The Banach Tarski
Paradox
The topic that I have chosen is a paradox called the
Banach Tarksi Paradox. The paradox has a
lot of elements to it, but in very simple terms it “states that it is possible to
take a solid sphere (a "ball"), cut it up into a finite number
of pieces, rearrange them using only rotations and translations, and
re-assemble them into two identical copies of the original
sphere.”(Kuro5hin) The mathematicians that proved this paradox to be true were
Stefan Banach and Alfred Tarski.
Banach was born in Poland during the
year 1892 and was one of the core members of the Lwow School of Mathematics.
Along with the Banach Tarski Paradox, he was known for three other theorems
also named after himself. Although these theorems were great pieces of work, he
was mainly known for a book called The Theory of Linear Operations. His
partner, Alfred Tarski, was also Polish but born in the year 1901. Not only was
he a great mathematician, he was also a famous philosopher and logician. He was
known for changing the face of logic in the 20th Century.
There is one thing that needs to be
made clear before trying to explain this paradox. When reading this you cannot
think about an actual object. It is obviously simple logic for someone to take
a golf ball, cut it up, and make two golf balls that are the exact volume of
the original golf ball. However, this paradox is not talking about a golf ball.
It is talking about a sphere with an infinite amount of points in a space with a
set of topical and measurable relations. The reason that this can’t be done
with a golf ball is because a golf ball is finite. If you try and cut up a golf
ball in you’re eventually going to run out of physical mass to cut. On the
other hand the sphere has no mass. It’s not possible to take a knife and
physically cut this sphere. Once you can wrap your head around this then the
paradox isn’t all that hard to understand. In order for this paradox to work
you must also assume that the axiom of choice is real. The axiom of choice is “an axiom of set theory equivalent to the statement that the
Cartesian product of a collection of non-empty sets is non-empty.”(Wiki)
Since I am not qualified to explain
to anyone about each equation that goes into this paradox, I am going to try
and get you to understand by using real world analogies and by trying to
explain the paradox in really simple terms.
One way this might be able to be
explained is by using a line segment. A line segment can be divided into an
infinite amount of sections. So if you cut the segment into two pieces then
both segments still have an infinite amount of points. Infinity equals infinity so you can say that
these two segments are equal to the original segment.
Another way that could make it a
little clearer is by using an analogy with a balloon. Say you have a balloon
and you empty out the air of the balloon into a container. So then you have an
empty balloon. Then the original balloon and fill it up with half the air it
used to have. Then fill up another balloon with the other half of the air so
you have two equally sized balloons. Both of these balloons are not the same
size as the original balloon, (however if you have any background of the laws
of gas) but if you decrease the pressure of the room in half, then the size of
each balloon will double and become the size of the original balloon. There is
a question about density when it you do this because the balloons have the same
volume as the original but only half the density. This is obviously correct in
the physical world, but in the mathematical sphere has an infinite density. If
you were to cut infinity in half it would still be infinity. So in the
mathematical world of the sphere the density is almost irrelevant.
So obviously understanding the
concept of infinity is crucial to understanding this. Another key factor in this
paradox is using immeasurable sets. The sphere that is being split up doesn’t
have a finite amount of atoms in it. So it is immeasurable. Every object in the
real world has a finite amount of atoms, therefore making each object
measurable. One amazing fact that has been proven is that this paradox can be
accomplished by cutting the sphere into a mere five pieces. It is impossible to
mathematically these pieces though. We just have to know they exist and they
have a strange re-assembling property.
I obviously wish I was smart enough
to understand all of the equations that go into this paradox. However even
without understanding them it’s really interesting to think about. It’s tough
for us to wrap our heads around this topic because we’re so used to dealing
with concrete objects. Again the most important thing to understanding this
theorem is that we are talking about a mathematical sphere that has no mass. It’s
mass is infinite. So therefore with an infinite mass it is possible to make a
two spheres identical to the original sphere, out of the original sphere.
Works Cited
"Alfred Tarski." Wikipedia. Wikimedia Foundation, 11 Oct. 2014. Web. 11 Nov. 2014.
"Banach-Tarski Paradox." Wikipedia. Wikimedia Foundation, n.d. Web. 9 Nov. 2014.
"Layman's Guide to the Banach-Tarski Paradox || Kuro5hin.org." Layman's Guide to the Banach-Tarski Paradox || Kuro5hin.org. N.p., n.d. Web. 9 Nov. 2014.
Li, Sean. "A Layman's Explanation of the Banach-Tarski Paradox." A Reasoners Miscellany. N.p., n.d. Web. 9 Nov. 2014.
MarkCC. "The Banach-Tarski Non-Paradox." Good Math Bad Math. N.p., n.d. Web. 9 Nov. 2014.
"Stefan Banach." Wikipedia. Wikimedia Foundation, 11 Oct. 2014. Web. 11 Nov. 2014.
This is an interesting paradox. It was a little hard for me to think of this sphere not being a part of the physical universe, but just more of a concept. I found a video to help me more understand the topic: https://www.youtube.com/watch?v=Tk4ubu7BlSk. But what i'm still confused about is why was this proven? What utility does it serve in the mathematical world?
ReplyDeleteAlthough I was a bit confused while first reading about the paradox, I actually found it amazing when I read that this paradox made it possible to take a sphere and cut it into finite pieces to make two of that original sphere and that they would both be the same volume as the original. It was also interesting to learn that mass is infinite because I would've never guessed that was true with measuring mass in the past.
ReplyDeleteThis is definitely the most interesting paradox I have ever heard of or read about. I understand your difficulty in explaining the paradox to the class because of the great mathematical skill set it requires. I like how even though you couldn't explain the paradox in mathematical terms, you still explained the paradox with real world examples. I appreciated the summary and although it was tough to visualize the sphere I felt I still grasped the concept you were try to explain to us.
ReplyDeleteI have never heard about this paradox before but it is very interesting. I would like to know how you found the topic? I was very interested throughout your whole post. I thought you explained the concept well but it was hard to visualize. It still amazes me that you can cut a sphere into so many pieces and create two identical ones!
ReplyDeleteAlthough this may have been the most confusing summary I have read I still found the essential idea of this paradox very interesting. I can see how this paradox becomes very difficult when explaining it to someone because not only is the mathematical aspect difficult to understand but the whole idea of this "sphere". I had found that to be the most confusing part since I had felt that may have been the hardest part to prove to people. And yet without understanding that part you may never understand the entire paradox. Overall though I thought it was a great summary with a great paradox to explain.
ReplyDeletewhile this paradox seems original in its application with the idea of a sphere, many parts of it remind me of other concepts and paradoxes in math. The main concept that comes to my mind is a fractal. fractals are infinitely repeating and therefore must have infinite properties making any subsets or multiples of the original, equal to the original. having this sort of background knowledge I'm left begging the question as to whether or not this concept was explained correctly or whether it IS simply just a different application of previous concepts.
ReplyDeleteI kind of understand the concept, as it seems rather simple, its just that it's something not concrete. I think you did a good job trying to explain it as it's not something easy to grasp. I think that maybe if you showed some of the equations to get a sense of what it is you are talking about would've helped, even if the equations themselves were not easy to understand. After reading this, I'm left wondering, "What purpose does this paradox serve?"
ReplyDeleteI sort of follow but not quite through with this paradox. Its hard for me to visualize it so how it operates still remains vague to me. If you had put a few examples of applications of this then I would have a better picture. Also with the balloon analogy, you mentioned that the balloons with half the air will have the same volume as the original balloon with the full air (twice as much as the other balloons); now I don't understand how they would have the same volume when obviously the size (/R) of the original balloon is bigger than that of the two half blown balloons, except if the balloons have infinite volumes too, in which case it would have a different effect on their densities.
ReplyDeleteI don't really understand how you can cut a sphere into an infinite amount and create the same shape into two without actually cutting it with a knife. I also thought it was cool that you can cut a golf ball into pieces and with those pieces create a two new golf balls. I found that interesting.
ReplyDeleteThis paradox is very interesting, I've just never heard about it before! Cutting a sphere into pieces and getting small spheres blows my mind, very cool! You did a good job explaining this topic. Good job!
ReplyDeleteThe real world examples really helped out with the comprehension of this paradox. It's hard to just visualize a sphere being cut into x amount of pieces then using those to create two new ones. I really liked how you used the balloons as an example with the gas laws. Being a chemistry major, it really helped as an example.
ReplyDeleteThrough our eyes, it is a bit difficult to imagine a sphere with no mass. Trying to visualize on how to rearrange and put the pieces back to normal is quite hard. Once I continued reading about the line segment and how if it is cut in half, both sides will still equal to the line segment because they are infinite.
ReplyDeleteThe examples you gave helped with understanding this paradox. I liked how you explained how you can cut the sphere into a bunch of pieces and create two identical ones. I found your topic to be somewhat confusing but it was interesting.
ReplyDeleteThe idea behind this paradox is amazing to think about. It was really hard to understand at first, and even going through it a second time was still somewhat confusing. But I worked on it, and can comprehend the paradox. It was really shocking when I read in the introduction that you get two spheres. This was a really great blog post.
ReplyDelete