infinite or none? does a circle has infinite corners or none? 5

[Spoonful]

Hi All:

infinite or none? does a circle has infinite corners or none?

I guess this could be a interesting or pointless discussion?

Can we say a shape with infinite number of corners, if it is not infinitely large, it has to be a circle? Then it becomes have no corners?

If true, how can one from linearly increasing number of certain property (in this case corners) to become none of that property?

 

[IDS]

Ho-hum
Watching grass grow is more interesting than this oversubscribed boring exercise in futility.
Why don't we let it go?

Why do people who have no interest in a topic take the time to open it, read through it, and then post an insulting response?

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/

 

[patprimmer]

Because we come here expecting a certain std of discussion. We often cannot tell if a thread is pointless, boring or inappropriate before we open it and read at least a little. By then some of our time has been wasted on stuff that should not really be here.

If this where in the pub it could be appropriate as the pub is kinda for this type of thread, but this philosophical non engineering really does not belong in a serious forum, well at least in my view, so people do have a right to complain and even red flag it.

Regards
Pat
See FAQ731-376 for tips on use of eng-tips by professional engineers &

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[Spoonful]

Hi All,

Thank for all the response.
I agree this question was a bit out of the scope of this forum, but I thought here I can get some intelligent response from some intelligent engineers. (even tho I do agree this topic is kinda of pointless of what we doing)

I thought my original question was more philosophic rather than mathematical. I was trying to emphasise on the question of how come when you increase something to infinite, at the end you could get zero(none).

When I say corner, I didn't mean corner by its narrow meaning, what I mean is more like a vertex formed by two (tangent lines). Starting with a basic shape with 3 corners(edges), a triangle, by increase the number of corners(edges) we get rectangle and so on. when your number of corners(edges) reaches infinite, geit becomes a circle, where all your increased number of your geometrical characteristic (corner,edges) has all gone.

Or perhaps you could never call a circle formed in such a way it a "true" circle?

To sum up, the question was, if something has a characteristic of A, and it has got so many of this characteristic, to an extend of infinite amount, can we say it has got none (or almost none) of that characteristic

 

[IRstuff]

OK, now that really sounds like a pointless exercise. Consider a tablespoon of salt; dissolve in a cup of water and the water is quite salty. Dissolve it in Lake Tahoe and you can't even tell if anything has changed. So what? Dilution is hardly a complicated or mysterious process.

Consider your many sided polygon with 10000 vertices; that makes the interior angles equal to 180-360/10000 = 179.964° Not much of "corner" already, and still a long ways from infinity. You'd be hard pressed to tell that it wasn't just a straight line. Again, so what? There is nothing magical or inherent about the "corners" in this problem. The "corners" of a triangle don't really have anything to do with the corners of a square. They didn't morph into the corners of the square. From a topological math perspective, all polygons are essentially the same closed shape, like a rubber band. I think you have too much time on your hands ;-)

TTFN
faq731-376

 

[Walterke]

how is kate's pregancy ? (wtf cares ??)

The Australians

(too soon?)

NX 7.5.5.4 with Teamcenter 8 on win7 64
Intel Xeon @3.2GHz
8GB RAM
Nvidia Quadro 2000

 

[rb1957]

not nice ! i'm sure the DJs thought they were having a bit of joke and didn't foresee the unintended.

as for "if something has a characteristic of A, and it has got so many of this characteristic, to an extend of infinite amount, can we say it has got none (or almost none) of that characteristic" i agree with IR ...

 

[TheTick]

Circles are not polygons.

 

[rb1957]

this is the song that never ends ...

 

[Cockroach]

A circle is a polygon of infinite sides.

Regards,
Cockroach

 

[Cockroach]

Okay, enough of this. Here is the definitive proof that a circle is a polygon of infinite sides.

I start with a equi-sided polygon, the distance from the centroid to the vertex being 0.50000 units. Clearly I am referring to a equilateral triangle which is inscribed within a circle of radius 0.50000 units. I ask the question, "What is the perimeter of this figure?". Using the Law of Sines and noting that the sum of angles equals 180 degrees in a triangle I get a side length of 0.86603 units. So the perimeter around the equilateral triangle is three times this amount or 3 X 0.86603 units = 2.59808 units. Because I know that the circumference of a circle is PI times diameter, the value of PI is simply the perimeter of a polygon divided by twice the radius which is 0.50000 units. In other words, the value of PI is the perimeter of the polygon divided by "1.0" or my polygon perimeter, 2.59808. I claim this is the first approximation to PI.

Increase the number of sides by one. I am now referring to a square which would be inscribed within a circle of radius 0.50000 units. I ask the question, "What is the perimeter of this figure?". Using Pythagorous Theorem and noting any two sides are of equal length, I get a side length of 0.70711 units. The perimeter of the square is therefore 4 X 0.70711 = 2.82843 units. Therefore following the logic above, PI equals the perimeter of the polygon divided by "1.0", the second approximation to PI is 2.82843.

Increase the number of sides by one. I am now referring to a pentagon inscribed inside a unit circle. Same question, find perimeter. Noting the composite triangle of the radius between any two successive verticies and a side, the central angle is 360/5 = 72 degrees. The other two angles noting they are equal and triangle interior angles sum to 180, I get 54 degrees. So Law of Sines using one of the known radii of 0.50000 units, side length is 0.58779 units, perimeter is 5 X 0.58779 units - 2.93893 units. This is the third approximation to PI, 2.93893.

Increase the number of sides by one. I am referring to a hexagon inscribed inside a unit circle. Noting the logic used in the pentagon, side length is 0.50000 units and the perimeter is 6 X 0.50000 units = 3.00000 units. This is the fourth approximation to PI, 3.00000. It also is the proof that a hexagon is the only regular polygon that can be inscribed inside a circle using a compass and straight edge because the sides of the hexagon equal the radius of the circle to which it is inscribed. The fourth approximation to PI is 3.00000.

Increase the number of sides by one. This is now a heptagon or seven sided polygon inscribed in a unit circle. I find the angle between two verticies subtended by a side with the centre of the figure to be 360/7 = 51.42857 degrees. Noting the angle sum to 180 degrees in this composite triangle, I get a half angle vertex to be 64.28572 degrees. Apply the Law of Sines, a side equals 0.43388 units. Therefore the perimeter for the heptagon is 7 X 0.43388 units or 3.03719 units. The fifth approximation to PI is 3.03719.

So in this fashion, I continue increasing the sides to the polygon inscribed in the unit circle. Finding the perimeter will converge to the value of PI. But we will never get to PI as 3.1415692...simply because I can never stop adding a side to the polygon. For an infinitely large number sided polygon, the value of PI would be just that, PI.

THEREFORE it comes to pass that a circle is a polygon of equal sides because the value of the perimeter of that infinitely many sided polygon is PI. In this special circumstance, the perimeter is called the circumference and the circumference of the circle is PI times diameter.

And now you have it.

Regards,
Cockroach

 

[TheTick]

No, that does not "prove" that a circle is an infinite-sided polygon. It is only a method of calculating pi.

By definition, a polygon is composed of straight segments. Segments are of finite length (therefore not infinitely small).

Next batter...

 

[Cockroach]

The sides of the polygon form "tangents to the circle" when circumscribed to said circle. As usual, engineers muck up the mathematics and are poor at arriving at a proof. If I have a polygon of one million parts, surely this is much more infinite than a three sided polygon. Therefore increasing the population of sides to well past one million, I get a number approaching but never reaching Pi.

You can't see the forest because of the trees. The method of calculating Pi in this fashion is sufficient to understand a circle is an infinitely sided polygon. Your freedom to note otherwise is noted, and wrong.

Regards,
Cockroach

 

[TheTick]

A segment is defined as a portion of a line between two points and all points between (therefore not "infinitely close" or coincident).

A polygon is a planar closed chain of segments.

A circle is defined as the set of points in a plane equidistant from a given point.

A segment can not be fully coincident with any portion of a circle's circumference, as it will always have points that are not equidistant from the circle's center.

So... no, a circle is not a polygon.

 

[patprimmer]

A circle is one continuous curved line where all points are equidistant from its centre.

If it where a collection of infinitely short straight lines, it would also have infinitely small variations in distance from the central point and would therefore fail the definition of a circle.

Regards
Pat
See FAQ731-376 for tips on use of eng-tips by professional engineers &

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for site rules

 

[Cockroach]

Infinite number of sides of a polygon = infinite number of points or verticies for that polygon = a circle on which it is inscribed. Perhaps it is a visual exercise and harder to describe lexigraphically.

But that's fine, you agree to disagree. I'm good with it.

Regards,
Cockroach

 

[KENAT]

Wow, this topic just keeps going round and round.

Posting guidelines faq731-376

http://eng-tips.com/market.cfm?

(probably not aimed specifically at you)
What is Engineering anyway: faq1088-1484​

 

[IRstuff]

Ah, the circle of life...

TTFN
faq731-376

 

[Cockroach]

Actually, it is a very interesting topic, one which would be expected to be discussed in the mathematical forum, not an engineering one. But you can see the process in the video display of CAD packaging as an end user increases the resolution of the display. SolidWorks shows this quite well, for example.

I've shown the convergence to Pi asymptotically from the left using inscribing of polygons of increasing sides. You could do the same thing by circumscribing that polygon and using the midpoint of the side as the tangent point to the circle. The arithmetic changes slightly, but it is the same process. In that case, you would converge on Pi from the right.

So we're showing the sum of an infinite series converging to a limit. The fact that circumference is Pi times diameter as a circle property falls out from the perimeter of the polygon. But the process, obvious to some in this forum, is the point I wanted to make.

I just wasn't expecting such a parochial mentality of some readers, which on occasion is refreshing, in an intuitively obvious demonstration. But I'm not in the faith conversion business, but I digress.

Regards,
Cockroach

 

[TheTick]

Anyone who was even only mildly lucid in Calculus I understands that method of calculating pi (which is not "proof" that you have made a circle from segments). Anyone who was merely present in high school geometry understands that it is still not a circle.

As for lack of imagination, I suppose I would defer to those that do not have the ability to imagine curvature and are still stuck pasting straight segments together. Then again, there are no curves, but only straight lines travelling through curved space.

 

[Guest102023]

The line describing a circle is infinitesimally close to being a straight one.