations may be either clockwise or counterclockwise.

All dances involve the movements of dancers in different directions, except a few cases. Dancers perform the same set of actions in diagonal directions for the same lyrics. A typical order of direction is straight, right diagonal and then left diagonal. Imagine a dancer is executing the actions facing the right diagonal and making an angle of rotation of 90 degrees in the counterclockwise to face left diagonal. The process is similar to the rotation transformation of shapes in mathematics. The size and the shape of the body of the dancer are maintained the same as the rotation takes place. 

A typical example of rotation transformation is in the top view. 

Translation: A general transformation that involves the movement of an object in a straight line without changing the direction. The size and shape of the object remain the same before and after the competition of the transformation.

A rectangle ABCD moves in a straight line to become A’B’C’D’. The size and the shape of the two rectangles remain the same. A’B’C’D’ is known as the image of the object ABCD.

 

The Periya adavu is one of the most beautiful and difficult groups of steps to learn and also to execute. One of the Periya adavu involves the translation movement. The hands of the body are diagonally opposite showing Pathaka and shikara hasta respectively on right and left sides. The dancers draw an imaginary line to maintain the strict line of movement. The size of the dancer in terms of height is the same before and after the translation.

A dilation is a transformation that produces an image that is the same shape as the original but is a different size. (The image is similar to the original object). Dilation is a transformation in which each point of an object is moved along a straight line. The straight line is drawn from a fixed point called the centre of dilation.

Alarippu, the first dance lesson in Bharta Natyam, is the best example of dilatation. It begins with the movements of the hands and legs in a standing position, and the same set of movements are performed in sitting. During each stage, this height of the dancer is reduced to half of its original height, I.e. From full standing to half-sitting, from half-sitting to full sitting. The dilations are observed when the movement of half-sitting is repeated in a full sitting. 

An illustration of dilations during Alarippu.

 Geometry exists everywhere, hence the name geo and meter. The measurement of the earth. Then how can dance be an exception? Mathematics is a difficult subject, so do dance. Combining them together is the birth of a new subject. Dancometery, The geometry of dance, or the measurement of dance.