So, for example, given x = 2 and y =3,
2^2 + 3^2 - 1 = 0
Okay, let’s see:
4 + 9 - 1 = 0 hence 13 - 1 = 0???
Confusing! 
I think I need to build up to that. :rolleyes:
Now I’m* really *lost. :smack:
This is true for two points on a plane surface. The most direct route from one point to another on a curved surface is a curved line.
Not quite. You don’t choose any random pair of numbers. You have to insert one and solve for the other to get a coordinate on the unit circle. (2,3) does not solve the equation, so it is not on the unit circle. It may be simpler to write the equation as such: y=±sqrt(1-x^2). That is, y equals the positive and negative square roots of 1 less the square of x. Just plug in x.
They’re not really tamed, merely biding their time, until they can wreak further havoc… Renormalization - Wikipedia
Consider adding a series of fractions
1/2 + 1/4 + 1/8 + 1/16 + 1/32 + 1/64 + …
Each term in the series is just half of the one before it. There is no smallest number, so the series never ends. You could keep writing out more terms, 1/128, 1/256, etc., but you’ll never write out the whole series.
And if you take the sum of the series, it gets closer and closer to 1 the more terms you add.
1/2 + 1/4 = 3/4
1/2 + 1/4 + 1/8 = 7/8
1/2 + 1/4 + 1/8 + 1/16 = 15/16
…
In a sense, you’re dealing with competing visions of infinity. As the series adds more and more terms (approaching infinity) the difference between the sum and 1 gets smaller and smaller (approaching infinitesimally small). In a way, they sort of cancel each other out, and the sum of the infinite series is said to be 1. There’s a bit of a conceptual leap, but it works out okay in the end.
The way you describe a circle is essentially the same. A triangle, a square, a pentagon, a hexagon, etc.; each one has more sides than the one before it, but the sides are smaller. Continue that process to an infinite degree and you have a circle. It’s kind of the same leap as the infinite series. You have to recognize that the infinite series becomes something that no finite sequence ever can, and we can describe the properties of the whole even if we can’t take it apart into individual pieces.
And then you really have invented calculus.
Sorry. The answers to math questions cannot be given in words. They can only be given in math. Rigorous math.
The most famous attempt to do this came in Principia Mathematica by Alfred North Whitehead and Bertrand Russell. The need to eliminate all the wriggle room was infamously brain-busting.
And yet it wasn’t enough. Gödel and others showed that any mathematical system has true statements which cannot be proved inside that system. That does not invalidate the system, as some people think. It just requires a larger, more encompassing system, which itself contains unproveable statements. And so on up forever.
Hundreds of the best mathematicians have spent hundreds of years refining math to deal with the points you so casually tossed aside. Words are how math is translated in non-rigorous analogy to people who don’t speak math; they are not a substitute for symbolic math.
This all implies that numbers exist in their own Platonic Ideal. A “perfect circle” exists in the same way other Ideal Forms exist*.
Okay, abashed - now invent philosophy!! 
*obligatory reference to the band of that name that’s a side project of Maynard James Keenan, the vocalist for Tool.
The way I learnt geometry, a point is dimensionless (has no length, width or height). IIRC, contrary to common sense that a dimensionless objects cannot exist, it was explained to me that a point’s existence is an Axiom.
A circle is the locus of a point which moves such that it’s distance is fixed from another fixed point (on a plane). So you use an axiom to develop another concept.
In reality, concepts of chaos theory come into play into natural shapes. For example, the length of the coastline of Florida depends on the scale used for measuring. You can have finite area with an infinite perimeter (fractals).
Not really, x and y are the co-ordinates for a single point on their axes.
So basically… the point (2,3) does not make up part of the circle.
Many years ago I asked about this, and my geometry teacher said to think about a circle as a series of equally off-set points.
I am too. I agree that a true circle is impossible to make in the real world, but I don’t understand what the irrationality of pi has to do with it.
You’re struggling because deep down inside, you’re thinking of “infinity” as “a really, really big number.” And that’s flat wrong. It’s not an ordinary number, just bigger. It’s a *qualitatively *different sort of beast, not just a quantitatively larger ordinary number.
When you approximate a circle as 4 short straight segments it looks more like a square. Because it *is *a square. When you approximate a circle as 100 short straight segments it looks like a pretty smooth circle. Substitute 10,000 shorter straight segments and it *looks *smoother. At one quadro-zillion segments it looks really, really smooth.
The jump from there to infinity is sort like the Millennium Falcon jumping into hyperspace: everything is different. Suddenly it doesn’t look smooth; it *is *smooth. Suddenly instead of gazillions of short straight segments you don’t have any straight segments at all. Instead you have points. etc. There is not a smooth transition from big-but-finite to infinite; there’s a hyperspace jump there and your brain needs a warp drive to make it.
You can’t apply “10, 100, 1000, … therefore …” logic to infinities. You need completely different mental tools. About all we can say here is go take a calculus class or the online equivalent. We can give you tastes and snippets, with each of us pointing out one or another small patch of the elephant in our individual way. That way probably lies abject confusion for you. You might get an Ah Ha moment, but that’s not the way to bet. You sure won’t get a coherent whole elephant out of it.
Yes, you do.
But rather than building up to that, that’s the very foundation of your very next baby step. As I said above. Right now you’re jumping several big leaps past your good knowledge and doing so in an unproductive (but very typical student) direction. Then you’re asking where you went wrong, and pushing back when we tell you it was way back at the beginning, not out at the end of the logic chain where you think the glitch is.
He’s trying to say that the physical world isn’t made of an infinitely large number of infinitely small particles. Instead it’s made of a very large number of very small particles.
An uppercase-C Circle is an Idea made of an infinite number of infinitesimal Points. A lowercase-c circle in the real world is a crude model or mock-up of a Circle. The crude mock-up is made of a large number of small particles. No matter how small the particles or how great the number of them, they aren’t mathematical Points and there aren’t infinitely many of them.
Are you seeing a pattern here to these comments? The other posters’, yours, and mine?
This is all true. Usually “points”, “lines,” and “planes” are taken as primitive terms. And the existence of some points is part of the axioms.
This remark is what led me to comment. Why ‘chaos theory’??
As for fractals, they are perfectly good geometrical figures, albeit far from those considered by Euclid. They are a good example to bring up in this thread, of something that does not look smooth when you look closely at it, unlike a circle which does.
This is true. But if you think of a curve (like a circle or a parabola) as made up of infinitely many straight line segments that are each infinitessimally small, you’re doing intuitively what calculus does in a more rigorous and mathematically coherent way. Hence friedo’s comment.
I tend to read a thread such as this until my eyes glaze over, then move on to the next.
Is that the limit of your wit? 
Actually, nice one. Well played!
You’ll be pleased to know we tackled this very issue only 3 weeks ago: http://boards.straightdope.com/sdmb/showthread.php?t=830451.
As you might expect, the answer is "Spheres aren’t within the usual definition of ‘Platonic Solid’. But they’re right nearby and there’s lots of sound ways to argue that a more generous definition of a PS is A) even more useful than the traditional one and B) of course includes spheres.
The OP may be interested in the Koch Snowflake which relates to the circle question.
The Koch Snowflake is something that has a finite area but an infinite perimeter.
Taken a bit further this is used in the Coastline Paradox which shows that depending how small of a unit you use to measure a coastline it could be infinitely long (e.g. the resolution you use as mentioned in the OP).