# Thread: Sides of a Circle- the mystery solved or unsolved?

1. 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********

2.

3. Originally Posted by arun karthik
Am here posting my theory on sides of a circle..
So, how many sides a circle has?
How are you defining 'side'?

4. side- any of the flat surfaces or edges of a solid or a figure

5. Originally Posted by arun karthik
side- any of the flat surfaces or edges of a solid or a figure
Then a circle has 0 (zero) sides; a circle doesn't have any flat surfaces.

6. You should not post this in multiple threads.

7. Infact, circle's sides approaches to be infinite, which makes it appear as though it doesnt have one.. Consider our planet earth.. from outside, it is a sphere, but on its surface, it looks as though it is a flat surface.. the more bigger the circle, the better that you can appreciate this flat surface..

8. it is mathematics and it is a hypothesis and it could be a general discussion.. so, is ther anything wrong.. am doing this mostly to get comments of people of different views.. so, some new talents could speak on this.. some new ideas.. so on..

9. 0*infinity is not zero (if it is an uncountable infinity), it is an indeterminate form. This invalidates a huge proportion of your argument. Also, the number of sides of a figure, if it is defined, is a fixed value. It can't "approach" things. Technically a circle does not have any sides, although it can be treated as a regular polygon with an infinite number of sides. This can be shown by noting that there cannot be three collinear points on a circle. Suppose there were three collinear points equidistance from a circle, say, C, D, and E all equidistant from A with C, D, and E being collinear with D between C and E. (draw a diagram). Since AC = AD = AE, then triangle ACD must be isosceles, and triangle ADE is also isosceles. We know that the base angles of an isosceles triangle are always congruent, acute angles. We also know that angles CDA and EDA are supplementary since they make a linear pair. However, these are base angles of the isosceles triangle they make, meaning they must both be acute, and two acute angles can never add up to 180 degrees. This is a contradiction, and thus there cannot exist three collinear points on a circle. This means there cannot be any straight line segments, so by the formal definition of sides, there are none. Although, on the other hand, if you take the limit as the number of sides of a regular polygon approaches infinity, you get a circle and you don't lead to any paradoxes.

10. Originally Posted by arun karthik
it is mathematics
You need to understand the concept of limits before you can make a claim like that.

A circle doesn't have any number of sides. However, it can be approximated (as closely as you want) by a polygon of N sides of length L as N tends to infinity and N tends to zero. In the limit, this will be a circle with circumference of L * N. Obviously, this doesn't work for N = or N = 0; which is why you need to use limits.

11. Originally Posted by arun karthik
Infact, circle's sides approaches to be infinite, which makes it appear as though it doesnt have one..
No. A circle has no flat surface - therefore no sides.
That is from your own definition.

Originally Posted by arun karthik
Consider our planet earth.. from outside, it is a sphere, but on its surface, it looks as though it is a flat surface..
It doesn't matter how it appears.
A globe does not have a flat surface.
Optical illusions cannot be used to support mathematics.

12. actually however large a number, if it is multiplied by 0, its nothing.. k, as you said, if 0*infinity is indeterminate, then it means you cant predict the outcome.. but a circumference is a finite, limited value which could be defined.. Nextly, assume a small circular wheel standing on a flat floor, it may cast just a point as its impression on the floor.. but if the circular wheel is very large and if this wheel is standing on a flat floor, it should cast an impression of atleast a very small line, which defines a side.. in a circle, i guess every point as a side, because the sides are very numerous that it almost never shows off a flat surface.., and those numerous sides are oriented in a different direction each.. it almost bends at every point..

13. And yeah, that doesnt work for N= infinity and L=0. but am stressing N is close but not equal to infinity and L is close but not equal to o... so, those values are close on both extremes(but not the extremes themselves).. which by multiply, they always give a finite number, which is the circumference..

14. Originally Posted by arun karthik
Nextly, assume a small circular wheel standing on a flat floor, it may cast just a point as its impression on the floor.. but if the circular wheel is very large and if this wheel is standing on a flat floor, it should cast an impression of atleast a very small line, which defines a side.. in a circle
You are confusing an imperfect construction (the wheel) with a perfect geometrical shape (the circle).

Imperfect wheels have many flat sides.
Perfect circles have no sides.

15. Originally Posted by arun karthik
And yeah, that doesnt work for N= infinity and L=0. but am stressing N is close but not equal to infinity and L is close but not equal to o... so, those values are close on both extremes(but not the extremes themselves).. which by multiply, they always give a finite number, which is the circumference..
Exactly. That is how limits work: you can get as close to a circle as you want, but never actually use the values infinity (which isn't a number) or zero. So you can work out what the circumference will be, in the limit, and not surprisingly it is the same as the circumference of a circle.

16. i would agree with your point that a wheel is an imperfect construction.. but by wheel, i mean a circle.. anyway, so what is your idea about a circle.. its bent at every possible direction that could exist... almost at every point, you could draw a tangent and that each tangent indicates a direction... and how come a constant value 'pi' is characteristic of a circle... circle circle circle... there's a lot about it.. and which puzzles..

17. i really have to thank u guys for a decent valid discussion.. and would expect many more ideas and comments..

18. I guess I accidentally posted the same thing twice in posts #8 and 9. Sorry for this, I thought that it originally didn't post.

19. Originally Posted by arun karthik
i would agree with your point that a wheel is an imperfect construction.. but by wheel, i mean a circle..
But that simply makes your analogy flawed.

Originally Posted by arun karthik
anyway, so what is your idea about a circle.. its bent at every possible direction that could exist...
That pretty much sums it up, yes.

Originally Posted by arun karthik
almost at every point, you could draw a tangent and that each tangent indicates a direction...
Almost at every point....

Originally Posted by arun karthik
and how come a constant value 'pi' is characteristic of a circle... circle circle circle... there's a lot about it.. and which puzzles..
I expect it is simply an emergent property of the mathematical construction of a circle.

20. I would like to stress here.. Firstly, a geometric shape- circle is more of a quality.. but in reality we deal them with two wings.. quality and quantity. a circle's trace, however big or small, always changes direction at every possible point.. and however big or small, the 'pi' value holds.. this is the quality of a circle.. Coming to quantity, it is the radius which is the quantifying part.. the bigger the radius of a circle, the larger its circumference is, but smaller does the curvature.. on the other hand. the smaller a circle, its curvature is bigger.. so as a bigger circle shrinks into a smaller one, its much more bent at every point.. and u should find difficulty allocating its directions.. that the SPACE around it, is much more bent compared to the space around a bigger circle..

Secondly, Mathematics is mathematics, only if it is applicable in real life.. If it could only theoretically exist, then i dont think math is useful.. the notion of maths is to give a baseline framework upon which physics and everyday living could depend on.. yes, the wheel is an imperfect construction and the circle is a perfect geometric shape.. However, the difference between reality and geometry is insignificant, that you really ought to consider both apparently equivalent.. and really you cant deny that, in reality many constructions and even in nature, circles (or wheels whatever you call) exist.. and for them we still really apply the formulae and properties of a circle, and we dont bother the tiny flaws that could not resemble the geometry..

Thirdly, no theory is correct in this world.. really, we come across many theories, but how they are'nt true..! yes, human could only propose theories, they are not real, but they are provisional, in the sense that it is only a hypothesis and you can never prove it. They are just a model, by which all our observations are accurately described and supported. the time the observations disagree the predictions of a theory, then that theory must be sent to trash.. so u could keep proposing and accepting that theory until it is proved false by a disagreeing observation. Here i would give an example of two great theories of physics.. one is quantum mechanics and the other is relativity.. They individually support many observations each, but contradict themselves.. Just for this reason, v dont ignore these theories.. they are competent on their part. and scientists are trying to form an unified theory that could satisfy both quantum mechanics and relativity.. Till that, they would remain as partial theories and would continue to be used..!

21. Originally Posted by arun karthik
Secondly, Mathematics is mathematics, only if it is applicable in real life..
You haven't heard of "pure mathematics" then?

Thirdly, no theory is correct in this world..
Mathematics is pretty much the only area where we can prove things to be correct, or otherwise.

I would seriously recommend you study some basic math.

22. That sounds like you trying to equivocate about what a circle is.

You need to decide what you are talking about: either a physical wheel or a mathematical circle.
You can't pick and choose different aspects from each.

23. pure mathematics, once used to describe only abstract things, now practically used in ares like navigation, and so on.. and s, in math either it s true or false.. in the point no:3 of my last reply, i deviated towards physics, rather than sticking with math.. and i admit that..

24. Originally Posted by arun karthik
pure mathematics, once used to describe only abstract things, now practically used in ares like navigation, and so on..
But pure mathematics still exists as an important discipline.

And, in case you are wondering why pure math is valuable, we need to keep the mathematicians off the street for their own good:
http://xkcd.com/356/

25. Friends, i have posted my another theory on circular paths in the physics page... In fact, this theory is one of my oldest, at least good that i post it now... it is titled " Circle again!!- the nature's trace??" in the physics page.. please do visit it.. and i really expect some good discussion... thank you all...

26. Originally Posted by Strange
Originally Posted by arun karthik
pure mathematics, once used to describe only abstract things, now practically used in ares like navigation, and so on..
But pure mathematics still exists as an important discipline.

And, in case you are wondering why pure math is valuable, we need to keep the mathematicians off the street for their own good:
xkcd: Nerd Sniping
Hmmm...resistance is not only futile, it's hard. But here's the answer (well, except for the actual evaluation of the integrals...), normalized to one ohm for the resistors in the grid:

Just trying to keep mathematicians from getting run over...

27. arun karthik:
Also in other areas, one deals with theories that are (at least for now) completely disconnected from reality. I'm thinking of String Theory in Theoretical Physics. For decades, it yielded results that could not be verified by experiment. The thing is that you can't keep people from pursuing things that they are interested in, be it that it has direct relevance to reality, or not.

And Strange actually stated the relevant thing here: Limits. A lot of mathematics depends on it. In many cases, you can't handle "infinity" as such. That's why you often have -arguments and need to take limits. The right words would be "arbitrarily big" and "arbitrarily small" and paying attention of how taking limits. It is probably a purely semantic thing to take as equal or not the two statements "This IS a circle" and "This is an object which, with an arbitrary degree of accuracy, approaches a circle".

28. Originally Posted by arun karthik
Friends, i have posted my another theory on circular paths in the physics page... In fact, this theory is one of my oldest, at least good that i post it now... it is titled " Circle again!!- the nature's trace??" in the physics page.. please do visit it.. and i really expect some good discussion... thank you all...
give link plz. you sound interesting

29. In mathematics, a point has no dimensions i.e. has no size.

A flat surface would occur between two separate points, a set distance apart.

A circle is the set of points in a plane that are a fixed distance from a specific point in the plane.

If you pick two points in the plane to define your flat surface, I can always pick a point, using more decimal precision than you, somewhere between the points you choose, that will lie on the locus of the defined circle.

In this manner, you will not be able to define a flat surface on the circle that someone cannot 'curve' by selecting a specific point that lies between them.

30. Originally Posted by NoCoolAvatar
In mathematics, a point has no dimensions i.e. has no size.

A flat surface would occur between two separate points, a set distance apart.
That would be a line, not a surface?

31. Originally Posted by Strange
Originally Posted by NoCoolAvatar
In mathematics, a point has no dimensions i.e. has no size.

A flat surface would occur between two separate points, a set distance apart.
That would be a line, not a surface?
Exactly. As you point out, Strange, surface does not exist on a circle. I got spirited away with the OP and typo'd.

Love the Alan Moore quote BTW.