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?

 

[telecomguy]

PI=4 proof:

http://www.lolblog.co.uk/wp-content/uploads/2010/11/1290616506315.jpg

 

[Cockroach]

This pretty much summarizes the method.

Asymtotic convergence from the left uses verticies of a polygon of increasing sides inscribed from a common point or centre of curvature. That from the right are circumscribed polygons of increasing sides that use the midpoint of a side tangent to a curve with common centre. Because of symmetry, such points, whether they are polygon verticies or midpoints to the sides of said polygon lying tangent to the curve, are equi-distant from the centre of curvature. Hence the notion of a "circle".

So a circle can be regarded as a polygon of infinite sides, no matter which way you choose to describe it. Points on a curve or tangent lines that are subtended at the midpoint of each side, convergence to a constant defined by perimeter divided by twice the radius of curvature, is a unique circular poperty that falls from the logic of the method. Regardless, the concept is a polygon of increasing sides thus geometrically forming a circle with points lying equi-distant from a common centre of curvature.

End of story.

Regards,
Cockroach

 

[Sparweb]

Cockroach,
"I get a number approaching but never reaching Pi."
Most of us don't have the patience to wait for an infinite series to reach its asymptote.
We'd rather just reach for the pi.

STF

 

[patprimmer]

Just because calculus uses a very close simulation to a circle, that does not make it a real circle, just an infinitely close approximation.

Just because a CAD program cannot actually draw a circle, that does not mean it's best attempt at an approximation is a circle.

Regards
Pat
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[electricpete]

I haven't read every reply in the whole thread. I don't think anyone here is confused about any of the concepts.

I think both "sides" (no pun intended) would agree with a statement worded as follows: In the limit as N approaches infinity, an N-sided polygon approaches a circular shape.

Personally I think the notion of a limit is a necessary part of that statement. The phrase "polygon with an infinite number of sides" does not seem mathematically precise and perhaps creates ambiguity which results in disagreement when people interpret the ambiguous phrase differently.

Sorry if someone else already said the exact same thing.
Also sorry for prolonging this. Barring any negative reply to my comments, I'm done.


=====================================
(2B)+(2B)' ?

 

[electricpete]

Correction in bold. Should have been:
"In the limit as N approaches infinity, an N-sided regular polygon approaches a circular shape."



=====================================
(2B)+(2B)' ?

 

[btrueblood]

Pretty sure the circle only has two sides, inside and outside. And with that, I'm steppin' outside.

 

[JohnRBaker]

You mean it's a 'half-space'?

Now that should really confuse people

John R. Baker, P.E.
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[WolfHR]

Cockroach, re. your definition/proof of circle as polygon with infinite sides (and I say this half serious)- it's all well but the first polygon (triangle) you mention was already inscribed in something called circle... So, by that definition a circle is already well established fact when we start inscribing polygons into it. On a more serious note, as others have noted before me- when a push comes to shove, a circle (which is defined as a set of points in a plane equidistant from a given point that is its centre) can be approximated by a polygon whose number of sides converges towards infinity (the greater the number of sides, the closer approximation to the actual circle).

Re. CAD analogy someone drew to Solidworks is not as to the point as someone did to AutoCAD (with answer being 42), because Solidworks and other modern CAD packages use 'proper' circles, and the limiting factor is resolution of the display, whereas AutoCAD actually (prolly for purposes of speed of display, &c) actually approximates circles with polygons (number of sides is set in options) to display, but calculates as it should with circles...

As for the intention of the OP, I must say the (intended) irony escaped me at the first, but I do see it. Admittedly, it's indeed not an engineering question per se, but interesting none the less. Increasing the number of sides of a polygon until there are none.

 

[Cockroach]

Actually WolfHR, I pointed out that the vertices in the case of a polygon, or the midpoints of the sides of the polygon, lie equi-distant from the centroid of said polygon. This is important, I drew the circle in dashed lines in Oder to accent this feature but more importantly to show that as the number of polygon sides increase, the figure starts to move closer and closer to a circle. I also pointed out that if possible to reach any large number, the true circle will never be obtained, for I can't count to infinity. What I am doing is to populate points equi-distant from a centre of area of a n sided polygon, the path of which is a circle. I never presupposed a circle and could of started with a polygon of any size, then determine the distance of these points from a theoretical centre. Don't let that detract from the mathematical argument.

ElectricPete brings to mind the notion of limit. I am not summing and allowing infinitesimal accumulations like an integration process. Rather, I an incrementally increasing the number of sides in a polygon from three to a very large number. The concept of circularity evolves out of the process as the circumference of a circle equals Pi times diameter. These fall out of the argument.

But two well thought out points, well presented, so I thank you. Try it for yourself, start with an equilateral triangle of any size and locate the centre. Then increase the sides of the polygon by one and recompute. You'll get the very same result whether you use the vertex of said polygons or midpoint of the sides. It is a beautiful proof! Try it for yourself over a few beers.

Regards,
Cockroach

 

[WolfHR]

I have no doubt, Cockroach, and I entirely see your point and concur*. Although... I'm a lazy git, so I'd start with triangle and just because of convenience increase the number of sides twofold. Admittedly, if we were going all the way to infinity, my method would not be any faster than the one you suggested (but both of us know we'd never get there), but it seems to be faster to reach large numbers, and halving the angles also seems more convenient.

* the point in previous post was an attempt at a bit of joke- we show that polygons tend to approximate circle more closely as number of sides increases, and start doing it by inscribing a triangle in a circle... so the intended joke was why would we approximate the circle if we took a compass and have already drawn it?

 

[FACS]

I'd look at it this way:

If my mom put me in a round room, and told me to go stand in the corner for 10 minutes..... I would be in there for the rest of my life.

Charlie

http://www.facsco.com

 

[patprimmer]

I would say you would be there to eternity.

Regards
Pat
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[IRstuff]

Well, your body would, YOU, on the other hand...

TTFN
faq731-376

 

[patprimmer]

OK. The non volatile decomposition products from the body would

Regards
Pat
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[Tmoose]

http://imgc.allpostersimages.com/im...nless-i-m-really-drunk-new-yorker-cartoon.jpg

 

[mjcole]

Well Cockroach I almost gave you a star and would have if your attempt weren't flatly wrong. You just may be more into math and proofs than actuality.

The more points you make the closer you will be getting to the correct answer or will you?
A Circle is a Circle now a computer may come awfully close to representing a circle, but if you get a pushpin/thumbtack a piece of string and a pen and make two tiny loops the size of the pushpin diameter and the other the size of the pen tip. Push the pin through the pin hole into a piece of paper. stretch the string to it's max extent and put pen to page and go 360º around so you end where you start and you'll have a circle. Of course no one can draw a perfect circle although it may be possible it's highly not probable.

Everyone who thinks of a circle as an infinite sided polygon, let's have a race. you can start by drawing your polygon and I can start walking halfway to the nearest wall.

No one will win because you'll never have infinite number of sides because that number does not exist and I'll never reach the wall.

Cheers everyone and thank you for the mindlessly inane and quite stupid imho discussion.
...............
The atom smiley is a Gif which is the closest that file type can get to a circle unless you make it ∞x∞ pixels and you may come close but never achieve a perfect circle.
Even with the best cad system and graphics card, if you zoom in far enough to a circle you'll see the linear approximation even with a Cray supercomputer. Pretty sure SolidWorks has a limitation to how many sides it can have in a polygon. If I find that out I'll post it on the SolidWorks forum which I recognize a few forum posters from.

One more thought before I leave the discussion Can you get a perfect circle from an etch a sketch with precisely programmed stepper motors at the knobs?

P.S. patprimmer If it's curved it's not a line at least not any longer.

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

Note: quote has been spell corrected were was "where"
But I completely agree with your point.

"It's not the size of the Forum that matters, It's the Quality of the Posts"

Michael Cole
Boston, MA
CSWP, CSWI, CSWTS
Follow me on !w¡#$%
@ TrajPar - @ mcSldWrx2008
= ProE = SolidWorks

 

[patprimmer]

Well if you want to be pedantic, if it's been changed it is no longer a quote.

http://www.merriam-webster.com/dictionary/line

It seems Merriam Webster disagrees with your definition of a line around points 7, 8 and 9

Just to be pedantic about it.

Regards
Pat
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[JohnRBaker]

What's all this crap about computers NOT being able to represent a circle except as some sort of approximation. It's just plan BS! A circle, besides being a periodic curve (as I've already pointed-out), can be represented using a conical form, that is, a mathematically determinant object. Meaning that NO approximation is required when performing mathematical operations involving circles or for that matter, any other shape that can be represented as a conical form, such as a line, ellipse, sphere, cylinder, cone, tori, etc.

I suspect that where this perception that somehow computers need to approximate a circle is when it needs to display one a computer screen, but this has absolutely NOTHING to do with how the computer software represents a circle mathematically. It's purely a 'mechanical' limitation of the display technology, but even that is not always what you'd think. There was a time before raster displays (like your TV) when we were using direct view displays, that is the displayes were not scan line based but with objects being directly drawn on the screen. However these were limited to monochromatic displays, the most common being White on Black with Display Refreash Devices...

...or Green on Black with Direct View Storage Terminals...

...but circles were circles, period.

John R. Baker, P.E.
Product 'Evangelist'
Product Engineering Software
Siemens PLM Software Inc.
Industry Sector
Cypress, CA
Siemens PLM:
UG/NX Museum:

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[IDS]

Where can I get one of these computers that allow methematical operations to be carried out with no approximation, rather than using floating point numbers with a finite number of digits?

Doug Jenkins
Interactive Design Services

http://newtonexcelbach.wordpress.com/