Question: Why can't you extend the complex numbers to 3D space?

Answer: Because the hairy ball theorem says so.

Yes, that's right. There really is a theorem called "the hairy ball 
theorem". Isn't that wonderful? :-D

Perhaps even more satisfyingly, what this theorem /says/ isn't some 
obscure exotic thing that only a mathematician could understand. 
Actually, it just says you can't comb the hair on a ball flat. You 
always end up with at least one tufty bit. (You /can/ comb the hair flat 
on a flat plane, a torus, or a number of other 3D shapes. Just not a 
sphere.)

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Le 07/05/2012 23:09, Orchid Win7 v1 nous fit lire :
> Question: Why can't you extend the complex numbers to 3D space?
> 
> Answer: Because the hairy ball theorem says so.
> 
> Yes, that's right. There really is a theorem called "the hairy ball
> theorem". Isn't that wonderful? :-D
> 
> Perhaps even more satisfyingly, what this theorem /says/ isn't some
> obscure exotic thing that only a mathematician could understand.
> Actually, it just says you can't comb the hair on a ball flat. You
> always end up with at least one tufty bit. (You /can/ comb the hair flat
> on a flat plane, a torus, or a number of other 3D shapes. Just not a
> sphere.)

It's even more deeper. The theorem says you cannot comb a sphere of odd
dimension, but that even dimension is ok.
(2D sphere == circle, 4D sphere is left as an exercise)

That's also why, even if complex cannot be extended in 3D, they can in
4D. (look at quaternion...

Notice that there is a meteorological application of the theorem: there
is always on earth at least one place where the horizontal speed of wind
is null.

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Orchid Win7 v1 escreveu:
> Question: Why can't you extend the complex numbers to 3D space?
> 
> Answer: Because the hairy ball theorem says so.

is it so hairy that numbers got scared to death and complexed? :p

-- 
a game sig: http://tinyurl.com/d3rxz9

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On 5/7/2012 14:09, Orchid Win7 v1 wrote:
> Question: Why can't you extend the complex numbers to 3D space?

Someone also figured out that you can do probability with 1D numbers and 2D 
numbers, but not any higher dimensions. The speculation is that's why 
quantum effects have the same math as waves and not something either "common 
sense" or more baroque than they are.

I.e., you can do math on "the probability that X happens given that Y 
happens" and deal with "and" and "or" and Bayesian probabilities, using 
either real numbers or complex numbers, but not 3D numbers.

I don't understand the proof myself, but so I'm told.

-- 
Darren New, San Diego CA, USA (PST)
   "Oh no! We're out of code juice!"
   "Don't panic. There's beans and filters
    in the cabinet."

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Le 08/05/2012 03:53, Darren New nous fit lire :
> Someone also figured out that you can do probability with 1D numbers and
> 2D numbers, but not any higher dimensions.

Are you talking about random movement having a non-null (aka about 100%)
probability of returning to your starting point (sort of) ?

In 1D (moving along a line), with whatever balanced distribution, it
seems rather obvious.

In 2D (moving in a plane), it's less obvious but you will cross your
previous path at a moment or another.

but once in 3D, you're lost and would quasi-never cross any points from
your previous path.

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Le 08/05/2012 09:29, Le_Forgeron nous fit lire :
> Le 08/05/2012 03:53, Darren New nous fit lire :
>> Someone also figured out that you can do probability with 1D numbers and
>> 2D numbers, but not any higher dimensions.
> 
> Are you talking about random movement having a non-null (aka about 100%)
> probability of returning to your starting point (sort of) ?
> 
> In 1D (moving along a line), with whatever balanced distribution, it
> seems rather obvious.
> 
> In 2D (moving in a plane), it's less obvious but you will cross your
> previous path at a moment or another.
> 
> but once in 3D, you're lost and would quasi-never cross any points from
> your previous path.

Oh, I forgot also a difference as dimensions are raised:
1D & 2D: you can evaluate the length/surface of any grid-aligned
line/polygon by counting the number of element of the grid inside the
limit. (you need to count twice: one for grid's nodes fully inside, and
another for grid's node also on the perimeter)

That computation fails once in 3D or more!

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>> Yes, that's right. There really is a theorem called "the hairy ball
>> theorem". Isn't that wonderful? :-D
>>
>> Perhaps even more satisfyingly, what this theorem /says/ isn't some
>> obscure exotic thing that only a mathematician could understand.
>> Actually, it just says you can't comb the hair on a ball flat. You
>> always end up with at least one tufty bit. (You /can/ comb the hair flat
>> on a flat plane, a torus, or a number of other 3D shapes. Just not a
>> sphere.)
>
> It's even more deeper. The theorem says you cannot comb a sphere of odd
> dimension, but that even dimension is ok.

Indeed.

> That's also why, even if complex cannot be extended in 3D, they can in
> 4D. (look at quaternion...)

No. Quaternions do not form a field. Neither do the hypercomplex 
numbers, nor any of the other 4D generalisations.

> Notice that there is a meteorological application of the theorem: there
> is always on earth at least one place where the horizontal speed of wind
> is null.

Interesting...

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On 5/8/2012 1:31 AM, Invisible wrote:
>
>> That's also why, even if complex cannot be extended in 3D, they can in
>> 4D. (look at quaternion...)
>
> No. Quaternions do not form a field. Neither do the hypercomplex
> numbers, nor any of the other 4D generalisations.
>

That's irreverent to Le_Forgeron's point. The application of the hairy 
ball theorem here does not depend on the numbers forming a field.

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On 5/8/2012 7:15 AM, Kevin Wampler wrote:
> On 5/8/2012 1:31 AM, Invisible wrote:
>>
>>> That's also why, even if complex cannot be extended in 3D, they can in
>>> 4D. (look at quaternion...)
>>
>> No. Quaternions do not form a field. Neither do the hypercomplex
>> numbers, nor any of the other 4D generalisations.
>>
>
> That's irreverent to Le_Forgeron's point. The application of the hairy
> ball theorem here does not depend on the numbers forming a field.

I should clarify, I took the fact that the extension wouldn't be a field 
to be implicit in Le_Forgeron's point, since the Hairy ball theorem's 
use doesn't depend on the numbers forming a field.  Not saying your 
reading was technically incorrect.

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On 5/7/2012 6:53 PM, Darren New wrote:
>
> I.e., you can do math on "the probability that X happens given that Y
> happens" and deal with "and" and "or" and Bayesian probabilities, using
> either real numbers or complex numbers, but not 3D numbers.
>

Do you happen to remember the name of this theorem?  I'm curious how 
fierce a devil is in the details.

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>>>> That's also why, even if complex cannot be extended in 3D, they can in
>>>> 4D. (look at quaternion...)
>>>
>>> No. Quaternions do not form a field. Neither do the hypercomplex
>>> numbers, nor any of the other 4D generalisations.
>>>
>>
>> That's irreverent to Le_Forgeron's point. The application of the hairy
>> ball theorem here does not depend on the numbers forming a field.
>
> I should clarify, I took the fact that the extension wouldn't be a field
> to be implicit in Le_Forgeron's point, since the Hairy ball theorem's
> use doesn't depend on the numbers forming a field. Not saying your
> reading was technically incorrect.

By "extend the complex numbers", I implicitly meant to extend it to make 
a larger /field/. Apparently that is impossible. There are of course 
several ways you can extend them if you don't mind the result failing to 
be a field...

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On 5/8/2012 8:23 AM, Invisible wrote:
>> I should clarify, I took the fact that the extension wouldn't be a field
>> to be implicit in Le_Forgeron's point, since the Hairy ball theorem's
>> use doesn't depend on the numbers forming a field. Not saying your
>> reading was technically incorrect.
>
> By "extend the complex numbers", I implicitly meant to extend it to make
> a larger /field/. Apparently that is impossible. There are of course
> several ways you can extend them if you don't mind the result failing to
> be a field...

This is certainly true, but your options are more limited in 3d than in 
4d.  I was just assuming that Le_Forgeron was already well aware that 
quaternions aren't a field.  Operating under this assumption I took his 
point to be that your options for extending the complex numbers to 3d 
are more limited then when extending them to 4d, even though obviously 
neither extension will result in a field.

Your interpretation of Le_Forgeron is of course also consistent with 
what he posted.  However I'm confused how you reconcile this "any 
extension must be a field" view with your original post.  In the initial 
post you say:

"Question: Why can't you extend the complex numbers to 3D space?

Answer: Because the hairy ball theorem says so."

But that seems a pretty weak explanation of why such an extension isn't 
possible if you're insisting that the extension be a field.  After all 
such an extension isn't possible to 4d, even though the hairy ball 
theorem *doesn't* apply in that case.  If you're going to take this view 
why bother with the hairy ball theorem at all?  You can prove more by 
other means.

If, on the other hand, you allow the extension to not form a field, then 
it makes more sense why you'd use the hairy ball theorem, as it helps to 
explain why the quaternions are possible in 4d but not 3d.  Not saying 
your view is wrong in a technical sense... it just seems a little 
hodge-podge to me.  Perhaps I am missing something?

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Invisible <voi### [at] devnull> wrote:
> No. Quaternions do not form a field. Neither do the hypercomplex 
> numbers, nor any of the other 4D generalisations.

  The reason is that it's not possible to make multiplication commutative
for them.

-- 
                                                          - Warp

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On 08/05/2012 06:23 PM, Warp wrote:
> Invisible<voi### [at] devnull>  wrote:
>> No. Quaternions do not form a field. Neither do the hypercomplex
>> numbers, nor any of the other 4D generalisations.
>
>    The reason is that it's not possible to make multiplication commutative
> for them.

The hypercomplex numbers possess commutative multiplication. What they 
do /not/ possess is multiplicative inverses. (Not for all elements, anyway.)

In short, you can construct lots of 4D number systems, all of which fail 
to be a field in a different way.

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On 08/05/2012 04:43 PM, Kevin Wampler wrote:

> "Question: Why can't you extend the complex numbers to 3D space?
>
> Answer: Because the hairy ball theorem says so."
>
> But that seems a pretty weak explanation of why such an extension isn't
> possible if you're insisting that the extension be a field. After all
> such an extension isn't possible to 4d, even though the hairy ball
> theorem *doesn't* apply in that case. If you're going to take this view
> why bother with the hairy ball theorem at all? You can prove more by
> other means.

OK, point taken.

It's just that the "other means" are usually incomprehensible, whereas 
"you can't comb a sphere flat" is a pretty intuitive description of why 
you can't do something.

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On 5/8/2012 7:34, Kevin Wampler wrote:
> On 5/7/2012 6:53 PM, Darren New wrote:
>>
>> I.e., you can do math on "the probability that X happens given that Y
>> happens" and deal with "and" and "or" and Bayesian probabilities, using
>> either real numbers or complex numbers, but not 3D numbers.
>>
>
> Do you happen to remember the name of this theorem? I'm curious how fierce a
> devil is in the details.

I know where I read about it, I think. I'll try to check if it's actually 
named when I find the book. :-)

-- 
Darren New, San Diego CA, USA (PST)
   "Oh no! We're out of code juice!"
   "Don't panic. There's beans and filters
    in the cabinet."

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On 5/11/2012 9:43 AM, Darren New wrote:
>> Do you happen to remember the name of this theorem? I'm curious how
>> fierce a
>> devil is in the details.
>
> I know where I read about it, I think. I'll try to check if it's
> actually named when I find the book. :-)
>

If you have the name of the book and don't feel like looking for it 
yourself, that'd probably be enough for me to go on.

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On 5/11/2012 11:07, Kevin Wampler wrote:
> If you have the name of the book and don't feel like looking for it
> yourself, that'd probably be enough for me to go on.

It was mentioned somewhere when I was reading about quantum physics, either 
wikipedia or Feynman's QED or something. I can't really find it, but it's 
basically the complex probability amplitude equations, and a mention that 
you don't get the multiplcation/addition of probabilities working out to 
(for example) always come out as "1" if you go higher than two dimensions.

-- 
Darren New, San Diego CA, USA (PST)
   "Oh no! We're out of code juice!"
   "Don't panic. There's beans and filters
    in the cabinet."

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