Have You Heard Of This Banach-Tarski Paradox?

Can you get your mind around THIS??

The concept is very bizarre and the ever inquisitive “VSauce” walks us through.  It is hard to get one’s mind around this one but he gets into depth in the details thoroughly and it involves thinking about 3 dimensional plains in a new way.  Once commenter even says:

I lost you after like 6 minutes

Some one else has a similar comment:

17 minutes in and i’m just here like, what is he talking about…

Here is a bit more information from wikipedia on the background of this theorem:

The Banach–Tarski paradox is a theorem in set-theoreticgeometry, which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjointsubsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them, without changing their shape. However, the pieces themselves are not “solids” in the usual sense, but infinite scatterings of points. The reconstruction can work with as few as five pieces.[1]

Let’s find out more about this amazing paradox in this video on page 2

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158 Comments

  5. Kyle Judge said:

    Well in the height of it’s use before it’s upgrade, the scientists working on the large hadron collider were finding subatomic particles at such an alarming rate that they would often joke that the nobel prize will go to the scientist that DOESNT find a new particle. Now they might be exaggerating to some extent (I wasn’t there so I wouldn’t know that), but it still brings me to this idea: what if subatomic particles are the countable infinite points in real matter. If you could make a collider or another device that was capable of extracting the infinite particles, could you then reform them and duplicate that matter? If so this means you would have the ability to reform basic elements at the very least (since you’re duplicating atoms you would only duplicate the element it forms, as far as I figure). Now I am certain this has been thought of before by someone, but I still think this would be an amazing achievement. At the very least we could create hydrogen atoms infinitely out of one atom, maybe even extending the life of a star out of it (if you could effectively duplicate the hydrogen within the core). It’s all speculation on my end, so please enlighten me if you can as to whether I am on the right track or not. Oh, and please, no berating me if I am off in any way. I am passionate, not a professional.

    Reply
  12. Rory Gallagher said:

    It’s because they’re working with infinite sets. The revolutions of the hyper sphere don’t require energy because the points already exist, we’re just identifying them.
    If you think about it more like a number line it makes me sense. It’s like I have an infinite line that I cut, both ends still continue infinitely. Extreme over simplification, but I think it works.

    Reply
  13. Rory Gallagher said:

    Your high school math teacher is wrong. To determine size with countable infinite sets is itself a paradox. The sets are infinite, forever doesn’t have a size because you can always have more.

    Reply
  14. Kaitlin Eileen Bourassa said:

    But there are different sizes of infinity. If you take the set of all real numbers and compare it with the set of rational and irrational numbers one is going to be much larger than the other even though they’re both infinite

    Reply
  21. Girish Nair said:

    A very rigorous/beautiful way to prove something that is intuitively obvious when you consider any object to be made up of a countably infinite number of points. I wonder how does Planck length and/or “quantum foam” affect any applicability of this to real world? Can we really have true points, or is there an indivisible space-time volume/string? If so, our random walk across the sphere will result in repeating arrivals at the same piece and so we will not be able to make 2 from 1.

    Reply
  31. Jay Farmer said:

    Jeremy Cantor this is a good example of something we were talking about a while back; a “paradox” of higher math that I think is really just an inability to consolidate mathematical and semantic verbal descriptions of the same concept. They are made out to be weaknesses of math and science but I think they are usually just weaknesses in language.

    Reply
  32. Jeremy Cantor said:

    I’m going to take a page from Bob Dylan, and not criticize what I can’t understand. But I will present Tom Weston’s paper on the subject here. If the set theory is beyond you, as it is beyond me, I suggest skipping immediately to the last page. I read about this long ago in an article called “The Crisis in Intuition.” A particularly brilliant friend of mine (who could only explain his ability to easily handle proofs that I could not possibly hold all of in my mind at one time by saying “I think in smooth white forms…”) said, “It said ‘The Crisis in Intuition, but it should have said ‘The Crisis in Rigor.”

    people.math.umass.edu/~weston/oldpapers/banach.pdf

    Reply
  35. Robert Powell said:

    DMT was what made me see the science in spirituality. It made me understand the infinite power of energy. It never started, it never ends.

    Reply
  36. Jay Farmer said:

    That was fabulous. Ironically, the original link here does the best job wording the issue, essentially that infinity is treated like a number on the number line when it is really a description of all the numbers on the number line. But then it goes on to use it as a number, run into a bit of nonsense, and then pretends to be shocked by the whole thing.

    Reply
  44. Trevar Wilson said:

    in a much bigger picture, this may suggest if you could walk over every surface on our earth, and never retread the same direction, you would never end up back in the same spot, and could never finish walking the entire world, which would make our world infinite despite being contained in a sphere

    Reply
  45. Trevar Wilson said:

    I also believe it would explain why there is no center of the universe when it explains the hotel theory, an infinite number of rooms with one guest filling every room is never full, guest one is simply moved to room b and opens up room a for the new guest, hence the center of the universe(or where our universe began, or was born) simply no longer exists, you could never fly a ship to that place

    Reply
  50. Bobby Ko said:

    How does Gödel’s Incompleteness Theorem play into this? It seems as though they may be describing the same paradox from two different positions. What am I missing? Probably a lot, as I am certainly no mathematician. But with a very elementary understanding of both, it seems as though these are relating to the same paradox.

    Reply

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