Argumentized Aramandi

Have you ever wondered what the most outstanding work of art in the world is? Is it Monalisa, The Taj mahal? Or the human body?" Human body is the most significant work of art". What would have happened? If a human had three ears, one hair, multiple legs, and one ear. I am sure it sounds like an alien straight out of a Hollywood movie. But thanks to the creator of the humans, someone had patience in designing the human body with sheer dedication and proportion. Science says the balance of the human body is due to the two ears present in the human body. Fun fact, humans are the only living animals who can change the morphology of their bodies. That's why dancing is the knowledge humans have mastered over the centuries.

Aramandi is the basic posture of Bharata Natyam dancer. It's the most used posture in Bharata Natyam, in which the dancer compresses their height to 3/4th of the original height. The word Ara means half, and mandi means knee. The compression of height is due to the bending at the knee. The reason why it is, referred to as Ara or half is the central area of discussion. Mathematically, one can see why it's called Ara or why the dancer's height stands compressed to exactly 3/4th of the original height; after all, the human body is the best proportion work in the universe.

On November 2017, a Saturday morning, I started a session by writing the headings "Area and perimeters", a part of the geometry section in mathematics. The task was to understand and estimate the perimeter and area of each given geometrical figure. The chapter expected the students to understand the change in the area as the figure transformed. The class landed on a question, which expected the students to find the change in the circle's area when it's transformed into a square. I guess the one who prepared the question was not aware of the debate that would generate in my class that fine day. The class was divided based on two opinions, one it's possible to find and another section of the class it's not possible. I loved it. I knew it was time to enlighten these young minds about the practical application of geometry than just a piece of textbook knowledge. I told the class, "stretch back because you are about to understand something very complicated. "

"Have you ever heard of a term called Squaring the circle?"

The question followed a small silence in the classroom. I told, squaring the circle is a classical geometrical problem proposed by bygone Greek-Roman geometers. It is an exigent task the involves the construction of a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. Like the class, there were two sections in the ancient mathematicians. The first group tried hard in proving the possibility, whereas the rest said it was an impossible task.

Vitruvius was a Roman author whose work on human body proportion has inspired the renaissance drawing by Leonardo DaVinci called Vitruvian man. Vitruvius explained the human body in terms of fractions and explained the measurement of the ideal human male body. According to Vitruvius, the ideal human body will have the following proportions.

1. Four fingers equal one palm

2. Four palms equal one foot

3. Six palms make one cubit

4. Four cubits equal a man's height

5. Four cubits equal one pace

6. 24 palms equal one man

In the second block of text, he describes the model body as fractions:

1. "From the roots of the hair to the bottom of the chin is the tenth of a man's height."

2. "From the bottom of the chin to the top of his head is one-eighth of his height."

3. "From the top of the breast to the top of his head will be one-sixth of a man."

4. "From the top of the breast to the roots of the hair will be the seventh part of the whole man."

5. "From the nipples to the top of the head will be the fourth part of a man."

6. "The greatest width of the shoulders contains in itself the fourth part of the man."

7. "From the elbow to the tip of the hand will be the fifth part of a man."

8. "From the elbow to the angle of the armpit will be the eighth part of the man."

9. "The whole hand will be the tenth part of the man."

10. "The beginning of the genitals marks the middle of the man."

11. "The foot is the seventh part of the man."

12. "From the sole of the foot to below the knee will be the fourth part of the man."

13. "From below the knee to the beginning of the genitals will be the fourth part of the man."

14. "The distance from the bottom of the chin to the nose and from the roots of the hair to the eyebrows is, in each case, the same, and like the ear, a third of the face".

Inspired by the details of human body proportion, Leonardo da Vinci proved the squaring or circle with the help of a diagram called Vitruvian man. It's a diagram on paper by ink, depicts a superimposed position of a man stretching his arms and legs, inscribed in a circle and square. Thereby solving the problem of squaring the circle.

 

DaVinci has combined the square and circle in such a way that, instead of keeping the square exactly in the middle of the circle, or circle in the middle of the square, he pushed the square a little down so that the base of the square touches the circle a particular point as shown. Consider the two diagrams shown below.

 

The circle's diameter is 2cm; hence the radius of the circle will be d/2 = 2/2 = 1cm. Superimposing the circle and square at the base of the square, just like Leonardo did. The side of the square measured as; side = 1.571 cm.

Perimeter of circle = 2*pi*r = 2*(22/7)*(1) = 6.28cm

Perimeter of square = 4*side = 4*1.571cm = 6.28cm

Consider the second figure, where the circle a superimposed, but the edges of the square lie outside the circle. In this case, if the circle's radius is 1cm, then the side of the square will be 1.772cm.

Area of circle = Pi*r*r  = (22/7)(1*1 ) = 3.142 cm2

Area of square = Side * Side = 1.772cm*1.772cm = 3.14 cm2

This way Leonardo DaVinci could solve the squaring of the circle, by integrating mathematics with art. What helped him most is the human body proportion by Vitruvius.

Vitastrya antaritau paadau  krutva tu chatursrakau . 

Tiryak kunchita janubhyam sthithirayath mandalam  //263 //

According to the 263rd verse of Abhinaya Darpana, the Arda mandala or ara mandi is defined as the "Standing in Chaturasra, bending the knees slightly and obliquely and keeping a distance of Vitasati between the two feet."

Another critical aspect of the ara mandi is a hunch less body posture. Every time a dancer takes the form, the teacher reminds you to keep the body erect and straight. Dancer decides their height now. The height is adjusted so that the distance from the navel to the head should be equal to the distance from the navel to the ground. Remember the 12th and 13th of proportions explained by Vitruvius!

It's interesting to note how the human body helped Leonardo DaVinci solve the problem of squaring a circle and me to understand the concept of ara mandi. 

 Observe the above figure of Vitruvian man by Leonardo DaVinci and the human body fractions by Vitruvius. The distance from the foot to the knee and the distance between the knee to genitals are precisely equal. Together they constitute precisely half of the human body. When a dancer sits in ara mandi, the knees remain expected to compress down, and the distance between the knee and the group will be 1/4th instead of half (1/2). Hence the name ara mandi. But that doesn't provide enough information to mathematically prove the meaning of the word Arai (1/2).

Consider the following two positions of the dancer, the first one is full sitting,  known as Muzhumandi. and the other one is full standing, known as Natyarambe position.

  

Consider the first diagram in which the vertical distance between the navel to the dancer's feet is d1, and the horizontal distance from one end of the knee to the other end in the second image is d2. When the dancer is standing, their exits only d1 and d2 are very small. When the dancer makes the ara mandi, the length of d1 reduces. As the dancer's height decreases and the length of d2 increases, as the knees are spread outward in the opposite direction. 

   

Carefully observe, there is the formation of Rhombus structure at the ara mandi position. The area is defined as the extent of a two-dimensional figure. Now, it was time for me to combine the ideas Leonardo gave in finding the area, Vitruvius human body proportion and the concept of ara mandi.

 

The area of the Rhombus will be:

A = 4 × area of ∆ AOB

= 4 × ½ × AO × OB sq. units

= 4 × ½ × ½ d1 × ½ d2 sq. units

= 4 × 1/8 d1 × d2 square units

= ½ × d1 × d2

Therefore, the area of a Rhombus = A = ½ × d1 × d2. The (1/2) here indicates that the area is half of the length of the lines d1 and d2. And we know half means Arai means half or ½.