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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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