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Posted on 2005-04-22

I'm interested in answers on this one. Way back when I was in highschool I argued with my geometry teacher about a circle being a polygon and I still think I am right on this, but it brings up a potential limit of a logical device like a computer.

The definition of a line any 2 connected points, and the definition of a circle is all points (infinite in number) equidistant from a center point, so I say that a circle is just a special polygon. I say it's an equalateral polygon of infinate sides where each side is exactly 2 points long and the angle between one side and the next is infintesimally smaller than 180 degrees. Thus a true circle is really an abstract concept that I don't believe can be programmed into a computer.

That said, can a computer draw a circle? I say no because a computer has a finite number of points it can display/print/whatever and if it tried to draw a circle and had the accuracy to do so, it would never finish since it would be drawing an infinate number of points.

Thoughts? Feel free to comment on both the theoretical capability of a computer to draw a circle as well as a circle just being an infinately and equally sided polygon.

The definition of a line any 2 connected points, and the definition of a circle is all points (infinite in number) equidistant from a center point, so I say that a circle is just a special polygon. I say it's an equalateral polygon of infinate sides where each side is exactly 2 points long and the angle between one side and the next is infintesimally smaller than 180 degrees. Thus a true circle is really an abstract concept that I don't believe can be programmed into a computer.

That said, can a computer draw a circle? I say no because a computer has a finite number of points it can display/print/whatever and if it tried to draw a circle and had the accuracy to do so, it would never finish since it would be drawing an infinate number of points.

Thoughts? Feel free to comment on both the theoretical capability of a computer to draw a circle as well as a circle just being an infinately and equally sided polygon.

30 Comments

I'll use "I" for infinity since I have n ogoogle key.

I*0 = pi*r^2

0 = pi*r^2

0/pi = r^2

0 = r^2

0 = r

Conclusion:

Either your definition is incorrect or correct only for zero radius circles, the formula I used for computing perimeter distance is incorrect, or the formula for the circumference of a circle is wrong. I'd put money on your definition being incorrect, although I've heard circles described that way many many times. (well, as a polygon w/ infinite sides, not as sides "2 points long", but the same thing applies either way)

I don't agree that there must be a line simply because 2 points wouldn't occupy the same location...I believe 2 separate points can occupy the same location without being "the same point". Besides, a point has no dimensions therefore takes up exactly zero space, therefore there could be an infinite number of points at the same location and they would still occupy zero space ;) This is getting off topic tho, so......

I believe a computer could accurately represent a circle for computational purposes, however....provided the computer displays it's computational results in terms of pi. Obviously, since no computer can accurately calculate, display, or otherwise perform mathmatical operations on a number with infinite decimal places, pi could not be used as an actual number in any computations. As long as pi is kept as a representation of pi and not attempted to be converted into an actual number, then a computer could in fact represent a circle and perform operations related to circles of specific theoretical dimensions.

I guess it comes down to how you define "draw". If you mean "draw" as in "represent for computational/informationa

looks kind of like this º

:-)

depending on your font may distort the image.

the only problem with the argument that EVERY circle is a POLYGON, is the polygon has clear defined points, where at the point there is an angle. Sure, _some_ circles may in fact represent a polygon better, but a perfect circle can be drawn,

If you look at the sun, or even the moon, which are both perfect spheres, from where we are, we seem in 2D form which is the representation of a perfect circle.

If we can program machines to cut metals into perfect circles (as we have been doing for years), the same program can have a computer display an infallible circle.

Doctor, do you concur? :-)

As above, if it's on a monitor, then no because the pixels are on or off and of a fixed dimension. Zoom in far enough and you get [ instead of (.

I'm not aware of any pixel-less display technology although I suppose one could exist...but I believe in analyzing the technology deep enough, you would still be working with a fixed resolution thereby having the same pixel problem.

As for printed media, same thing...soom into the smoothest paper printed on by the highest resolution plotter, and you will find very jagged edges. Look at a pencil line on a piece of paper under a microscope...or even a marker on glass. Under high enough magnification, the line itself will look more like mountain terrain rather than a smooth arc. A circle is made of arcs, not mountains. When it comes right down to it, there isn't even any such thing as a flat surface...if viewed under an electron microscope. Without a flat, contiguous surface to display on, a flat, contiguous image cannot be displayed.

In fact, if you zoom in to a molecular level, there is no actual physical contact ("I swear I never touched him, officer! It was an electromotive force!") between any 2 physical objects....like an ink molecule and a paper molecule. A circle by definition cannot be jagged, yet zoom in far enough to anything physical and you will find jaggedness. Therefore nothing, including a computer, is able to draw a circle.

Is that good enough for ya? :)

Since the pixel arrangement on a computer screen is rectilinear, how could we ever hope to draw a line at an angle let alone an arc. We can make good approximations but that is about it.

mount an arm of some fixed legnth to an electric motor so that it becomes the radius of the intended circle.

attach a drawing instrument to the end of the arm in a manner that allows it to contact the media to be drawn upon.

interface this drawing apperatus to the computer, and have the computer activate the motor.

be sure to stop it after one or more rotations.

The resulting drawing will be as true of circle as allowed by the mechanical tolerances of the setup, and significant environmental and legal factors.

The computer itself, can only utilize the outputs that are made available to it. When that output device is an array of squares, you will never be able to get a circle. Hmm... wouldn't a single drop of ink from an inkjet printer also be circular? Or the resulting pattern on a CRT monitor when only a single point is allowed to energize its phospers?

By the classical definition, a circle cannot be a polygon.

All CRT screens are inherently analog. We choose to rasterize them, but it wasn't always the case.

In the days of analog computers and analog oscilloscopes, the recipe for a perfect circle was pretty simple:

Apply Vx = sin ft to the x-deflection circuit

Apply Vy = cos ft to the x-deflection circuit

>>

>> Apply Vx = sin ft to the x-deflection circuit

>> Apply Vy = cos ft to the x-deflection circuit

I still don't believe you could create a *perfect* circle that way...miniscule differences in component tolerances and temperatures and even the rediculously small temperature difference in the room from one nanosecond to the next would change the analog response time and therefore the display. It may not be a measurable difference without completely insane levels of measuring devices, but it would be a difference nonetheless, thereby producing a less than perfect approximation of a circle. Certainly, it'd be good enough for many purposes, but from a 100% literal and exact mathematical definition of a circle carried out to enough degrees of preciscion, it would be slightly off.

Same could be said for a square, actually...if measured to enough preciscion, the sides would not be *exactly* the same legnth.

Same goes for parallel lines or any pure mathematical concept applied to what many refer to as "the real world".

From a practical standpoint, reasonably accurate representations of most objects can be created and copied. From a pure standpoint, no two physical objects can have exactly the same dimensions and no physical object can be rotated exactly at one point around one axis. There will always be variables that influence the rotation and/or radius (temperature change from air resistance, tensile resilience from going from rest to moving, gravitational attraction ever so slightly pulling the rotation mechanism off center, even electromagnetic radiation affecting neighboring electrons at different times since even the speed of light isn't infinite). That's my only point here...pure math cannot be replicated perfectly (i.e. to infinite preciscion) in the physical world. Period.

Guess it's one of those "can be reasoned, but never proven" situations...like proving <blah> doesn't exist.

Yes ... but !

When 2 points are so close together then the distance "d" ... (d---->0)

then the line that cross this points obey to the 16th dimension...

The line is no more a line. It twists. And we have the circle...

Superstring Theory:

"All regular polygons, irregular polygons, a circle, an ellipse, etc. are all considered the same shape, which for simplicity, is represented by the circle. In other words, they are topologically equivalent."

That does not make sense. Between any two geometric points there are an uncountably infinite number of intermediate points.

> Can any physical object have an irrational dimension?

If the sides of a square have rational dimension, the diagonals will have irrational dimension.

But, that could just mean that no physical object can be at the corners of a perfect abstract square.

Any physical measurement has uncertainty, so you could never know whether the dimension was rational or irrational. (although with a continuous spectrum it would be infinitely likely to be irrational)

On the othet hand, it may be that space is quantized at the scale of the Planck length,

with area and volume having only discrete values. But the area eigenvalue would still seem to involve square roots, so you would still have irrational ratios

>>"We can not do anything infinite nor anything that we can create/program, etc can be infinite. "

I'm a programmer, I have seen infinite loops, while not necessarily a circle, we can create infinity.

depends on your definition of a circle and also infinity :P

cos if you have

Dim infinity As Integer

infinity = 100

Do Until counter = infinity

Loop

Then according to that code, infinity is 100, but then that would be wrong because that is by no means anywhere near infinity :)

I went here :

http://dictionary.referenc

and found this :

2. <programming> The largest value that can be represented in

a particular type of variable (register, memory location,

data type, whatever).

So taking that into consideration, that would or could mean that infinity would be the maximum amount a variable can hold so if it was an integer then that would mean it would be aprox 65536 or w/e the max amount an integer can store !!

Correct me if I am wrong :)

Circles have many definitions and some are offered above. Dictionary.com says a circle is "A plane curve everywhere equidistant from a given fixed point, the center.” Plane refers to the circle existing in a single plane and a curve defined as "A line that deviates from straightness in a smooth, continuous fashion." In that sense yes a circle can most definitely be displayed.

Since there is no way to realize every point that can exist on a circle (there being an infinite number of possible points), in life or computers, then it seems odd to define a circle as every possible point equidistant from a center. It is purely a theoretical device, a wisp of imagination. If we really want to define a circle as such then no one has ever seen one since we are incapable of perceiving an infinite set.

A circle, in any practical sense, should be defined as all perceivable points equidistant from a center. Then the definition becomes a relative matter. If we should suddenly become able to perceive infinite sets, then we are still able to enjoy the existence of the circle

As for a circle being a polygon, I don't think it can be construed as such. Polygons are made of lines, but no matter how small a line we make there exists a midpoint of that line. Since that midpoint exists linearly from its origin it does not form a curve and thus the midpoint does not lie on the circle. The circle must constantly deviate to be a circle, infinite or not.

I would take the same statement and say "A line that deviates from straightness in a smooth, continuous fashion." In that sense yes a circle can most definitely *not* be displayed.

I would say that due to the reasons I previously posted....zoom in to the displayed circle (be it on a screen or physical media like a plotter) and it will be quite apparent that there is nothing smooth and contiuous about the jagged pattern you'd find under, say, a microscope.

then there is this device called a protractor. We should all know how those work.

The sun, just like the earth, are not perfect sphere's nor are their atmospheres. THey are sphere-like but the sunward side of earth has a bulgier atmostphere due to the expansion of hot gas. The sun's distribution of mass is not perfectly symetrical and the sun has sun-spots around which the later layers of atmosphere are going to be bulged out some, plus there are the flares which definately disrupt the sphere. Even without any flares or sun-spots the sun wouldn't be perfectly spherical.

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