The Earth is a projective plane.

Firstly, I don't agree that Earth is non-orientable. Surface distortions does not effect if it is orientable or not, unless you are using a different definition of it.

The Earth is non-orientable because it cannot be perfectly embedded in Euclidean space. It cannot be perfectly embedded in Euclidean space because our space is minutely curved. For clarity, with which part of that chain of reasoning do you disagree?

Even if it was non-orientable, that doesn't make it a projective plane.

True, but it greatly narrows down the options. What further shows it is a real projective plane is that it is one sided. Do you disagree with that?

A projective plane is one example of a non-orientable object. There are others, such as mobius strips and klein bottles.

You are probably correct that I may be being too restrictive by asserting it is a 'real projective plane' rather than simply a projective plane (although I have aforementioned reservations). But can you suggest a fourth type of non-orientable object that could be equal in contention than a projective plane, unless you believe it is feasible for the Earth to be a Mobius strip or a Klein bottle?

You are trying to pretend Earth is flat by blatantly misusing concepts and making things seem far more complex than needed and that this complexity magically makes Earth flat.

I have yet to claim that this process of reasoning I am employing proves the Earth is flat, there are a few further steps needed. However, I do not believe I am misusing these concepts, and I do not believe you have adequately established I am.

Quote from: JackBlack on June 19, 2018, 03:00:22 PM

Firstly, I don't agree that Earth is non-orientable. Surface distortions does not effect if it is orientable or not, unless you are using a different definition of it.

The Earth is non-orientable because it cannot be perfectly embedded in Euclidean space. It cannot be perfectly embedded in Euclidean space because our space is minutely curved. For clarity, with which part of that chain of reasoning do you disagree?

Yes, I agree that the space-like component of spacetime is minutely curved, but from a practical point of view we need to know how much.

In this document, Wm. Robert Johnston, has calculated the difference between the "relativistic" and "non-relativistic" diameters for the Sun and Earth.
Calculations on space-time curvature within the Earth and Sun by Wm. Robert Johnston and he probably knows what he is talking about, Wm. Robert Johnston, Ph.D. (Physics), M.S. (Physics), B.A. (Astronomy).

I'll let you look at the details and just quote his conclusion:

9 Conclusion
This exercise produced several expressions of relativistic curvature for solar system objects. The true diameters of the Sun and Earth are 4.1 km and 4.4 mm
greater, respectively, than one would expect from applying Euclidean geometry (C = πd) to the observed surface of these bodies.

The relative errors are about 1 in 174,000 at the surface of the sun and a little over 1 in 3 billion at the surface of the earth.
In other words, forget any idea of a non-Euclidean space near the earth.

Yet, so many seem to complicate matters to unnecessarily and drag in relativistic mechanics quite unnecessarily.
Sometimes I think that the only purpose in this is to show off their deep knowledge of physics.

Quote

9 Conclusion
This exercise produced several expressions of relativistic curvature for solar system objects. The true diameters of the Sun and Earth are 4.1 km and 4.4 mm
greater, respectively, than one would expect from applying Euclidean geometry (C = πd) to the observed surface of these bodies.

The relative errors are about 1 in 174,000 at the surface of the sun and a little over 1 in 3 billion at the surface of the earth.
In other words, forget any idea of a non-Euclidean space near the earth.

Yes and no. I've already agreed that space is minutely curved, but I don't think we should 'forget' this if it will lead us to a more accurate understanding of the Earth's shape. That 4.4mm difference is all it takes to ensure the Earth is non-orientable.

Yes and no. I've already agreed that space is minutely curved, but I don't think we should 'forget' this if it will lead us to a more accurate understanding of the Earth's shape. That 4.4mm difference is all it takes to ensure the Earth is non-orientable.

But it totally immaterial in any discussion of whether the earth is an almost perfect and or an almost perfect plane.

Quote from: Tessa Yuri on June 19, 2018, 11:45:16 PM

Yes and no. I've already agreed that space is minutely curved, but I don't think we should 'forget' this if it will lead us to a more accurate understanding of the Earth's shape. That 4.4mm difference is all it takes to ensure the Earth is non-orientable.

But it totally immaterial in any discussion of whether the earth is an almost perfect and or an almost perfect plane.

Sorry, I don't understand your question. Could you clarify?

Quote from: rabinoz on June 20, 2018, 12:07:56 AM

Quote from: Tessa Yuri on June 19, 2018, 11:45:16 PM

Yes and no. I've already agreed that space is minutely curved, but I don't think we should 'forget' this if it will lead us to a more accurate understanding of the Earth's shape. That 4.4mm difference is all it takes to ensure the Earth is non-orientable.

But that 4.4mm difference is totally immaterial in any discussion of whether the earth is an almost perfect and or an almost perfect plane.

Sorry, I don't understand your question. Could you clarify?

It wasn't a question, but I did have a small omission, sorry so I amplified it a bit..

My meaning was that this 4.4 mm is completely insignificant compared to the earth's diameter of over12,700 km.

Space exploration orbital mechanics involves the highest velocities and largest change in gravitational potential likely to be met in the solar system.
These calculations are bad enough using Newtonian mechanics let alone General Relativity and the difference is far less than any likely measurement or control error.

As an example this reference calculates how little effect GR has on an Earth-Moon transfer orbit.
Could we send a man safely to the Moon in a rocket without knowledge of general relativity?
The first answer concludes that:

So that's how much difference including general relativity makes to the calculated Earth-Moon transfer orbit - about 1.3 cm!

When one considers that an error in the injection burn time of Apollo 11, I think, lead to a velocity error of about 3 m/s, this 1.3 cm after well over 384,000 km is totally I significant.

Do you wonder the those doing orbital calculations use Newtonian mechanics and not GR? In the few cases where relativistic calculations are necessarily, they usually added as corrections later.

Though GR is incorporated into the predictions of the positions of planetary locations and for Mercury fly-bys the difference is a few metres and a few seconds. See:

Accounting for general relativity at Mercury
So, the answer is yes, the MESSENGER navigators did take relativistic effects into account. But do they really have to? Taylor tried out some navigational predictions with and without the relativistic corrections included. He said that normally, for MESSENGER, they include the relativistic parameters for the Sun, but stick with the mathematically simpler Newtonian description of the motion of all the other planets. So, first of all, he turned on relativity for all the planets ("and Pluto too!" he said), just to see if there was any change. He said he "demonstrated that this made a difference only to the meter level when integrating a trajectory starting last September to the Mercury flyby on January 14th. I suspect that even that tiny amount of dithering was due to numerical precision errors."

From: Accounting for general relativity at Mercury

Even there Newton was not far off the mark!

Of course, for particle physicists it's a completely different story.

But that 4.4mm difference is totally immaterial in any discussion of whether the earth is an almost perfect and or an almost perfect plane.

I'll assume you mean an almost perfect sphere or an almost perfect plane.

I agree that this 4.4mm is not what shows the Earth is flat, but it does show it is non-orientable. Because it is non-orientable and one-sided it is a projective plane. Is this incorrect?

The Earth is non-orientable because it cannot be perfectly embedded in Euclidean space. It cannot be perfectly embedded in Euclidean space because our space is minutely curved. For clarity, with which part of that chain of reasoning do you disagree?

The minute curvature of our space does not mean Earth is not capable of embedded in Euclidean space.
A simple example of extreme curvature is a spherical shell in a curved space. That is still capable of embedded in Euclidean space.

True, but it greatly narrows down the options. What further shows it is a real projective plane is that it is one sided. Do you disagree with that?

Yes.
The other simple examples I gave are also both one sided.

unless you believe it is feasible for the Earth to be a Mobius strip or a Klein bottle?

I don't believe Earth to be an orientable surface.
This is because you can take an object, such as a pawl, move it around Earth however you wish, keeping it restricted to the surface, and still having the same allowed rotation. You cannot reverse it by moving it along the surface (as you can with projective planes and mobius strips and so on) and thus it is an orientable surface.

I have yet to claim that this process of reasoning I am employing proves the Earth is flat, there are a few further steps needed.

That was implied by your comments, regardless of any further steps needed, these steps don't help as they are factually wrong.

However, I do not believe I am misusing these concepts, and I do not believe you have adequately established I am.

So is that a veiled admission that you are and a claim that I just haven't proven it yet?

Quote from: Tessa Yuri on June 19, 2018, 03:28:28 PM

The Earth is non-orientable because it cannot be perfectly embedded in Euclidean space. It cannot be perfectly embedded in Euclidean space because our space is minutely curved. For clarity, with which part of that chain of reasoning do you disagree?

The minute curvature of our space does not mean Earth is not capable of embedded in Euclidean space.

But surely it does? There is a 4.4mm deviation, as rabinoz has explained, from Euclidean space. Earth can be immersed in Euclidean space, but not perfectly embedded, because of this deviation.

I don't believe Earth to be an orientable surface.
This is because you can take an object, such as a pawl, move it around Earth however you wish, keeping it restricted to the surface, and still having the same allowed rotation. You cannot reverse it by moving it along the surface (as you can with projective planes and mobius strips and so on) and thus it is an orientable surface.

This is true of the Earth in reality, but not true of the Earth in perfectly Euclidean space. The 4.4mm deviation would cause an overlap.

Quote from: Tessa Yuri on June 19, 2018, 03:28:28 PM

I have yet to claim that this process of reasoning I am employing proves the Earth is flat, there are a few further steps needed.

That was implied by your comments, regardless of any further steps needed, these steps don't help as they are factually wrong.

I'm sorry that my comments implied to you that assertion. I'm also sorry your opinion of me is so low you believe you can dismiss any future arguments of mine as 'factually wrong' before I have even presented them.

Quote from: Tessa Yuri on June 19, 2018, 03:28:28 PM

However, I do not believe I am misusing these concepts, and I do not believe you have adequately established I am.

So is that a veiled admission that you are and a claim that I just haven't proven it yet?

No, it is not.

Seriously.
You found a 3×10^-10 amount of error between netown and einstein?
All of RE is false then and can come into question?
Come on.

But surely it does?

No it doesn't. Not in the slightest.
You can happily embed a surface from one space in another.

This is because the surface itself can be defined by itself, without needing to appeal to the space it is in.

You appear to be equating embedding with embedding and having everything be exactly the same relative to the space it is in. The 2 are not equivalent.

This is true of the Earth in reality, but not true of the Earth in perfectly Euclidean space. The 4.4mm deviation would cause an overlap.

No. The deviation would not cause any overlap.
It would result in distortion of the shape, nothing more.
You are trying to put Earth in, and then put in a distorted form as well.

Again, you can easily take a pawl (or other handed, planar object), move it all around Earth (trapped in the surface), and still have it in the same orientation. There is no way to bring it back to the same point but not have the same orientation.

Conversely, for a non-orientable object, you can do that. For example by moving such an object along a mobius strip until it returns back to its starting point, it will be reversed.

I'm also sorry your opinion of me is so low you believe you can dismiss any future arguments of mine as 'factually wrong' before I have even presented them.

I meant the steps you had already presented.
It doesn't matter if you want to build upon the foundation of the arguments you have already presented as your foundation is factually wrong and any argument built upon a false foundation will be flawed.

Quote from: Tessa Yuri on June 20, 2018, 02:56:27 AM

But surely it does?

No it doesn't. Not in the slightest.
You can happily embed a surface from one space in another.

This is because the surface itself can be defined by itself, without needing to appeal to the space it is in.

Okay, I see your point here. But don't you contradict yourself here:

It would result in distortion of the shape, nothing more.

?

I suppose my idea of the perceived shape and the real projective planar nature have been fairly divorced from each other. I'll have to have a bit of a rethink here, although I still agree with the crux of what I've said. Thanks for pointing out the flaws!

You found a 3×10^-10 amount of error between netown and einstein?

It's amazing how much a single question can be wrong.

You found

I theorised it was there, but it was rabinoz (a round Earther) that actually found it.

a 3×10^-10

3.45x10^-10

amount of error

It's not an amount of error, but a deviation.

between netown and einstein?

Not even in the slightest. The deviation is between Euclidean space and the space that the Earth inhabits in reality.

You found a 3×10^-10 amount of error between netown and einstein?

No. Please make some attempt.

Okay, I see your point here. But don't you contradict yourself here:

Nope.

You can measure a surface in many ways. In some cases that will be relative to the space, in others it will be purely based upon the surface.

A spherical surface is a good example of this.
In Euclidean space, it is a spherical shell, with a particular area and a certain radius, and it curves relative to the space.
Putting it in other spaces (or taking it from them and what they would be there) it may have a different radius, but the same area, due to contraction or stretching along the radius. It may not curve relative to the space.
It is still the same surface, it just appears different (or distorted) relative to the space.

Meanwhile, a real projective plane cannot be embedded in 3D Euclidean space as it self intersects.
It is similar in this regard to a klein bottle.
This does not happen with Earth.
The surface does not self-intersect.

Quote from: Themightykabool on June 20, 2018, 11:04:36 AM

You found a 3×10^-10 amount of error between netown and einstein?

It's amazing how much a single question can be wrong.

Quote from: Themightykabool on June 20, 2018, 11:04:36 AM

You found

I theorised it was there, but it was rabinoz (a round Earther) that actually found it.

Quote from: Themightykabool on June 20, 2018, 11:04:36 AM

a 3×10^-10

3.45x10^-10

Quote from: Themightykabool on June 20, 2018, 11:04:36 AM

amount of error

It's not an amount of error, but a deviation.

Quote from: Themightykabool on June 20, 2018, 11:04:36 AM

between netown and einstein?

Not even in the slightest. The deviation is between Euclidean space and the space that the Earth inhabits in reality.

Quote from: Themightykabool on June 20, 2018, 11:04:36 AM

You found a 3×10^-10 amount of error between netown and einstein?

No. Please make some attempt.

Wow.
So whats YOUR point then?
Mine was that the number was so small and insignificant to matter.
Ok 3.45 vs 3 x 10^-10.
Whoopdee do.   
My point is that extrapolating a round 3d surface area just so you can show it in 2d math does not make the 3d object magically no longer 3d in reality.

Im exaggerating the FE typical fallacy of discreditting RE math just because newton and einstein differ on a few concepts (of scale).
If you think that 4.5x^-11 has any bearing for estimating the flattness of earth then you neednt go to sapce to see the curve. 
You can climb up on a 10ft ladder and know all that you can perceive as flat.

As per jackblack you definitely are misusing.
If you claim you are NOT claiming, then stop inserting nonapplicable comments.
Jane and her nonrelevant comments...

Quote from: Tessa Yuri on June 20, 2018, 02:26:10 PM

Okay, I see your point here. But don't you contradict yourself here:

Nope.

You can measure a surface in many ways. In some cases that will be relative to the space, in others it will be purely based upon the surface.

A spherical surface is a good example of this.
In Euclidean space, it is a spherical shell, with a particular area and a certain radius, and it curves relative to the space.
Putting it in other spaces (or taking it from them and what they would be there) it may have a different radius, but the same area, due to contraction or stretching along the radius. It may not curve relative to the space.
It is still the same surface, it just appears different (or distorted) relative to the space.

Meanwhile, a real projective plane cannot be embedded in 3D Euclidean space as it self intersects.
It is similar in this regard to a klein bottle.
This does not happen with Earth.
The surface does not self-intersect.

Ahh ok thanks, I have no idea why I forgot about distortion, one of my own key points. Remind me not to post so early in the morning.

I'll have to get back to you about your last sentence though.