Can i say there are infinite number of corners in a circle?

Question

There are 3 corners In an equlateral triangle, 4 corners in a square, 5 corners in a pentagaon, .... 8 corners in octagon. Like that if I increase no.of sides in the polygon, the no.of corners also increase. But when the no.of sides in the polygon reaches infinite, the polygon looks like a circle and hence there are infinite corners in a circle.

Answer 1

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yes Chaos' lil bro Order09-17-2006, 02:09 AM

A thought occurs...

In a world devoid of curved lines, can we create a circle using only straight rigid lines? If we take a square as our starting block, then use 4 straight lines to cut off its 4 corners at 45 degree angles, we now have a perfect octagon. If we now take 8 straight lines and use them to cut off each of the 8 octagonal corners, we now have a 16 sided shape. You can see where this is going... If we continue this process of using straight lines to 'shave' off the corners of our shape, it become more and more circular. In fact, after only a few sets of 'shavings' the shape becomes a circle for all subjective purposes.

One caveat would be that, even though we start to see this shape subjectively as a circle, we know that it is not because upon detailed measurement we would see that its radius to circumference ratios would only be true for a very small number of the shape's radii, vs. the shape's total radii. My question arises. If we can continue shaving the corners of this circle-like shape to the nth degree, our shape becomes more and more circular, but never reaches the shape of a true circle as defined by the radius-circumference ratio requirement. But, how can we know for certain that all circles as we know them, are not these so-called 'squares shaved to the nth degree?' After say, 1 million sets of shaving, the corners of this circle-like shape would be spaced so closely together that we would not have a measuring device capable of measuring all the radii sandwiched inbetween each of these respective corners. So then, how can we logically prove that our 'square shaved to the nth degree' is in fact not a circle?

This writer thinks we cannot. Please prove me wrong!

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Galileo09-17-2006, 04:26 AM
If you don't have a circle, then not all points will have the same distance from the center.
For any nth degree approximation you mentioned, it will simply not be a circle by the definition of a circle.


The idea you mentioned is great for approximating circles though and can be used to prove the area of a disc is \pi r^2. Archimedes used the same idea (http://www.ugrad.math.ubc.ca/coursedoc/math101/notes/integration/archimedes.html).

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Chaos' lil bro Order09-18-2006, 06:08 AM
Yes but we can never know if any circle is a perfect circle because of the limitations we have in measuring each tiny radii. For example, if we consider a circle with a trillion trillion trillion spokes, we cannot measure the radii between 2 adjacent spokes. Therefore, this tangent could be flat and not curved and we could never know.

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Chaos' lil bro Order09-18-2006, 06:30 AM
Let me further illustrate my point will the attached diagram that i drew:

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radou09-18-2006, 06:43 AM
This is practically equivalent to the diagonal paradox. Read this thread: http://www.physicsforums.com/showthread.php?t=80796 Matt Grime's first reply explanes everything.

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Galileo09-18-2006, 08:18 AM
Yes but we can never know if any circle is a perfect circle because of the limitations we have in measuring each tiny radii. For example, if we consider a circle with a trillion trillion trillion spokes, we cannot measure the radii between 2 adjacent spokes. Therefore, this tangent could be flat and not curved and we could never know.
You give me a regular n-sided polygon (n is very large) and I`ll give you two points on it having different distances to the center.

The point is, circles are abstract mathematical objects and don't have to correspond to objects in physical reality.
If I ask you what a circle is, you could draw one on a blackboard. But no mathemetician would be foolish enough to define a circle as a bunch of chalk particles on a blackboard.
Do you get what I`m trying to say?

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HallsofIvy09-18-2006, 09:52 AM
You said, to begin with that your world "has no curves". Given that, it cannot contain a circle. Polygons that are arbitrarily close to a circle, yes, but no true circle.

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Data09-18-2006, 10:42 AM
Yes but we can never know if any circle is a perfect circle because of the limitations we have in measuring each tiny radii. For example, if we consider a circle with a trillion trillion trillion spokes, we cannot measure the radii between 2 adjacent spokes. Therefore, this tangent could be flat and not curved and we could never know.

That's true!

But mathematically, circles are circles, and an n-sided approximation for a circle is not a circle. It's certainly true that it's impossible for us to draw a perfect circle. But that doesn't make any difference to mathematics.

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DaveC42691309-18-2006, 11:06 AM
I don't know how many decimal places pi has been calculated to so far but at some point, your n-gon's perimeter will diverge from pi. It might be the trillionth decimal place.

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net_nubie09-18-2006, 11:22 AM
But mathematically, circles are circles, and an n-sided approximation for a circle is not a circle. It's certainly true that it's impossible for us to draw a perfect circle. But that doesn't make any difference to mathematics.
You are missing a point here. Entire calculus is based on approximations.
The goal of mathematics is to solve problems and in doing so some steps are required to be taken. In Archimedes' method for finding the area of a circle, the circle has to become a polygon of infinite sides, but to prove that all diameters of a circle have the same length, we need to fall back to the original definition i.e. it is a locus of points equidistant from a given point. Both the definitions of a circle are correct. We can choose either depending upon which is more convenient.

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net_nubie09-18-2006, 11:28 AM
But mathematically, circles are circles, and an n-sided approximation for a circle is not a circle. It's certainly true that it's impossible for us to draw a perfect circle. But that doesn't make any difference to mathematics.
You are missing a point here. Entire calculus is based on approximations.
The goal of mathematics is to solve problems and in doing so some steps are required to be taken. In Archimedes' method for finding the area of a circle, the circle has to become a polygon of infinite sides, but to prove that all diameters of a circle have the same length, we need to fall back to the original definition i.e. it is a locus of points equidistant from a given point. Both the definitions of a circle are correct. We can choose either depending upon which is more convenient.

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radou09-18-2006, 11:44 AM
You are missing a point here. Entire calculus is based on approximations.
The goal of mathematics is to solve problems and in doing so some steps are required to be taken. In Archimedes' method for finding the area of a circle, the circle has to become a polygon of infinite sides, but to prove that all diameters of a circle have the same length, we need to fall back to the original definition i.e. it is a locus of points equidistant from a given point. Both the definitions of a circle are correct. We can choose either depending upon which is more convenient.

There is a difference between a infinite countable set and an uncountable one. So, if both definitions are correct, we could, at some point, consider the cardinality of the set of natural numbers and the cardinality of the continuum to be equal. Sounds like rubbish, doesn't it? :biggrin:

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Data09-18-2006, 01:10 PM
In Archimedes' method for finding the area of a circle, the circle has to become a polygon of infinite sides, but to prove that all diameters of a circle have the same length, we need to fall back to the original definition i.e. it is a locus of points equidistant from a given point. Both the definitions of a circle are correct. We can choose either depending upon which is more convenient.

A "polygon of infinite sides" is not a definition for anything. It's meaningless. Archimedes' argument can be made rigorous using limits. It doesn't need a different definition for what a circle is.

Entire calculus is based on approximations.

Give me an example of what in calculus is "based on approximations."

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HallsofIvy09-18-2006, 03:41 PM
You are missing a point here. Entire calculus is based on approximations.
The goal of mathematics is to solve problems and in doing so some steps are required to be taken. In Archimedes' method for finding the area of a circle, the circle has to become a polygon of infinite sides, but to prove that all diameters of a circle have the same length, we need to fall back to the original definition i.e. it is a locus of points equidistant from a given point. Both the definitions of a circle are correct. We can choose either depending upon which is more convenient.

Twice you posted this! It's unfortunjate that it is completely untrue. "Calculus" (which is what I think you mean by "entire calculus") is not based on approximations. You may be confusing the "difference quotient" which can have many different, approximately the same, values with the derivative itself which is a limit of difference quotients and is exact.

Archimedes never "defined" a circle as "a polygon of infinite sides"- he did show that (a primitive form of) the limit of the areas of such polygons was the area of the circle. Since even in modern mathematics, there is no such thing as "a polygon with infinite sides" no, you cannot "choose either".

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net_nubie09-19-2006, 10:13 AM
sorry guys. and thanks. it got posted twice 'cause my connection got cut-off when i clicked submit the first time.

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Robokapp09-19-2006, 01:17 PM
area of a circle is \pi r^{2}
area of a regulated convex polygon is S*a where S=semiperimeter and a is apothem.

a=\sqrt{r^{2} - \frac{l^{2}} {4}}

where l is the side and r is radius of circle that polygon is inscribed in.

the higher the number of sides, the shorter the sides...r is a constant because we're inscribing this...let's say in a Unit Circle... so it's obvious the more sides, the shorter, so the \frac{l^{2}} {4} decreases, leaving a closer and closer to \sqrt{r^{2}}.
bkoz

Answer 2

Statements about infinity are always dodgy.

It's a good approximation to say that a circle is a polygon with an infinite number of corners. Technically, a circle is the limit of a many sided regular polygon as the number of sides (and number of corners) increases towards infinity.

But it is only an approximation, in that a circle does not have any corners or straight sides.

Answer 3

listen if u see a circle very carefully u will see that what u were saying is a truth but if u see it fram a distance u will see that it is a circle and it has no corners but if u say that u r right then why r u asking this question in yahoo answers,ask it in the front of whole world make it a theory

Answer 4

Circle is a set of points which are equidistant from a fixed point called center of circle. Hence it is a set of infinite points. Each point can be treated as corner w.r.t. other points.

Answer 5

yes, you are right. You can say it makes up of infinite number of triangles with the centre of the circle as one of the vertex and the other two on the circumference.

Answer 6

yes,u can say there are infinite number of corners in a circle.

Answer 7

ye of course you can say. as triangle is made with the help of joining three points ,square with four points and so on ..........
simmilarly cirle are made with joining infinite no of point hence infinite corners

Answer 8

yes absolutely as it's approaching to unlimited, then u can say that a circle is a polygon consisting of unlimited corners


trust me!!!

Answer 9

Those are not exactly corners. But if the examples given by u are taken into consideration, we can say that.....

Answer 10

Yes of course

Answer 11

yes,you can draw infinte corner in asingle circle