1) Does a unit square contain the same amount of points as a unit line?
(We are talking about real numbers here.)

2) If yes, that means there has to be a 1-to-1 mapping between those
points. Give a function that expresses such a mapping.

3) If the answer to the first question is yes, then it follows that
the amount of points inside a unit cube is also the same as the amount
of points on a unit line. The same for a four-dimensional hypercube,
and so on. Can you give a generic function that gives a 1-to-1 mapping
between a unit line and an n-dimensional unit cube?

4) So the next question is: Does a countably-infinite-dimensional unit
cube contain the same amount of points as a unit line? If yes, can you
give a 1-to-1 mapping between them?

5) And the logical extreme: Does an uncountably-infinite-dimensional
unit cube contain the same amount of points as a unit line? Explain why.
(Also explain how the number of dimensions can be uncountably infinite.
That seems to defy the definition of "dimension".)

-- 
                                                          - Warp

Post a reply to this message

On 13/07/2013 2:50 PM, Warp wrote:
> 1) Does a unit square contain the same amount of points as a unit line?
> (We are talking about real numbers here.)

I would say no. They are different orders of infinity.
I recommend Rudy Rucker's fiction book White Light as a good entry to 
thinking about infinity. It is a bit strange but easier to read than 
Edwin Abbott Abbott's Flatland.

http://en.wikipedia.org/wiki/White_Light_%28Rudy_Rucker_novel%29


-- 
Regards
     Stephen

Post a reply to this message

On 13/07/2013 02:50 PM, Warp wrote:
> 1) Does a unit square contain the same amount of points as a unit line?
> (We are talking about real numbers here.)

Weirdly, yes.

(As an aside, if the answer were no, the rest of the questions all 
become in applicable, which kind of gives the game away...)

> 2) If yes, that means there has to be a 1-to-1 mapping between those
> points. Give a function that expresses such a mapping.

Given the 2D coordinates of a point on the unit square, you can 
interleave their decimal digits, which always yields a unique point on 
the unit line. For example,

   0.3425
   0.2183 -> 0. 32 41 28 53

> 3) If the answer to the first question is yes, then it follows that
> the amount of points inside a unit cube is also the same as the amount
> of points on a unit line. The same for a four-dimensional hypercube,
> and so on. Can you give a generic function that gives a 1-to-1 mapping
> between a unit line and an n-dimensional unit cube?

This amounts to interleaving N sequences of digits rather than just two.

> 4) So the next question is: Does a countably-infinite-dimensional unit
> cube contain the same amount of points as a unit line? If yes, can you
> give a 1-to-1 mapping between them?

Now I'm lost. I can neither prove nor refute that the cardinalities match.

Post a reply to this message

Le 13/07/2013 15:50, Warp nous fit lire :
> 1) Does a unit square contain the same amount of points as a unit line?
> (We are talking about real numbers here.)
> 
Yes. It must be, from Cantor, something bigger than Aleph-0. (and is not
countable)


> 2) If yes, that means there has to be a 1-to-1 mapping between those
> points. Give a function that expresses such a mapping.

IIRC, Peano curve would do. Expressing the position of any point using
(x,y) can be replaced with the ratio (between 0 and 1) over the Peano
curve of the length to reach the position over the total length of the
curve (and the precision of the ratio number can be used to infer the
order of the peano curve, or you can use any very big order of Peano
curve as it suits you)

You can also consider Wunderlich curves. Same approach. All real numbers
in [0,1) maps to the points on the unit square.

> 
> 3) If the answer to the first question is yes, then it follows that
> the amount of points inside a unit cube is also the same as the amount
> of points on a unit line. The same for a four-dimensional hypercube,
> and so on. Can you give a generic function that gives a 1-to-1 mapping
> between a unit line and an n-dimensional unit cube?

Be careful, using infinite and common sense might yields some surprise.
A basic approach would be to convert the first two coordinates (both in
[0,1) ) into a single coordinate using either Peano or Wunderlich, and
then repeat.

 X,Y -> a
 X,Y,Z -> a,Z -> b
 X,Y,Z,T -> a,Z,T -> b,T -> c

 C0,C1,C2,... Cn -> (C0,C1)->p0,C2,... Cn

> 
> 4) So the next question is: Does a countably-infinite-dimensional unit
> cube contain the same amount of points as a unit line? If yes, can you
> give a 1-to-1 mapping between them?

Yes, apply the answer of point 3, once the dimension value has been
mapped (because countably-infinite) to the set of natural number , in
order to have the number of original coordinates.

The mapping (of dimension) is obvious, because it is the definition of
the countably-infinite. ("obvious" as in math, after 10 to 2000 pages of
demonstration and 40 minutes or 500 years of intense thinking)

> 
> 5) And the logical extreme: Does an uncountably-infinite-dimensional
> unit cube contain the same amount of points as a unit line? Explain why.
> (Also explain how the number of dimensions can be uncountably infinite.
> That seems to defy the definition of "dimension".)
> 
No.
Dimension can be uncountably infinite if it is not mappable to the set
of natural number. Such case can be the set of real number in [0,1),
(see Cantor's diagonal).
The number of dimension can be uncountably infinite as soon as the
number of dimension is no more restricted to a value mappable to the set
of natural integer. Fractal object have such dimension.

Post a reply to this message

Stephen <mca### [at] aolcom> wrote:
> On 13/07/2013 2:50 PM, Warp wrote:
> > 1) Does a unit square contain the same amount of points as a unit line?
> > (We are talking about real numbers here.)

> I would say no. They are different orders of infinity.

You would have to explain that in more detail.

Remember that, for example, one could easily think that there are "more"
rational numbers than there are integers. Yet that's not correct. There
are equally many. (That's because it's possible to construct a one-to-one
relationship between every rational number and every integer.)

If you can construct a function that gives a one-to-one mapping between
the points on a unit line and the points on a unit square, that means
that both sets have the same size (as unintuitive as that might sound.)

-- 
                                                          - Warp

Post a reply to this message

Orchid Win7 v1 <voi### [at] devnull> wrote:
> > 2) If yes, that means there has to be a 1-to-1 mapping between those
> > points. Give a function that expresses such a mapping.

> Given the 2D coordinates of a point on the unit square, you can 
> interleave their decimal digits, which always yields a unique point on 
> the unit line. For example,

>    0.3425
>    0.2183 -> 0. 32 41 28 53

That doesn't work because it's not a one-to-one mapping. Ie. the mapping
is not unambiguous.

-- 
                                                          - Warp

Post a reply to this message

On 13/07/2013 05:44 PM, Warp wrote:
> If you can construct a function that gives a one-to-one mapping between
> the points on a unit line and the points on a unit square, that means
> that both sets have the same size (as unintuitive as that might sound.)

Nothing in set theory is intuitive. ;-)

Post a reply to this message

On 13/07/2013 05:47 PM, Warp wrote:
> Orchid Win7 v1<voi### [at] devnull>  wrote:
>>> 2) If yes, that means there has to be a 1-to-1 mapping between those
>>> points. Give a function that expresses such a mapping.
>
>> Given the 2D coordinates of a point on the unit square, you can
>> interleave their decimal digits, which always yields a unique point on
>> the unit line. For example,
>
>>     0.3425
>>     0.2183 ->  0. 32 41 28 53
>
> That doesn't work because it's not a one-to-one mapping. Ie. the mapping
> is not unambiguous.

That would imply that two distinct 2D points exist which map to the same 
1D point. Can you provide such a counter-example?

Post a reply to this message

Le_Forgeron <jgr### [at] freefr> wrote:
> > 5) And the logical extreme: Does an uncountably-infinite-dimensional
> > unit cube contain the same amount of points as a unit line? Explain why.
> > (Also explain how the number of dimensions can be uncountably infinite.
> > That seems to defy the definition of "dimension".)
> > 
> No.
> Dimension can be uncountably infinite if it is not mappable to the set
> of natural number. Such case can be the set of real number in [0,1),
> (see Cantor's diagonal).
> The number of dimension can be uncountably infinite as soon as the
> number of dimension is no more restricted to a value mappable to the set
> of natural integer. Fractal object have such dimension.

Note that this might be a bit of a trick question. It's talking about
an "uncountably-infinite-dimensional *unit cube*". As you say,
dimensionality being uncountably infinite requires non-integral dimensions.
Thus answering the question correctly would first require demonstrating
whether a non-integral-dimensional unit cube can exist. (I think it can't,
because it goes contrary to the definition of "cube".)

Of course for the sake of the question we can loosen up the requirement
of the object being a unit cube and simply say that it's an arbitrary
object of a unit "volume".

-- 
                                                          - Warp

Post a reply to this message

Orchid Win7 v1 <voi### [at] devnull> wrote:
> On 13/07/2013 05:47 PM, Warp wrote:
> > Orchid Win7 v1<voi### [at] devnull>  wrote:
> >>> 2) If yes, that means there has to be a 1-to-1 mapping between those
> >>> points. Give a function that expresses such a mapping.
> >
> >> Given the 2D coordinates of a point on the unit square, you can
> >> interleave their decimal digits, which always yields a unique point on
> >> the unit line. For example,
> >
> >>     0.3425
> >>     0.2183 ->  0. 32 41 28 53
> >
> > That doesn't work because it's not a one-to-one mapping. Ie. the mapping
> > is not unambiguous.

> That would imply that two distinct 2D points exist which map to the same 
> 1D point. Can you provide such a counter-example?

By using a decimal notation you are equating the set of real numbers
with the set of integers, thus making the assumption that the set of
real numbers is countable. Not all real numbers can be represented with
digits, because digits can only be used to represent a countably infinite
set, which the set of reals isn't.

-- 
                                                          - Warp

Post a reply to this message