Defining a circle as having every point on the circumference equidistant from its centre is sufficiently rigorous for no
other information to be required. So while it can also be defined in terms of infinite sides it is not actually necessary
to do so. The first definition is way simpler and easier to understand which is why it has survived for as long as it has
It is redundant if all you want to do is determine whether any given shape is a circle or not. We can do that without the concept of side. The concept of side does not help us in any way. That’s Occam’s Razor. However, if you want to do something more with these shapes, say measure their circumference, then it is certainly not redundant. The polygon approach is more complete in the sense that everything you need is there, in one place. Note that polygon approach builds on top of point approach. Polygon approach is simply point approach with points being connected by lines.
Yes. There is also trigonemtric approach which involves sine waves. And finally, there is my favorite approach, which involves easing functions ← that approach is the most analytical approach of all.
Normal brains process information synthetically (a.k.a. holistically.) This means they focus on the big picture – they don’t focus on the details. Or more precisely, they focus on the details but the amount of time they spend focusing on the details is minimal. They spend more time shifting their attention than they spend maintaining it. You can say they “glance over the details”. You cannot see the whole if you are focusing on the parts for too long – you will only see a lot of parts. You cannot see the forest if you are focusing on the trees for too long – you will only see a lot of trees. So a global thinker will see a circle where a local thinker won’t see a circle (e.g. UrWrong’s picture.)
When you focus on the big picture you lose the sight of the details and vice versa. When we look at UrWrong’s picture we see a circle. We don’t see its details, and hence, we don’t see the imperfections. If we switch out attention away from the big picture and towards the details, then yes, we would be able to see the imperfections, but this won’t make us abandon our earlier claim. If we did abandon the claim and if we adopted the attitude of spending disproportionately long time looking for imperfections then our understanding of circles would become so strict that most, if not all, of the shapes that we normally consider to be circles would no longer be seen as circles. That’s horribly counter-inuitive.
The problem is that there are times when the data that we possess regarding the world around us does not have what is necessary in order to be able to guide us in choosing what to assume regarding the unknown. This is the situation when every choice is as good as any other a.k.a. equi-probability. You cannot approximate (i.e. choose the closest relative) in such a situation because every candidate is equally similar to the original.
Paraphrasing what I said in another thread:
Intelligence is the search for the closest relative of some data set d within some set of data sets S.
In effect, the task of intelligence is to approximate.
(Of course, pedants will be quick to say that this is not intelligence but intuition. And not even intuition but induction. And not even induction but something else – something I made up.
My point will be that if you want to understand intelligence then you have to understand intuition. And if you want to understand intuition then you have to understand induction. In fact, you have to understand whatever is fundamental to the process. You need to understand the foundation of intelligence. Any other approach will be horribly superficial.)
When a data set has more than one closest relative then we’re speaking of randomness.
If we have a data set such as {A, A, B, B} that represents “2 occurences of A and 2 occurences of B” and if we want to find out how many occurences of A and how many occurences of B there will be at 5 occurences of either-A-or-B we will have no choice but to conclude either {A, A, B, B, B} or {A, A, B, B, A}. This is because the two data sets are equally similar to the original data set. That’s randomness. The greater the number of closest relatives, the greater the degree of randomness. On the other hand, if what we want to find out is how many A’s and how many B’s there are at, say, 6 occurences of either-A-or-B then we will be able to give a definite answer: 3 A’s and 3 B’s. This is because this data set is more similar to the original data set than all others. That’s order. The lower the number of closest relatives, the greater the degree of order.
12-side polygons look like this:
Squares look like this:
Circles look like this:
We can easily see, using our intuition, that the 12-side polygon has more in common with the circle than it does with the square. So the right way to call it is certainly not “circular square” but “square-ish circle”.
Your claim that this shape – the 12-sided polygon – is merely more circular than a square quite simply wrong.
It is A LOT MORE circular than a square.
Note that I narrowed the context. The 12-sided polygon is allowed only two relations: one with circles and one with squares. This is why I had to conclude that the best way to describe the polygon is as “squarish circle”. That’s true but only within that limited context. And that is sufficient to make my point – all I wanted to say is that describing it as “circular square” is very bad. Inuitively, however, describing the 12-sided polygon as “squarish circle” sounds rather strange. This is because intuitively we consider a broader context. The best way to describe the shape is as “spiky circle”. Even describing it as “triangular circle” sounds better. But we’ll notice that the similarity with triangles is only in the details.
The argument isn’t that a circle is a 12 sided polygon anyway. It’s an infinite sided polygon.
In order to test whether any given shape is a circle (or not) we need to decide how many points there are on the boundary. This is not an objective parameter. The boundary of the shape under the test does not have a finite number of points. Instead, it is us who has to decide, entirely subjectively, how many points there are. What this means is that testing whether any given shape is a circle or not is in actuality a test of how circular that shape is. No shape is inherently a circle. Instead, shapes are MORE OR LESS circular. When we choose a smaller number of points, we are testing for a lower degree of circularity. When we choose a larger number of points, we are testing for a higher degree of circularity. This is perfectly in line with the holistic stance that there are no absolutes, only degrees.
Sure, we can run binary true/false tests. Any given shape either passes the test or it fails the test. But what we are testing for can never be “the perfect circle”. Why? Because there is no such a thing. Can anyone here define what a perfect circle is? But without taking things out of context. Don’t just say “a perfect circle is every point in the plane that is equidistant from some fixed point”. Such a definition does not specify the number of points that have to be tested. If you say that the number of points does not matter, then the test becomes too lax . . . nearly every shape can pass it. The 12-sided polygon? It’s a perfect circle and James is wrong! On the other hand, if you say that the number of points is, say, quadrillion, then it becomes too strict . . . no shape can pass it. And the problem is not only strictness but the arbitrariness of choosing a number of points that defines the perfect circle. It’s entirely subjective. Why quadrillion and not, say, quintillion? The question cannot be answered without taking context into account. The problem is that analytical thinkers, such as James, don’t want to take context into consideration. They are absolutists.
When someone says that a circle is “an infinite sided polygon” what is meant is that the greater the number of sides a circular polygon (i.e. a shape with its key points being equidistant from the center) has, the more circular it is. That’s all it means. Pentagon is less circular than hexagon which is less circular than octagon which is less circular than 12-sided polygon which is less circular than 24-sided polygon and so on and so forth.
The problem is that with his interpretation of “side”, every polygon and every shape is infinite sided.
That’s the point though, Absolutism.
Some people are too absolute to admit mistakes in definitions and meaning. Like how James refuses to admit that a Chiliagon, 1000-sided shape, is a circle. He is too proud to admit a weakness in his position, his meanings. But James and Arc are missing the points. There will be many different approaches to definition, interpretation. Accuracy is important but there are gray areas wherein which some will say “this is a circle” and others will disagree and say it is not. This gray area, area of conflict, is when each side offers superior definition and increases the stringent requirements, leading to the perfect, absolute circle.
“My definition is the absolute perfect circle and YOURS ISN’T!!!”
That’s not what I’m talking about. But it was necessary to show a few people here they were wrong to completely reject my, ARBITRARY definition. As if a brief definition of a circle has to do with the greater topic at hand? How about a square? A square has 4-sides and 4 right angles. Are people here going to argue with that definition as well? In order to do what? In order to dispute that if I define a cause as this or that, that those definitions must also be absolute and perfect?
That’s the point, isn’t it?
That for there to be “cause” of anything at all, in existence, then those causes must be defined absolutely and perfectly? Nope, that was never my implication or intention, James.
Causes, like these definitions of shapes, require lots of effort and sophistication to apprehend and understand.
Like how an offshore earthquake keeps causing a coastal tidal wave. It keeps happening, everytime, one after the other. So ought we not then deduce, and operate from the premise, that offshore earthquakes cause coastal tidal waves???
My opposition has few directions to travel, to dispute, to doubt this. And you could even say “well so what, it doesn’t matter”. It doesn’t matter to you, but knowing the causes of things, can save lives, and to those who are swept away by coastal tidal waves, will absolutely know the difference of importance, and whether it was the ‘good’ thing to do, to attribute causes to natural phenomenon.
It’s not imaginary, a matter of the mind only. Some humans, a few at least, attempt to find, locate, and identify causes in existence. The patterns of reality, of existence. And in so doing, they gain advantages that other people do not have. The scientist who creates a device to detect offshore earthquakes, can save countless lives, whereas those other places and people who do not, will not save lives. If you are fine with that then so be it. But cause, as a general matter, is much larger than merely offshore earthquakes and chiliagon circles.
Yes, it is true, every polygon and every shape can be said to be infinite sided. Where I disagree is that this is a problem. I ask: what exactly is the problem? Why is it a problem?
We don’t see squares and other shapes as infinite sided simply because there is no need to. There is, on the other hand, very good reason to see circles as infinite sided.
Circles have sides that are related to each other in a very specific manner.
The only way that every polygon or shape could be described as infinite sided would be if a straight line was also described as such
But anything with edges cannot be infinite sided because they are the points where two sides converge and therefore begin or end
- It is pointless to say.
- It is rationally incorrect (circles have no “straight sides”).
Even a hectagon, a 100-sided shape, is very much a circle.
Polygons become difficult to distinguish from cicles once you give them more than 50 sides.
Another thing to note is that the official definition of circles does not state that circles are not polygons.
This is a common mistake.
If a definition does not state that a shape is a polygon, or that it has sides, it does not mean that the shape is not a polygon, or that it does not have sides.
Yes, which is why NOONE is saying it.
You mean that when we look at circles we see no straight sides?
Yes, that is true, but that is merely due to the manner in which our brains process information.
A polygon appears to be a circle from a distance, for example.
In fact, any polygon can appear to be a circle if you simply don’t pay enough attention to it.
If you infinitely divide a line, what do you get?
If you divide a line, you get two sub-lines. If you divide each of the two sub-lines, you get 4 sub-sub-lines. And so on. The angle between the adjacent sub- . . . - lines is always 180 degrees since the main line is straight.
How is this relevant?
Circles are not polygons because that would mean they have a finite number of sides
And then not every point on the circumference would be equidistant from the centre
A polygon cannot be a circle just because it looks like a circle from a certain distance
The definition of a circle is dependent upon mathematical logic not human perception
A hexagon is not a circle but a six sided polygon. A circle has infinite sides so cannot be a polygon. It also has no
sides so cannot be a polygon. So therefore saying it has infinite sides is exactly the same as saying it has no sides
Every shape is a collection of finite number of points informational atoms (analogous to how pixels are the atoms of computer screens.)
There are no shapes that have an infinite number of points.
The word “infinite” must not be taken literally.
When I say that a circle is a polygon with an infinite number of sides what I mean is that the greater the number of sides a polygon has the more circular it is.
That’s all that is meant.
When we’re testing a shape whether it is a circle or not (more precisely, how circular it is) we’re always checking a finite number of points.
As I’ve said, the larger the number of points we’re checking, the higher the degree of circularity we’re testing for. And vice verse, the smaller the number of points we’re checking, the lower the degree of circularity we’re testing for.
Once a shape is tested positive, it remains so EVEN IF we find out there are points on its boundary that do not obey the pattern of circularity.
This is NORMAL because we only tested for A SPECIFIC DEGREE of circularity. We DO NOT CARE if we find out that the shape fails at the test of higher degree of circularity.
I hope this is clear.
In that case, almost nothing is a circle.
The purpose of mathematical logic is to MAP human perception.
A young boy is asked, “What is a circle?”
The young boy then draws a circle in the sand with his finger. Is it perfect? Does it need to be? No, and no. It is a circle.
For those demanding absolute perfection, of definition, you’ve already lost the point about infinity and geometry. All shapes have sides. Whether or not humans perceive a certain level of detail, is a moot point. Engineers, mathematicians, architects, if you want to call them ‘authorities’ on what is or is not a circle, then each will give similar yet different definitions, according to their work, tools, equations, and functions.
Many in this thread are being, naive, simple, dull, childish, elementary. Even the young boy can exceed the intelligence demonstrated by some in this thread.
All you need to do is draw a circle in the sand.
The cause of this dispute, is ego, being wrong, being humiliated, admitting inferiority to the common sense of the young boy. If he can figure it out then why can’t you?
Start simple, and start with reality. Shapes are approximations. Rarely or never will any person need ‘perfection’. Save that for the engineers who use calculus, when it matters.
Here’s some logic for those clamoring for rationality:
- All shapes have sides.
- Circles are shapes.
- Therefore circles have sides.
For those claiming circles “have no sides”, you imply that circles are not shapes, which is false.