# 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]

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1. George Nelson said:

If it’s mechanically impossible, divides by 0, claims something is infinite or in any other way is logically incomplete, its not a paradox. It’s the ramblings of someone who thinks they’re smarter than they really are. See 90% of philosophers.

2. Jason Reitz said:

If this works from infinity, then if the possibility that the universe is infinite, could it be physically possible that such a thing has happened with the universe? Giving an infinite number of universes? Of course we would never know, because to detect something that is an infinite distance away is impossible.
Right?

3. Zachary K Stoner said:

Extremely interesting. The guys at Vsauce really know how to break down complicated subject matter for the lay person. Part of me says we simply are reaching and another says that the real world use would just involve eliminating or not bothering with extemporaneous matter when applying this principle but then does that make the whole concept fail… Cool video

4. Frank Myobfb said:

The explanation doesn’t seem valid. First the explanation of countable and uncountable seems more like word play. I’m guessing that it’s a simplification.
Next, in the first part he tells how infinity isn’t really a number but an amount, but then the explanation goes on to use infinity as number. I’m sure the math, if I could grasp it all, would explain it better, but the idea of subdividing an object via points that are infinite seems like an invalid example.
Indeed I believe I could simplify the explanation bu simply saying that a circle (or sphere) is made up of an infinite amount of points. You can remove an infinite amount of these points an infinite number of times and still have the original object, and an infinite amount of duplicate objects.

5. Jaime Guerra said:

Right on! Numbers and stuff. So this stuff is obvious to everyone or am I crazy to think this is common sense. Idk pudding pop. It’s cool and all.

6. Alex Wharfield said:

The way I’ve always imagined it is constantly zooming in. If you have something you should theoretically be able to zoom in on it.

7. William Walker said:

Seriously flawed. All numbers have a negative…infinity in both directions.

The alphabet starts with ‘a’ and goes forward, definitely not infinite. 1/2 of infinity?
The hyperdictionary starting with ‘a’ would never arrive at ‘b’. B would then have to be its own set of 1/2 of infinity. Because it will never arrive at ‘c’ and will never go back to ‘a’,

Further all ‘a’s can can be translated to ‘b’s or ‘c’s and so on ad infinitum.

Each would be it’s own dictionary.

8. Steve Reina said:

DO ALL POSSIBLE MATHEMATICAL REALITIES ACTUALLY EXIST? Some twenty five hundred years ago Zeno of Elea composed some parables or thought experiments about reality. In one of them Zeno said one could never reach even a finite destination because to reach that destination one would first have to travel half the distance to destination and then half again in infinite sequence. Because each new step is one half its previous step, one could never arrive at the destination, or so Zeno reasoned. More recently Roger Penrose wrote an excellent book detailing the known mathematical laws of creation. In thinking about matters like Zeno Penrose suggested that the test of a mathematical model was to see if it really modeled some aspect of reality. To the extent it did, it was useful and to the extent it didn’t it might then become something useful. At this point I think the jury is still out on Banach -Tarski but it’s still an intriguing idea.

9. Steve Reina said:

William Walker Interesting point. Until Einstein came around there really wasn’t any use for Riemannian geometry. Just because we can’t find a real world analogue now doesn’t mean one won’t get found in the future by someone with better insight.

10. John Hardin said:

Just to get this out of the way first… 1+1=1 doesn’t describe his illustration. What he illustrated is 1÷6=2. Also, he posed the question whether this paradox has a real world application to quantum physics. I say it absolutely does. Physics isn’t limited to the size of particles according to the standard model. Broader than that, physics includes measurements of distance (space) which IS uncountably infinite. One can calculate half the width of an electron to prove this. My question is, can this hypothesis be applied to the universe to mathematically prove Multiverse Theory?

11. Kyle Judge said:

One of which I never thought I’d understand, the other makes my imagination go haywire. I’m glad it at least made sense. Think it’ll at least seem sound in a science fiction story?

12. Logan Catron said:

I see that he has created a computational method of infinite complexity to generate a sphere. Of course he derives his computational methodology from the physical characteristics of a sphere. We know that every 3d object can be represented by information encoded on a 2d boundary comprising the surface of the object (theorized as gravitaional petrubtions along the event horizon of a black hole or the cmb radiation at the cosmological horizon). He has effectively created a set of information which represents this concept. Because of the nature of the sphere the master data set can be divided into subsets of data that adequately describe the same object from multiple orientations or orders of steps. I think this is due to the recursive nature of the data. I see the sphere as analogous to the universe. We as observers are at the center point. What we see encoded on the cosmological horizon (edge of the sphere) is sufficent to recreate the complex structure found within our cosmological horizon. Because of the uniform nature of a sphere how the information is arranged on that boundary is irrelevant (although the arangement is relevant to the structure of the universe because the universe is not perfectly uniform but still very close). However I think there could be strong applications within black hole physics or posible for describing gravity within 2 or 1 dimensional spaces

13. Logan Catron said:

He means like how a whole number has a derivative value and is this “countable”. Unlike irrational numbers that may continue infinitly with ever reaching the end numerical value. Like the value of ok. That would constitute an uncountable infinity

In a a nutshell: “one can manipulate ‘uncountable infinity’ out of ‘countable infinity.’
Or
“Peeling away the standard interpretation of numbers the difference between 1 and 0 is infinity.”

15. Joe Tritt said:

My 16 year old is very into topology…i encourage him but i think hes a bit brighter than i am : )

16. Rodrigo Peres Triana said:

Extension is just a consciousness phenomenon, Energy-Space interactions are extensionless because we can not have a real concept of them, in other words, we can not have a real concept of a real world but a kind of limited ‘palpable’ imagination…

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