How can a Circle have an infinite amount of points, that together and individually, hold no space, yet they create a finite enclosure?
If I created a mathematically perfect circle barrier in the real world, and dumped the smallest cosmic goo into it, would it escape?
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>If I created a mathematically perfect circle barrier in the real world
Nonsensical premise.
The circle wasnt defined by the points, so their properties are irrelevant.
Good question. What's even more surprising is that, according to the banach tarski paradox, a circle is actually mathematically equivalent to two circles
>Circles
Step aside kiddo.
That only works for spheres and it doesn't say they are equivalent.
It was meant to be a thought experiment. Theoretically, if possible, does the smallest cosmic goo drip from a true mathematical circle ?
That depends on if your autistic or not.
because 1/inf = 0
a circle has infinite "points," so it has 0 points.
Math is a tough bitch, isn't it.
Pop-math general?
>Theoretically, if possible, does the smallest cosmic goo drip from a true mathematical circle ?
No.
>because 1/inf = 0
1/inf = inf or undefined, your choice
Awesome, but why
Circles arent real, stop acting as if they are
Truth is just whatever your peers will let you get away with saying.
how can a line, infinitely thin, completely fill up a circle?
What does this to do with circles? Squares, triangles, anything with lines or curves had an infinite number of points. these are not comparable to real structures made of matter because matter is not infinitesimal. At best you can assume space is continuous and compare them to abstract boundaries in space such as the event horizon of a black hole
By zigzagging infinitely.
1/inf =/= inf
pls stop. That is wrong for so many reasons.
Wow, are you just now learning 3rd grade Geometry?
I don't believe that creating a mathematical boundary would be able to contain anything. As far as I can tell the universe was here before mathematics. We as a species developed maths to communicate the complexities and operations of our reality in a way our thought processes are able to grasp. Essentially maths is an abstract representation of the laws and rules of our environments and cannot influence matter.
The same way that any line (or line segment) has an infinite number of points that it covers. That's just how continuous works.
from the set theoretic point of view, a circle is nothing but a bunch of points
there is no area associated with it
when you get to measure theory, you learn how to associate length, area, and volume to certain objects
but i believe the ~actual~ answer is in geometric algebra
you take two one-dimensional segments (analogous to vectors) and create a new object, which "looks" like a parallelogram
geometric algebra provides some tools to assign numbers to these parallelograms, which are interpreted as area
and of course, this can be generalized to parallepipeds in higher or lower dimensions
here's the kicker: i literally just googled "mathematics of area and volume" and found 10 things which would immediately answer your question
i'm a biology major btw
How many points [math](a,b)[/math] can you create out of the interval [math][1,2][/math] if your domain of discourse is [math]\mathbb{R}[/math]?
stop
OP: a circle can be defined type-theoretically (homotopically) as being generated by a point and a single path from the point to itself. This definition says nothing about points.
However, a circle does have metric structure, where it's defined as the set of vectors of norm 1 (say, in a Banach space). This definition does use points but conceivably we could find a way to avoid that in the future. The point (no pun intended :^) is just that points are not such a primary concept in modern mathematics, as demonstrated by topos theory. Types *have* points but they are not defined by them. Only a set is completely determined by its points.
>This definition says nothing about points.
durr, I mean that it doesn't say a circle is "made of points".
>R
Stop right there Cantorist scum.
quantum-cloning?
>type theory
>(homotopically)
except then your circle is equivalent to a point
doesnt exist
no, it's not: mathoverflow.net
I meant one nontrivial loop, obviously (that's what generated means anyways).
aha got em friend
Now do one dancing on the head of a pin.
Because you can't have infinite distinct points in finite space. It's just a concept, ffs
it WAS our choice and everyone agrees it's undefined, shit fucks up when you say it's infinity, we've had this discussion