An Ideal Projection of the Non-Euclidean Flat Earth

Due to recent research on the non-euclidean flat earth, it has been determined an appropriate projection for the use of mapmaking is one which uses a Euler Spiral as its basis. The centre of the two spirals represent the poles.

To increase the accuracy of such a projection, one need only sample smaller areas at a time. This resolves the earlier mentioned issue with this model in that it provides an answer to complaints centering around Gaussian Curvature. It should be noted that this never needed to be shown anyways, as the complaint in the first place was invalid. However, it is now tabled.

In short we know nothing about the shape of the earth. All we have managed to do is figure out the period of sin. I may need to revise my thoughts on mathematical nominalism.

This resolves the earlier mentioned issue with this model in that it provides an answer to complaints centering around Gaussian Curvature.

Care to explain how? Or explain how it was invalid?

It seems to just try and avoid it by cutting the Earth into thin strips.

You can do the same with lots of projections. For example, the HEALPix projection:

All you need to do to remove the distortions is cut thinner and thinner strips.

None of these address the issue of Gaussian curvature which shows the surface of Earth isn't flat, in a way which doesn't care about the space it is in.

It makes it harder as you now need to figure out how a path will go across all the cuts, but it is still there.
The easiest way is to join it all together into a simple shape, and what shape is that? Roughly a sphere.


Nor does it address the more critical Gaussian curvature problem, the fact that non-Euclidean means non-flat, as non-Euclidean things have non-0 Gaussian curvature, meaning they are not flat.

So why not ditch the fancy naming and just call it a non-flat flat Earth, or even simpler, a round Earth?

Non-euclidean does not mean non-flat. Your overly narrow definition of non-euclidean might, but as you might notice it means that euclid's postulates are not used. I have a wonderful history text on non-euclidean geometry. If you'd care for a read I'd be happy to send it to you, so you can see how narrow your definition is even within its use in the mainstream.

Non-euclidean does not mean non-flat. Your overly narrow definition of non-euclidean might, but as you might notice it means that euclid's postulates are not used.

Yes, the postulates for flat geometry.

Can you provide a flat geometry which isn't Euclidean?

i.e. a geometry where the angle sum of a triangle is always 180 degrees (A simple measure of flatness) but where Euclid's postulates don't hold?

Then, show that this is the geometry of the surface of Earth, instead of roughly spherical geometry.
Because even if you can show that some niche non-Euclidean geometry is flat, that doesn't mean the non-Euclidean geometry required for Earth is flat.

In short we know nothing about the shape of the earth. All we have managed to do is figure out the period of sin. I may need to revise my thoughts on mathematical nominalism.

I thought measured distances, satellites etc. prove the shape of the earth. Not least, the WGS84 model used for navigation and mapping.

Am I missing something?

In short we know nothing about the shape of the earth.

Makes one wonder how China can possibly get their goods to America.

Some questions:

1. What was the research? 

2.  How was determined that the Euler spiral fits the data?

3.  Can you give us an example of coordinate transformation using this to explain how it works?  ie. Pick a location and show what happens to the numbers.

In short we know nothing about the shape of the earth. All we have managed to do is figure out the period of sin. I may need to revise my thoughts on mathematical nominalism.

What on Earth (or in Heaven or Hell for that matter) is "the period of sin"?

You might claim to know nothing about the shape of the Earth - that's your problem.

Many others, however, know all that is needed to chart the shortest course in 3D Earth-centred space covering thousands of kilometres.

Then they can program that course into the flight computer of the plane that they are flying and let it fly the plane precisely to the destination.

Now, what about a map that real people can use in the physical 3D world? I guess that you, like all other flat Earthers, simply do not have ONE.

What on Earth (or in Heaven or Hell for that matter) is "the period of sin"?

While not sure, I think he may have meant sine, as in sin(x).

Quote from: rabinoz on January 16, 2020, 02:02:27 PM

What on Earth (or in Heaven or Hell for that matter) is "the period of sin"?

While not sure, I think he may have meant sine, as in sin(x).

Probably but why then is John Davis about 1500 years out of date?

Indian mathematics
Some of the early and very significant developments of trigonometry were in India. Influential works from the 4th–5th century, known as the Siddhantas (of which there were five, the most important of which is the Surya Siddhanta) first defined the sine as the modern relationship between half an angle and half a chord, while also defining the cosine, versine, and inverse sine. Soon afterwards, another Indian mathematician and astronomer, Aryabhata (476–550 AD), collected and expanded upon the developments of the Siddhantas in an important work called the Aryabhatiya. The Siddhantas and the Aryabhatiya contain the earliest surviving tables of sine values and versine (1 − cosine) values, in 3.75° intervals from 0° to 90°, to an accuracy of 4 decimal places. They used the words jya for sine, kojya for cosine, utkrama-jya for versine, and otkram jya for inverse sine. The words jya and kojya eventually became sine and cosine respectively after a mistranslation described above.

But, hey what can we expect from a . . . . ?

Some of those Indian astronomers, especially Aryabhata were pretty smart about the shape of the Earth too!

Due to recent research on the non-euclidean flat earth, it has been determined an appropriate projection for the use of mapmaking is one which uses a Euler Spiral as its basis. The centre of the two spirals represent the poles.

To increase the accuracy of such a projection, one need only sample smaller areas at a time. This resolves the earlier mentioned issue with this model in that it provides an answer to complaints centering around Gaussian Curvature. It should be noted that this never needed to be shown anyways, as the complaint in the first place was invalid. However, it is now tabled.

Lol! Non-euclidean geometry is another name for spherical geometry. Well played, John Davis! A round earther masquerading in plain sight as a flat earther! 

All we have managed to do is figure out the period of sin.

You've been sinning for too long, John.  Repent now before it's too late.

Already available here: https://teespring.com/en-GB/euler-spiral-world-map?pid=652

More about the projection here:

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Great video
Nice voice.
Resilt is same problem as the "orange peel" prjoection.

What have you got against South America?

Nice. Glad to see it was replicated by a round earther. Usually makes the discussion part of it all a bit easier. I noticed after searching off that projection that Tom had a comment around this being basically a dipolar projection; I'm not sure I agree. Its easy to see some similarities though with my previous work around a fractally recursive geographic models which peaked my interest.

This particular case is extremely interesting by Trenton Vogt and Darin J. Ulness



Modifying a few variables of course, but as a general idea.

This particular case is extremely interesting by Trenton Vogt and Darin J. Ulness



Modifying a few variables of course, but as a general idea.

What about these:

So we're playing mathmatecal semantics?

Nice. Glad to see it was replicated by a round earther. Usually makes the discussion part of it all a bit easier.

Well it should make it easier, with you admitting it is a projection of a round Earth.
But you want to keep up the pretence of it magically being flat.

But if it was flat, then you wouldn't need any special projection, you would just scale it down.

Tom had a comment around this being basically a dipolar projection; I'm not sure I agree.

Yes, because it is a projection of a sphere, which has 2 poles.
The 2 small pluses in your diagram are the 2 poles.

Quote from: John Davis on January 21, 2020, 10:20:12 AM

Nice. Glad to see it was replicated by a round earther. Usually makes the discussion part of it all a bit easier.

Well it should make it easier, with you admitting it is a projection of a round Earth.

It can be a projection of two things.

But you want to keep up the pretence of it magically being flat.

But if it was flat, then you wouldn't need any special projection, you would just scale it down.

This is only true assuming Euclid was correct. We know he was not.

Quote from: John Davis on January 21, 2020, 10:20:12 AM

Tom had a comment around this being basically a dipolar projection; I'm not sure I agree.

Yes, because it is a projection of a sphere, which has 2 poles.
The 2 small pluses in your diagram are the 2 poles.

The difference being the rest of the space. If you refresh yourself with Wilmores projection, you'll see the difference instantly - namely a bunch of water.

Quote from: JackBlack on January 21, 2020, 12:14:52 PM

Quote from: John Davis on January 21, 2020, 10:20:12 AM

Nice. Glad to see it was replicated by a round earther. Usually makes the discussion part of it all a bit easier.

Well it should make it easier, with you admitting it is a projection of a round Earth.

It can be a projection of two things.

Quote

But you want to keep up the pretence of it magically being flat.

But if it was flat, then you wouldn't need any special projection, you would just scale it down.

This is only true assuming Euclid was correct. We know he was not.

Quote

Quote from: John Davis on January 21, 2020, 10:20:12 AM

Tom had a comment around this being basically a dipolar projection; I'm not sure I agree.

Yes, because it is a projection of a sphere, which has 2 poles.
The 2 small pluses in your diagram are the 2 poles.

The difference being the rest of the space. If you refresh yourself with Wilmores projection, you'll see the difference instantly - namely a bunch of water.

What's the problem with the WGS84 model?

It's a fine enough projection, but like any projection it doesn't necessarily match reality.

This is only true assuming Euclid was correct. We know he was not.

Again, can you explain how you can have a flat geometry which doesn't follow Euclid's postulates?
So far you have just asserted it.
But every non-Euclidean geometry I know of is not flat.

Can you also explain why a flat geometry would require any special projection to project it onto a flat surface.
What makes it special that you can't just put it on a flat surface by scaling it down?

Is it because your "flat" surface isn't actually flat?

The difference being the rest of the space.

I never said it was THE bipolar projection, just that it was a dipolar projection.
Yes, there is a big difference between the more common bipolar projection and this one.
This one cuts Earth apart into a thin, winding strip, while the normal bipolar projection just distorts Earth to keep it all connected.

It's a fine enough projection, but like any projection it doesn't necessarily match reality.

In what way?  Seems to work OK with eg. GPNSS everywhere.

Quote from: John Davis on January 21, 2020, 12:55:35 PM

It's a fine enough projection, but like any projection it doesn't necessarily match reality.

In what way?  Seems to work OK with eg. GPNSS everywhere.

It may provide a mathematically accurate way to predict certain phenomena, but that does not mean it is an accurate model of the earth. One can construct infinitely many as accurate projections and models that have equal predictive and historical confirmation powers, which leads any reasonable person to come to the conclusion that it is not necessarily a correct way to look at reality simply because it holds these traits. It must be shown that these infinitely many other views are incorrect to take it on face value that it is the correct representation.

This proves difficult.

Quote from: inquisitive on January 21, 2020, 01:08:29 PM

Quote from: John Davis on January 21, 2020, 12:55:35 PM

It's a fine enough projection, but like any projection it doesn't necessarily match reality.

In what way?  Seems to work OK with eg. GPNSS everywhere.

It may provide a mathematically accurate way to predict certain phenomena, but that does not mean it is an accurate model of the earth. One can construct infinitely many as accurate projections and models that have equal predictive and historical confirmation powers, which leads any reasonable person to come to the conclusion that it is not necessarily a correct way to look at reality simply because it holds these traits. It must be shown that these infinitely many other views are incorrect to take it on face value that it is the correct representation.

This proves difficult.

It does not 'predict' certain 'phenomena'. It is an accurate model of the earth, agreed and used internationally.

What does 'have equal predictive and historical confirmation powers' mean, why do you write in this way, it does not make what you say any more correct.

Can you please provide details of an accurate alternative that has the correct distances, maybe even a map of a flat earth.

That's not now things work, we do not prove everything we might come up with is wrong before we know what is correct.  However, nice try to confuse the discussion, millions base their navigation etc. on the WGS84 model and it works.  As requested, where are the errors in it?

Why this reminds me of Julia fractal?