douglasm wrote:Spoiler:In order for a construction similar to that to be accurate, its limit must correctly approach both position and direction (and maybe further derivatives too, I'm not sure). This one gets position right, but nothing else - the end result is tangent to the circle at exactly 4 points, and no amount of iterating or taking the limit of the process will ever change that.
This weirds me out more than the notion of pi being four, to be honest.phlip wrote:The upshot: the pointwise limit of those shapes is a circle, but the pointwise limit doesn't preserve many features of functions, including arclength.
The Mighty Thesaurus wrote:I believe that everything can and must be joked about.
Hawknc wrote:I like to think that he hasn't left, he's just finally completed his foe list.
Micali wrote:it's simply taxicab geometry. you can't 'infinitely' repeat it to infinity and therefore make the assumption it goes in a curved shape. rather it retains that jagged edge thus retaining the original length of the square, just at an infinitely small level. according to taxicab geometry, it will never be a curved line even if you take the limit to infinity.
Do I understand it correctly that 1) "after" infinitely many steps the resulting line is indeed identical to the circle, but 2) the perimeter is only 4 after finitely many steps, but "after" infinitely many steps it's "suddenly" PI?The EGE wrote:Sam Hughes has an excellent deconstruction at Things of Interest.
charonme wrote:Do I understand it correctly that 1) "after" infinitely many steps the resulting line is indeed identical to the circle, but 2) the perimeter is only 4 after finitely many steps, but "after" infinitely many steps it's "suddenly" PI?The EGE wrote:Sam Hughes has an excellent deconstruction at Things of Interest.
ameretrifle wrote:Magic space feudalism is therefore a viable idea.
zerkrox wrote:That is not an actual perfect circle.
zerkrox wrote:Regardless of the sides, no matter how many, a polygon will never become a circle, only a ∞-gon. That will always just be an infinitely close approximation.
uncivlengr wrote:A regular polygon with an infinite number of sides is a circle.
After infinity iterations, every point on the folded in square is on the circle, and vice-versa. The limit of the folded in squares is precisely a circle.tomtom2357 wrote:I think that the best way to resolve this is to realize that, even after infinitely iterations, there are still many points (uncountably many, actually) on the circle that never touch the folded in square. This means that 'most' of the circle do not touch the folded in square, so they are entirely different shapes.
This is a simple problem derivative from the "Zeno's Aporias". The guy is making equality between continuous set of points and discrete set of points. And this is wrong (as we already know from the quantum physics).
Problem: Which set has more points - the set of all natural numbers or the set of all even numbers?
Mike Rosoft wrote:And again, your objection misses the point; and the quantum physics has no relevance here, as it's a purely mathematical problem. The point-wise limit of the series of curves is exactly the circle; the problem is that the point-wise limit does not necessarily preserve the curve's length.
Mike Rosoft wrote:The answer is: The set of all natural numbers and the set of all even numbers have the same number of elements! Meaning that there exists a one-to-one correspondence between the two sets; namely, y=2*x. (Set theory 101.)
wattie wrote:Mike Rosoft wrote:The answer is: The set of all natural numbers and the set of all even numbers have the same number of elements! Meaning that there exists a one-to-one correspondence between the two sets; namely, y=2*x. (Set theory 101.)
Again wrong - infinities cannot be compared. You cannot count the number of elements in these sets.
wattie wrote:Yes, it's a math problem and yes - it has relation to quantum physics. The relation is that similar math problem gave birth to to the quantum physics. And the answer to the problem can be given in the Bulgarian high schools already - it is that "you cannot compare discrete (finite) set of points with continuous (infinite) set of points". End of story
wattie wrote:Again wrong - infinities cannot be compared. You cannot count the number of elements in these sets.
HungryHobo wrote:Am I the only one who'd love to see a troll childrens math book with lots of these kinds of proofs?
Dason wrote:Kewangji wrote:I confess I am actually scared of peanuts, and tend to avoid them, given how lethal they are to some people.
I'm not. I do my part in the fight against peanuts by destroying them with my powerful teeth. Take that peanut! How does being digested feel!?
baf wrote:Really, this should be obvious. Consider the sequence 1, 1/2, 1/4, ... 1/2^n ... Every single term in this sequence is positive, but the limit isn't.
Or, heck, consider the sequence 1, 2, 3... Every term in this sequence is finite. The limit isn't.
So, the troll mathematician gives us a sequence of curves. Each term in this sequence has length 4. We're supposed to conclude that the limit of the sequence also has length 4? Why? Should we also conclude that the limit is composed entirely of horizontal and vertical segments?
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