Am here posting my theory on sides of a circle..

So, how many sides a circle has? They say a circle has infinite number of sides. But i dont think it could hold..

We all know that the

perimeter of a figure = (number of sides of that figure) x (length of one side).------------------ FORMULA (1)

this formula holds for symmetric figures like squares, pentagons, hexagons, etc..

(N- number of sides of a figure; L- length of one side; C-circumference; r- radius of a circle). I wil use these symbols to represent those terms from now on..

The circumference of a circle is nothing but its perimeter. Now, if the circle has infinite no. of sides, then according to formula 1,

C= (infinity) x ( L).............................. (2)

Now, according to equation 2, if the length of one side of circle is, for example say, 1 unit, then

C= (infinite) x (1) = infinite

By this, it means that the circumference is infinite, but its not the reality.. we actually know that circles' circumference is finite and limited..

Now, if we argue that a circle's locus changes direction at every point that it could not have a well defined length for its side,then L=0,

substituting this L=0 in eq. 2, then,

C= (infinite) x (0) = 0

By this, it means that there is no circumference and it suggests that a circle wont exist, but merely a point could exist.. but we all see many circles, they exist...

So, whats actually going on with them???

My idea is that the no. of sides of a circle is not infinite, and the length of each side is not equal to 0..

rather, i would say the no. of sides approach infinity but not reached infinity... and the length of each side approaches 0, but do not reach 0.

If a person is asked to jog on a square field, he would initially jog on a square path, but as it goes on, he would finally finish with jogging on a circular path..

what he has done in mean time is that he has divided each side of a square into many sides(while running) and it goes on like that to form a circular path..

similarly, you can make a square into a circle by keep on dividing the sides into many small sides, each oriented in a different direction.. u keep working and working and you could still be able to divide any small side into many smaller sides, and again each in a different direction.. but you could do them only with difficulty as it keeps going.. By this work, the no. of sides increase and approach infinity and at the same time, the length of each side decreases and approaches 0, but it doesn't reach 0. hence, from equation 1,

C= (N approaching positive side of infinity) x (L approaching 0)

ie., C= (N---> +ve infinity) x (L---> 0)................................................ ........ eq (3)

By equation 3, the very large N value gets multiplied with very small L value and it now gives a finite number.. at last, the circumference becomes a finite number...!!

but... by the time i thought that the problem is solved, its really not yet.. Anyone could argue that there are many circles of varying sizes and hence with varying circumferences, but the eq 3 doesnt seem to care about the radius of a circle...( the usual circumference formula = 2 x pi x r). The value obtained from eq 3 is independent of the radius of a circle. and hence needs further modification..

We have to include a new factor called time..

When you are provided with two squares, one is small and other is big, and if you are asked to convert both into circles,

In the small square, if you start working, initially you could make many sides, but at one point of time, you would find difficulty in dividing small sides into still smaller sides. by that time your speed of conversion decreases, and hence the N increases slowly after that and hence the N approaches infinity slowly. but since it is a small square, the length of each side decreases quickly, and hence L approaches 0 quickly. therefore,

small circle C= (N slowly ---> +ve infinity) x ( L quickly ---> 0)

= (N slowly getting bigger) x ( L quickly getting smaller)

= small

In the big square, if you start working, even after some time, you could be able to divide sides into smaller sides since it is bigger compared to previous square. Hence, your speed of conversion remains at a higher rate, and so the N approaches infinity quickly. since it is a bigger square, L decreases slowly. therefore,

bigger circle's C= (N quickly ---> +ve infinity) x (L slowly ---> 0)

=(N quickly getting bigger) x (L slowly getting smaller)

= big

By this way, the radius is also involved finally.. And this is a theory probing into the circle's sides... And thats it about my theory... May be, i still have much of work left behind.. And i hope i would continue..

The locus of circle still makes me puzzle.. how could such a shape dominate the universe like this.. Not only the geometry of a circle, but the circular path which is traced by many objects still needs research.. but am a medical student.. and i hope a good mathematician could go on further... And finally... the mystery of circles could exist as far as mankind would..!! **** your suggestions and comments are welcome********