
Layman's Guide to the Banach-Tarski Paradox - carterschonwald
http://www.kuro5hin.org/story/2003/5/23/134430/275
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RiderOfGiraffes
This has been brought up here many times:

<http://searchyc.com/banach+tarski>

I actually think the wikipedia article is pretty good:

<http://en.wikipedia.org/wiki/Banach-Tarski_paradox>

I think the article lunk to here is not actually a very good explanation. It
mixes folksy and technical too freely, really being neither one nor the other,
and even though I actually know the proof quite well (and its 1D and 2D
relations) I still found it unhelpful and awkward.

YMMV - I'd be interested to see if anyone here finds it enlightening.

~~~
roundsquare
I think it was pretty good. Its not really for people who don't know math, but
for people (like me) who know _some_ math but have never looked into the
paradox. The key here is to explain what pieces means and the whole cloud
idea. With that, I sorta get it (I think so anyway, I'd need to understand the
proof better to see if my intuitive feeling is correct).

That being said, I notice a problem immediately. Why do we need 4 pieces? From
the explanation, its pretty much giving a sort of "algorithm" for creating the
4 (or 5) pieces. Essentially, start somewhere, and go around and around, and
put each "atom" into one of 4 sets. I.e. the first "atom" goes into the first
bucket, the second atom into the second bucket, etc... Then, take bucket 1 and
2, and translate them (really just to avoid intersection) and then rotate set
2 so that the atoms are "half way" between those of set 1, and do the same
rotation for bucket 4.

So... why not just call buckets 1 and 3 "bucket A" and buckets 3 and 4 "bucket
B" and just do the translation?

I know enough to know that my "algorithm" doesn't work, but its what the
intuition here suggests. So, there is still some gap in my understanding.

I'm curious how far an intuitive explanation can go. I'll take a look at the
link someone else gave below, but does anyone have an intuitive answer to my
question?

------
ezy
The initial description deceived me (e.g. the layman :-)). It used the word
"piece" in a very technical way. Piece means contiguous chunk in layman's
terms, it typically does not mean a set of fragments of the object (that would
be many pieces).

If someone told me you could, mathematically speaking, take 5 piles of
fragments of a sphere and combine them into two identical spheres, it would
still be slightly mysterious, but it would only take the words "infinite
density" to make it clear, instead of three paragraphs explaining what "piece"
means. :-)

Anyway, I still thought the following explanation was pretty clear. It was
probably not the absolute correct mathematical intuition, but what
summarization of extremely technical topics are completely correct?

------
yannis
I love Maths, but this one got me. I admit publicly it is beyond me. When I
was small my grand mother used to tell me this story:

One night as the priest was lying with his wife, he said, ‘Wife, if you love
me, get up and light a candle, that I may write down a verse which has come
into my head.’ His wife, getting up, lighted the candle, and brought him pen
and inkstand. The priest wrote, and his wife said, ‘O Master of my soul, won’t
you read to me what you have written?’ Whereupon he read, ‘Amongst the green
leaves methinks I see a black hen go with a red bill’.

To this day I have not grasped what she meant with this story.

Please can you remove the Layman from the title?

~~~
acangiano
> To this day I have not grasped what she meant with this story.

The story is from a 19th century Turkish book: "The Turkish Jester: Or The
Pleasantries Of Cogia Nasr Eddin Effendi"
(<http://www.gutenberg.org/etext/16244>).

~~~
swombat
For many more (and rather better written) Nasredin stories (all of which are
highly interesting, entertaining, illuminating, or all three at the same
time), see my father's story-blog:

<http://nasredin.blogspot.com/>

Start from the first one:

[http://nasredin.blogspot.com/2007/10/yes-there-is-nothing-
ne...](http://nasredin.blogspot.com/2007/10/yes-there-is-nothing-new-in-world-
but.html)

------
g__
Simple version of B-T in 1D:

[http://www.math.ucla.edu/~tao/resource/general/121.1.00s/tar...](http://www.math.ucla.edu/~tao/resource/general/121.1.00s/tarski.html)

------
carterschonwald
for those who want to have a go at a better/more mathematical exposition,
<http://www.math.hmc.edu/~su/papers.dir/banachtarski.pdf> is an excellent
rigorous exposition.

~~~
gjm11
And Stan Wagon's book "The Banach-Tarski paradox" is outstandingly good. (But
intended for mathematicians; look before you buy.)

------
hristov
I understood the paradox but did not buy the explanation at all.

~~~
frig
It's not a great explanation b/c it doesn't go into what "measure" and
"volume" mean.

Tough to do without going through a full course but a slightly-less-handwavy
explanation would go like this:

There's a mathematical tool (called measure) that formalizes-and-generalizes
the notion of "volume".

It generally behaves very much as you'd expect; your intuitions about how
"volumes" combine and intersect will generally apply.

There's a catch, though: the way the tool is constructed leaves open the
possibility of a non-measurable set, meaning a set for which the definition of
the tool leaves you unable to assign that set a well-defined "volume"; you
can't just assume that the measure of a set exists.

If you assume the Axiom of Choice then not only are such sets possible, but
you can construct non-measurable sets.

The core process in Banach-Tarksi looks like this:

(1) take the sphere (a 'nice' set, which we'll say has "volume" V)

(2) divide that sphere into some sub-sets that are non-measurable (are sets
for which our tool cannot supply a measurement) (2.a) Effect on total volume:
should have no impact, as the parts we have reassemble to an object of known
volume

(3) move the subsets around by sliding-and-rotating them (3.a) Effect on total
volume: should have no impact, as neither sliding nor rotating changes volume

(4) wind up with 2 spheres (both 'nice' sets, each with "volume" V)???

The paradox comes from getting double the volume through a sequence of
operations that are apparently volume-conserving.

You can go with this a couple different directions.

I'm not convinced this _should_ be an intuitive outcome.

An intuitive, hand-wavy explanation for what's "really going on" would be
something like non-measurable sets carry around infinite amounts of finely-
detailed structure (too finely-detailed to perceive using our measuring tool);
depending on how you position some sets relative to each other their finely-
detailed structure might either cancel out (adding no volume) or reinforce
each other (adding lots of volume).

That said I think the real lesson here is that your intuition is trying to
have its cake -- a non-measurable set -- and eat it too -- have the "volume"
of a non-measurable set be preserved under volume-preserving operations.

------
diN0bot
my first thought is: this happens all the time, if by tiny pieces you mean
atoms. gently heat a balloon until it swells to twice its size. am i missing
something?

~~~
jibiki
A balloon is mostly empty space. The paradox deals with solid spheres rather
than atomic matter.

~~~
diN0bot
true. what i meant was that my off-the-cuff physics radar did not
automatically think this was impossible. i used a gas as an example, but
solids can also be restructured into different volumes and shapes. for
example, diamonds, charcoal and graphite. i'm no doubt missing the precise
definition of the math problem...i was just responding from my practical
layman perspective.

------
anatoly
It occurred to me that a better way to approach explaining the B-T paradox to
a layman might be through the story of the infinite hotel. If you aren't
familiar with how an infinite hotel that is full can still make room for an
infinite number of guests, a good introduction is here:
<http://diveintomark.org/archives/2003/12/04/infinite-hotel>

Let's assume you know and understand the infinite hotel thing from one of
Martin Gardner's books, the link above or any other source. Visualize a number
line with the points 1, 2, 3, 4, ... on it marked as "rooms". Now when you
make room for just five new people, by moving existing guests 1->6, 2->7 and
so on, and freeing rooms 1-5, you can look at it as shifting all the "room
points" five units to the right. When you need to make room for an infinity of
new people, and you move guests 1->2, 2->4, 3->6 and so on, this isn't a
simple shift, because points move non-uniformly: the farther away, the farther
you move. But it turns out that that's just because you don't have much
freedom of movement, so to speak, in one dimension.

It's even more useful to look at the hotel process "in reverse": say you have
all the rooms taken, now people in rooms 1, 3, 5, 7... all move out and
renumber themselves, founding another hotel of the same kind, while people in
rooms 2, 4, 6, 8... squeeze together, each moving to the room half their
original number. So you start with one hotel and you get two identical ones,
again with all the rooms taken. And again, the manner of movement here is non-
uniform, but it's because one dimension is too crowded.

In two dimensions, there's a way to shift an infinity of points to become two
identical infinities, but all the movement is simple shifting together or
rotating together. You divide the points similarly to the even versus odd
numbers division in the hotel example, but because you have a lot more space
to move around in two dimensions, it turns out you can move all the "evens"
and all the "odds" uniformly with respect to each other, as if you were
shifting and rotating them together in the physical world. But the end result
is the same: two infinities where one was, and the basic idea is just the one
with the hotel rooms. The actual way you divide the points into two groups in
two dimensions and shift/rotate them is the tricky technical part of the proof
you'll have to take on faith here. It's not very complex math, but it does
require some abstract higher math knowledge, at about the level of a math
major college degree.

OK, so given all that, what do we do with a ball? In a ball, we first look at
just its surface - the sphere - which is really two-dimensional. You can take
an infinite mesh of points in two dimensions - the one we learned to
"duplicate" with the hotel process above - and stretch them over the sphere,
like a lattice. It's not too difficult to show that by wiggling around this
lattice of points you can cover the whole sphere with its copies, and the
tricky hotel-like rotation and shifting that you do with one lattice, you can
do with all of them together in sync. So it looks like we are breaking the
sphere down into two parts, and shifting/rotating them around to get two
spheres next to each other. Each part is the composition of one half of the
infinite lattice - one half of the "hotel rooms" in the two-dimensional hotel
- collected over all the wiggled lattices together. Only it turns out to be
more complex to unify them like that, so it requires four parts and not two.

Now, these four parts are unbelievably complex-looking. Just as with the
original hotel puzzle there's a break with intuition where you get two of the
same from one, here you do this infinitely many times at the same time, in two
dimensions. Nothing like that could be done in the physical world. You're
basically taking the sphere, breaking it down to individual points, and them
juggling them very intricately hotel-like in an infinity of configurations
together. The point is, any intuitive notion of "volume" or "space taken" by
the sphere breaks down in this process, becomes irrelevant. With the infinite
hotel, two hotels are also taking up twice more "space", but we don't perceive
_that_ as especially freaky on top of everything else, because they stretch to
infinity anyway. But they don't have to; there's an infinity of points inside
a fixed volume too. You could host the infinite hotel on a surface of a sphere
if you were willing to make the rooms really tiny (one point each), and this
is kinda what happens in the Banach-Tarski paradox. So the paradoxical sense
of getting something from nothing is because in the physical world, we can
never go to the scale of individual points, where the notion of "volume" loses
relevance. But in math we can.

Well, back to the ball - if I convinced you, with lots of handwaving, that you
can break down the sphere into four unbelievably complex-looking parts and
reassemble them into two spheres, balls are now easy. Every time you do
something to a point on the sphere, think of a ray from that point to the
ball's center, and do exactly the same shifts and rotations to all the points
on that ray. This way, you're sort of shifting and rotating many concentric
spheres at the same time, all the way from a single point in the center to the
surface of the ball. And each sphere gets reassembled into two of the same, so
the entire ball gets reassembled into two of the same. You do need a bit of a
special treatment for the very point in the center, and that's your fifth
"part".

~~~
drbaskin
Your response is a reasonable explanation of Hilbert's hotel paradox, but that
paradox is really different from the Banach-Tarski paradox. In particular, the
Banach-Tarski paradox is _false_ in two dimensions. (One technical reason for
this is that you can embed a nonabelian free group into SO(3), the rotations
on the sphere, but not into SO(2), the rotations on the circle.) This means
that you should not think of the Banach-Tarski paradox as being a Hilbert-
hotel-like statement about a two dimensional lattice.

------
modelic3
There is always a gap between formalism and intuition and sometimes quite
interesting things happen between these gaps. The Banach-Tarski paradox is
labeled a paradox because it says something about how geometry and topology
are formalized in set theory. There are other formalizations of geometry and
topology where such things are not possible.

------
lucifer
[http://www.biblegateway.com/passage/?search=Matthew+14%3A13-...](http://www.biblegateway.com/passage/?search=Matthew+14%3A13-21&version=KJV)

Now, quick: what happened to the 2 little fishies?

