Is ( ( 1,2 ) ) a vector bundle for that matter?

Allowing a point position such a label as X[n] where the n-ary components of X are obviously as in C programming array language so that the title of this thread would contain:
( X1, X2 )
where X1[0] = 1, X1[1] = 2, X2[0] = 2, X2[1] = 3
and if you don't like zero based indexing please freely shift the indices to your liking.

Of course these are just usual 2D Cartesian values in a list.
Am I missing something? I understand that there is more interpretation to come, but this is the vector bundle isn't it?

On Sunday, December 19, 2021 at 7:52:01 AM UTC-8, timba...@gmail.com wrote:
> Is ( ( 1,2 ) ) a vector bundle for that matter?
>
> Allowing a point position such a label as X[n] where the n-ary components of X are obviously as in C programming array language so that the title of this thread would contain:
> ( X1, X2 )
> where X1[0] = 1, X1[1] = 2, X2[0] = 2, X2[1] = 3
> and if you don't like zero based indexing please freely shift the indices to your liking.
>
> Of course these are just usual 2D Cartesian values in a list.
> Am I missing something? I understand that there is more interpretation to come, but this is the vector bundle isn't it?

You haven't said whet YOU mean by 'vector bundle'. Give us YOUR definition and maybe we can say.

By the common definition (since the 1050's), the answer is 'no'.

https://en.wikipedia.org/wiki/Vector_bundle
https://mathworld.wolfram.com/VectorBundle.html
https://www.math.stonybrook.edu/~azinger/mat566-spr15/vectorbundles.pdf
https://www.math.ucla.edu/~archristian/teaching/225a-f19/week-5.pdf
https://www.google.com/search?q=vector+bundle

On 12/19/2021 9:51 AM, Timothy Golden wrote:
> Is ( ( 1,2 ) ) a vector bundle for that matter?
>
> Allowing a point position such a label as X[n] where the n-ary components of X are obviously as in C programming array language so that the title of this thread would contain:
> ( X1, X2 )
> where X1[0] = 1, X1[1] = 2, X2[0] = 2, X2[1] = 3
> and if you don't like zero based indexing please freely shift the indices to your liking.
>
> Of course these are just usual 2D Cartesian values in a list.
> Am I missing something? I understand that there is more interpretation to come, but this is the vector bundle isn't it?
>

nope, it is your vector bungle.

On Sunday, December 19, 2021 at 12:30:56 PM UTC-5, FredJeffries wrote:
> On Sunday, December 19, 2021 at 7:52:01 AM UTC-8, timba...@gmail.com wrote:
> > Is ( ( 1,2 ) ) a vector bundle for that matter?
> >
> > Allowing a point position such a label as X[n] where the n-ary components of X are obviously as in C programming array language so that the title of this thread would contain:
> > ( X1, X2 )
> > where X1[0] = 1, X1[1] = 2, X2[0] = 2, X2[1] = 3
> > and if you don't like zero based indexing please freely shift the indices to your liking.
> >
> > Of course these are just usual 2D Cartesian values in a list.
> > Am I missing something? I understand that there is more interpretation to come, but this is the vector bundle isn't it?
> You haven't said whet YOU mean by 'vector bundle'. Give us YOUR definition and maybe we can say.

I definitely mean to go by your definition on this. I certainly have an intuitive sense of the term and I've posed this bundle of two vectors as a vector bundle, but apparently this simple minded approach cannot be taken.

If I specify a differential manifold that is flat with zero differential will that then suffice?
Your mappings simply evaporate, right?
They are trivial, let's say.
Okay?
pi maps M to V maps M to M maps V to V maps V to M maps ...
pi is sort of unified maybe.
the manifolds too.
Of course the term manifolds as many is old and yet claiming they are one is equivalent, no? I guess maybe I could defend this point on the universe being unified, and that some claim that the many are not one is falsifiable.
This would basically land us in ordinary Cartesian space in black and white point positions. Then from there whether segment traces are taken or platelets or whatever those are higher forms. I did see distinctions like sheaves and it seems legitimate to impose some higher structures but the simplest and least structured is a series isn't it? This is as the graphical depictions show it. I don't quite understand the vertical and the 's' but there are definitely discrete depictions taking place in those graphics. If those graphics are inaccurate then there would be a problem with those graphics. Can't we relax in the calculus sense on this? That the continuum will fill itself in? This then is the segment approach at which point my bundle becomes a path. Path dynamics are great. If you want more points that is great too, but I do think that two could do. Maybe even just one, though it will seem degenerate.

Descartes appreciated his magnitudes as general dimensional and really quite ray like and lo and behold the point as a zero dimensional entity requires two dimensions in this study. It's visibility is exactly that of a ray and this observer context might be useful. Occlusion is regularly felt so no claim of universal observation really should be expected. It is fine if we witness a shadow of the universe, but it is still one universe. It is etymologically correct. If not then there be miniature dragons breathing fire at you swooping out of the coffee can tomorrow morning.
Mind the lid.

>
> By the common definition (since the 1050's), the answer is 'no'.
>
> https://en.wikipedia.org/wiki/Vector_bundle
> https://mathworld.wolfram.com/VectorBundle.html
> https://www.math.stonybrook.edu/~azinger/mat566-spr15/vectorbundles.pdf
> https://www.math.ucla.edu/~archristian/teaching/225a-f19/week-5.pdf
> https://www.google.com/search?q=vector+bundle

So if one is seeking a very simple instance of the effect in say 2D reals, which are arguably first vectors in the traditional sense, how many vectors must there be? If two is not enough then could there be three? Isn't the requirement general?

I have reviewed the information mainly from wikipedia and off of a reference to Alexander Grothendieck:
https://en.wikipedia.org/wiki/Alexander_Grothendieck

On Sunday, December 19, 2021 at 10:52:01 AM UTC-5, Timothy Golden wrote:
> Is ( ( 1,2 ) ) a vector bundle for that matter?
>
> Allowing a point position such a label as X[n] where the n-ary components of X are obviously as in C programming array language so that the title of this thread would contain:
> ( X1, X2 )
> where X1[0] = 1, X1[1] = 2, X2[0] = 2, X2[1] = 3
> and if you don't like zero based indexing please freely shift the indices to your liking.
>
> Of course these are just usual 2D Cartesian values in a list.
> Am I missing something? I understand that there is more interpretation to come, but this is the vector bundle isn't it?

I was hoping for more help here.

On 12/22/2021 6:07 PM, Timothy Golden wrote:
> On Sunday, December 19, 2021 at 10:52:01 AM UTC-5, Timothy Golden wrote:
>> Is ( ( 1,2 ) ) a vector bundle for that matter?
>>
>> Allowing a point position such a label as X[n] where the n-ary components of X are obviously as in C programming array language so that the title of this thread would contain:
>> ( X1, X2 )
>> where X1[0] = 1, X1[1] = 2, X2[0] = 2, X2[1] = 3
>> and if you don't like zero based indexing please freely shift the indices to your liking.
>>
>> Of course these are just usual 2D Cartesian values in a list.
>> Am I missing something? I understand that there is more interpretation to come, but this is the vector bundle isn't it?
>
> I was hoping for more help here.

Too much time is required for your learning supporting subject material.

On Wednesday, December 22, 2021 at 8:56:36 PM UTC-5, Serg io wrote:
> On 12/22/2021 6:07 PM, Timothy Golden wrote:
> > On Sunday, December 19, 2021 at 10:52:01 AM UTC-5, Timothy Golden wrote:
> >> Is ( ( 1,2 ) ) a vector bundle for that matter?
> >>
> >> Allowing a point position such a label as X[n] where the n-ary components of X are obviously as in C programming array language so that the title of this thread would contain:
> >> ( X1, X2 )
> >> where X1[0] = 1, X1[1] = 2, X2[0] = 2, X2[1] = 3
> >> and if you don't like zero based indexing please freely shift the indices to your liking.
> >>
> >> Of course these are just usual 2D Cartesian values in a list.
> >> Am I missing something? I understand that there is more interpretation to come, but this is the vector bundle isn't it?
> >
> > I was hoping for more help here.
> Too much time is required for your learning supporting subject material.
According to Wolfram:
"The simplest nontrivial vector bundle is a line bundle on the circle, and is analogous to the Möbius strip."
but if I am right then my instance is simpler.

The requirement for continuosity through the concept of a neighborhood seems to be the sticking point.
In that this is done in the calculus sense, it should be the case that discrete systems will come in though their neighborhood analysis will be trivial:
at Xn the neighborhood gives back simply Xn.

In effect, if you can feel free to punch out a circle and call it a bundle you may as well punch out some points and call them a bundle as well.
That they connected in one case and did not in another: what breaks within the construction?

Nicely enough the terminology can take such a simple form and is appropriate really to physics and atomic relations. The atoms bounce around under standard thermal theory. It's nice to work on a continuum in fluid dynamics, say, but ultimately it is not true.

Standard geometry is a white piece of paper on which we put some black strokes. When that representation takes on arithmetic representation, like a pdf, and hopefully quite vector based and path oriented, points will ultimately be the basis. If a circle can be a fiber bundle then can a million equally spaced points in a circle become a fiber bundle? A thousand? A hundred? A humble two?

Their connection via a line is not a problem. But it is not a part of the specification. It comes along for free. That is the nature of coordinate systems and spaces. To claim that these things come at a higher level is not convincing.

On 12/24/2021 3:21 AM, Timothy Golden wrote:
> On Wednesday, December 22, 2021 at 8:56:36 PM UTC-5, Serg io wrote:
>> On 12/22/2021 6:07 PM, Timothy Golden wrote:
>>> On Sunday, December 19, 2021 at 10:52:01 AM UTC-5, Timothy Golden wrote:
>>>> Is ( ( 1,2 ) ) a vector bundle for that matter?
>>>>
>>>> Allowing a point position such a label as X[n] where the n-ary components of X are obviously as in C programming array language so that the title of this thread would contain:
>>>> ( X1, X2 )
>>>> where X1[0] = 1, X1[1] = 2, X2[0] = 2, X2[1] = 3
>>>> and if you don't like zero based indexing please freely shift the indices to your liking.
>>>>
>>>> Of course these are just usual 2D Cartesian values in a list.
>>>> Am I missing something? I understand that there is more interpretation to come, but this is the vector bundle isn't it?
>>>
>>> I was hoping for more help here.
>> Too much time is required for your learning supporting subject material.
> According to Wolfram:
> "The simplest nontrivial vector bundle is a line bundle on the circle, and is analogous to the Möbius strip."
> but if I am right then my instance is simpler.
>
> The requirement for continuosity through the concept of a neighborhood seems to be the sticking point.
> In that this is done in the calculus sense, it should be the case that discrete systems will come in though their neighborhood analysis will be trivial:
> at Xn the neighborhood gives back simply Xn.
>
> In effect, if you can feel free to punch out a circle and call it a bundle you may as well punch out some points and call them a bundle as well.
> That they connected in one case and did not in another: what breaks within the construction?
>
> Nicely enough the terminology can take such a simple form and is appropriate really to physics and atomic relations. The atoms bounce around under standard thermal theory. It's nice to work on a continuum in fluid dynamics, say, but ultimately it is not true.
>
> Standard geometry is a white piece of paper on which we put some black strokes. When that representation takes on arithmetic representation, like a pdf, and hopefully quite vector based and path oriented, points will ultimately be the basis. If a circle can be a fiber bundle then can a million equally spaced points in a circle become a fiber bundle? A thousand? A hundred? A humble two?
>
> Their connection via a line is not a problem. But it is not a part of the specification. It comes along for free. That is the nature of coordinate systems and spaces. To claim that these things come at a higher level is not convincing.

Fwiw, a bundle for 10 points on the unit circle:

((1, 0),
(0.809017, 0.587785),
(0.309017, 0.951057),
(-0.309017, 0.951056),
(-0.809017, 0.587785),
(-1, -8.74228e-08),
(-0.809017, -0.587785),
(-0.309017, -0.951056),
(0.309017, -0.951056),
(0.809017, -0.587785))

______________________________________________
void circle_index(
ct::plot::cairo::plot_2d& plot,
glm::vec2 origin,
float radius,
unsigned long n
) {
float angle_step = CT_PI2 / n;
float radius_scale = radius * .125;

std::cout << "(";

for (unsigned long i = 0; i < n; ++i)
{
float angle = angle_step * i;

glm::vec2 p_normal = { glm::cos(angle), glm::sin(angle) };
glm::vec2 p0 = origin + p_normal * radius;

plot.circle_filled(p0, radius_scale, CT_RGBF(1, 0, 0));

std::cout << "(" << p0.x << ", " << p0.y << ")";

if (i < n - 1)
{
std::cout << ", \n";
}
}

std::cout << ")\n";
} ______________________________________________

On Friday, December 24, 2021 at 5:34:29 PM UTC-5, Chris M. Thomasson wrote:
> On 12/24/2021 3:21 AM, Timothy Golden wrote:
> > On Wednesday, December 22, 2021 at 8:56:36 PM UTC-5, Serg io wrote:
> >> On 12/22/2021 6:07 PM, Timothy Golden wrote:
> >>> On Sunday, December 19, 2021 at 10:52:01 AM UTC-5, Timothy Golden wrote:
> >>>> Is ( ( 1,2 ) ) a vector bundle for that matter?
> >>>>
> >>>> Allowing a point position such a label as X[n] where the n-ary components of X are obviously as in C programming array language so that the title of this thread would contain:
> >>>> ( X1, X2 )
> >>>> where X1[0] = 1, X1[1] = 2, X2[0] = 2, X2[1] = 3
> >>>> and if you don't like zero based indexing please freely shift the indices to your liking.
> >>>>
> >>>> Of course these are just usual 2D Cartesian values in a list.
> >>>> Am I missing something? I understand that there is more interpretation to come, but this is the vector bundle isn't it?
> >>>
> >>> I was hoping for more help here.
> >> Too much time is required for your learning supporting subject material.
> > According to Wolfram:
> > "The simplest nontrivial vector bundle is a line bundle on the circle, and is analogous to the Möbius strip."
> > but if I am right then my instance is simpler.
> >
> > The requirement for continuosity through the concept of a neighborhood seems to be the sticking point.
> > In that this is done in the calculus sense, it should be the case that discrete systems will come in though their neighborhood analysis will be trivial:
> > at Xn the neighborhood gives back simply Xn.
> >
> > In effect, if you can feel free to punch out a circle and call it a bundle you may as well punch out some points and call them a bundle as well.
> > That they connected in one case and did not in another: what breaks within the construction?
> >
> > Nicely enough the terminology can take such a simple form and is appropriate really to physics and atomic relations. The atoms bounce around under standard thermal theory. It's nice to work on a continuum in fluid dynamics, say, but ultimately it is not true.
> >
> > Standard geometry is a white piece of paper on which we put some black strokes. When that representation takes on arithmetic representation, like a pdf, and hopefully quite vector based and path oriented, points will ultimately be the basis. If a circle can be a fiber bundle then can a million equally spaced points in a circle become a fiber bundle? A thousand? A hundred? A humble two?
> >
> > Their connection via a line is not a problem. But it is not a part of the specification. It comes along for free. That is the nature of coordinate systems and spaces. To claim that these things come at a higher level is not convincing.
> Fwiw, a bundle for 10 points on the unit circle:
>
> ((1, 0),
> (0.809017, 0.587785),
> (0.309017, 0.951057),
> (-0.309017, 0.951056),
> (-0.809017, 0.587785),
> (-1, -8.74228e-08),
> (-0.809017, -0.587785),
> (-0.309017, -0.951056),
> (0.309017, -0.951056),
> (0.809017, -0.587785))
>
> ______________________________________________
> void circle_index(
> ct::plot::cairo::plot_2d& plot,
> glm::vec2 origin,
> float radius,
> unsigned long n
> ) {
> float angle_step = CT_PI2 / n;
> float radius_scale = radius * .125;
>
> std::cout << "(";
>
> for (unsigned long i = 0; i < n; ++i)
> {
> float angle = angle_step * i;
>
> glm::vec2 p_normal = { glm::cos(angle), glm::sin(angle) };
> glm::vec2 p0 = origin + p_normal * radius;
>
> plot.circle_filled(p0, radius_scale, CT_RGBF(1, 0, 0));
>
> std::cout << "(" << p0.x << ", " << p0.y << ")";
>
> if (i < n - 1)
> {
> std::cout << ", \n";
> }
> }
>
> std::cout << ")\n";
> }
> ______________________________________________

Yes: consistent with the notion of a continuum in the calculus sense which is the continuum in the epsilon/delta sense. That we can do it discretely is no trouble at all. As to what we wish to do with this structure: this is simply constructive freedom. All the rules in the world mean little. True mathematicians will make their own. If those rules cohere somehow then possibly they will propagate. Whether academia follows this formula though: I am confident that politics has played a dangerous part in the accumulation as we witness it this day.