[edby]

The problem is that you cannot unfold a sphere into a flat surface without causing severe distortion complete with pac-man style wormholes. It's easier to see with a cube. I suspect most of us have built a paper cube before. You trace out the 6 faces on a flat surface, cut them out, and fold them into a cube. Unfold it again, and you'll see that many of the faces aren't anywhere near the edge they touched as a cube. Those edges are what I'm calling the pac-man style wormholes... you cross one boundary and appear instantly on another face of the cube. It works when it's assembled as a cube, but when you lay it flat, you need wormholes.

What this means is that the Earth cannot be both a sphere AND a flat surface. It must be one or the other (assuming we aren't going into higher-dimensional math and allowing for wormholes). I hope that's simple enough. It's either a globe or a plane - it can't be said that "either way works".

So if you make a flat map (any map really), we'll want to test the latitude and longitude of any city on your map against empirical observations. (These were outlined by a previous poster.) If the flat map does not match the known latitude and longitude at any point, we either need to prove the accepted latitude and longitude of that city is incorrect, or we must reject the map as inaccurate.

As I've just pointed out, there is no way for both the globe and a flat map to represent the exact same latitudes and longitudes without severe distortion. We also know that the standard globe model DOES accurately represent the latitude and longitude of every city on it. (Please let us know if you'd care to dispute this point.)

So... before making any FE map, you would first want to come up with some justification for why you are going to challenge the latitude and longitude of cities all over the Earth.

The same can be said for distances between these cities. If we can measure (or even estimate) the distance between any 2 points on the Earth, we'll need to make sure your new map reproduces those empirical results correctly. Once again, you'll first need to establish that there are a pair of cities on the globe which are represented inaccurately.

So if you have a specific, testable reason to challenge the latitude and longitude of any city on the Earth, please share that. (My apologies if this has been given before... if so, I have missed it and would appreciate being directed to it... thanks).

As far as I understand, no Flat earther challenges the measurement of latitude and longitude, which are essentially measurements of position. It's the distances between those positions that they challenge.

[thehumanitarian]

I use Google Maps to get around to places when I am in the States, in India, or even in the UAE and 100% of the time, the directions are accurate. I get to where I intended and it would not have been possible without Google Maps.

[SiDawg]

google maps represents an infinite repeating plane.

mmm not really, google maps represents a globe. Note below: the line of the distance measured is a perfectly straight line on a shere: google maps is "warping" the sphere in to a repeating flat plane: hence why that straight line appears curved in the image below. It's still aware of the sphere as you zoom in: Greenland is warped to all buggery, but as you zoom in, it's aware of the "actual" size



Check this out: i've zoomed in a fair way in to Greenland: check the scale size at the bottom right. Now i've simply "scrolled down" to cuba: i've moved the map without affecting the zoom level at all. The scale is half the size: i.e. Greenland is appearing a lot larger than it actually is compared to Cuba, because the sphere has been warped to appear flat.

[iamcpc]

I use Google Maps to get around to places when I am in the States, in India, or even in the UAE and 100% of the time, the directions are accurate. I get to where I intended and it would not have been possible without Google Maps.

Yet another testament to the concept that an ACCURATE MAP OF THE EARTH EXISTS.

[edby]

I use Google Maps to get around to places when I am in the States, in India, or even in the UAE and 100% of the time, the directions are accurate. I get to where I intended and it would not have been possible without Google Maps.

Yet another testament to the concept that an ACCURATE MAP OF THE EARTH EXISTS.

What Gauss discovered was that one could, in fact, determine the shape of the earth, or of any surface for that matter, just by looking at certain geodetic measurements. In particular, Gauss was interested in the sum of the angles of a triangle whose sides were all segments of shortest paths on the surface, like arcs of great circles on a sphere. Gauss noticed that on many curved surfaces, including spheres, the sum of the angles would be more or less than 180o. For angle sums in excess of 180o, Gauss described the surface as having ‘positive’ curvature, while ‘negative’ curvature corresponded to angle sums of less that 180o. He introduced the concept of what we now call the Gaussian curvature of a surface at a given point .. (Timothy G. Feeman, Portraits of the Earth: A Mathematician Looks at Maps, p.37).

Does Google maps have positive, zero or negative curvature?

Gauss's principle was the basis of all the great geodetic surveys from the early 18th century almost to the present. Geodetic measurement involves no assumptions about astronomy, or speed or satellites. You simply apply a tape measure to the earth, as it were.

[totallackey]

They have lines on them matching the flat paper on which they are drawn.

So are you comfortable that the flat maps you use are an accurate representation of distances on the earth's surface?

I came to understand all man made things are filled with imperfections.

Flat maps have existed long before the idea of a globe.

[edby]

They have lines on them matching the flat paper on which they are drawn.

So are you comfortable that the flat maps you use are an accurate representation of distances on the earth's surface?

I came to understand all man made things are filled with imperfections.

Flat maps have existed long before the idea of a globe.

Suggest again that you refer to Gauss https://en.wikipedia.org/wiki/Theorema_Egregium.

The theorem is that Gaussian curvature can be determined entirely by measuring angles, distances and their rates on a surface, without reference to the particular manner in which the surface is embedded in the ambient 3-dimensional Euclidean space. In other words, the Gaussian curvature of a surface does not change if one bends the surface without stretching it. Thus the Gaussian curvature is an intrinsic invariant of a surface.

[edby]

Spelling this out. What Gauss discovered was that even the inhabitants of a world which was perpetually covered in cloud could discover whether they were living on a flat surface or not, simply by measuring distances.

Thus, if we have a sufficient number of distances given by Google maps, and nothing else, we could tell whether this was the map of a flat surface or not. Get to it!

[edit]Some quick results from Google maps

Distances                km
London   Frankfurt   633.54
London   Nantes   490.17
London   Genoa   1,035.32
Frankfurt   Nantes   817.56
Frankfurt   Genoa   634.42
Nantes   Genoa   873.6

Now see if these can be fitted on a flat surface. If not, then either (1) Google maps is wrong, or (2) earth not flat.

Let's go!

[edby]

I tried again with

London Warsaw
London Bucharest
London  Madrid
Bucharest Madrid
Bucharest Warsaw
Madrid Warsaw

Get the distances on Google maps. (Right click on a place, and select 'measure distance', then left click on place to measure to).

Then plot the distances on a piece of paper. You will need compasses*. I find I can always construct two triangles between four of the places. But the last one always gives a wrong distance, as measured on the paper.

Since the earth is flat, I conclude that Google maps is wrong.

*Further instructions. Start with any two places, say London Warsaw, and draw a line length the scale you want. E.g. 1447km = 14.47 cm.
Then use the compasses to trace a small arc corresponding to the distance between Warsaw and Bucharest, 9.37cm. Then find the place on the arc that gives you the correct distance Bucharest London (20.9cm). You have successfully constructed the first triangle.

Then use the same technique to construct the triangle London-Bucharest-Madrid. It will always be possible to construct two such triangles.

Finally measure the final distance, here Madrid-Warsaw. You will find it doesn't match. So Google maps is wrong. There is no way you can map Google distances onto a flat surface.

[HorstFue]

Google shows a Mercator Projection. Straight lines on a Mercator Projection are Rhumb Lines or loxodromes.
What Google gives as "distance" are the Great Circle distances, in most cases smaller than the loxodrom distances.
It's a bit tricky: Great Circles on a Mercator Projection would be shown as arcs. On a first view these arcs are longer than the straight line, but consider the distortion implied by the Mercator Projection, than the distance following these arcs is smaller.

[edby]

consider the distortion implied by the Mercator Projection, than the distance following these arcs is smaller.

Forget 'distortion'. Do the distances reflect the shortest path across the surface (whatever its shape) from one point to another?

[TomInAustin]

Spelling this out. What Gauss discovered was that even the inhabitants of a world which was perpetually covered in cloud could discover whether they were living on a flat surface or not, simply by measuring distances.

Thus, if we have a sufficient number of distances given by Google maps, and nothing else, we could tell whether this was the map of a flat surface or not. Get to it!

[edit]Some quick results from Google maps

Distances                km
London   Frankfurt   633.54
London   Nantes   490.17
London   Genoa   1,035.32
Frankfurt   Nantes   817.56
Frankfurt   Genoa   634.42
Nantes   Genoa   873.6

Now see if these can be fitted on a flat surface. If not, then either (1) Google maps is wrong, or (2) earth not flat.

Let's go!

Of course, the problem is that Tom Bishop would have you believe these distances can't be correct as no one knows distances at all be it inches, feet, meters, or miles.


All kidding aside, a while back I took Google Sketch Up and tried to create a flat map using distances from both Google Earth and published airline flight distances.  As you can imagine it's not possible to lay them out on a flat map.   The published results were ignored.

[HorstFue]

consider the distortion implied by the Mercator Projection, than the distance following these arcs is smaller.

Forget 'distortion'. Do the distances reflect the shortest path across the surface (whatever its shape) from one point to another?

the shortest distance/connection between two points on a sphere is following the great circle through these two points.

[edby]

consider the distortion implied by the Mercator Projection, than the distance following these arcs is smaller.

Forget 'distortion'. Do the distances reflect the shortest path across the surface (whatever its shape) from one point to another?

the shortest distance/connection between two points on a sphere is following the great circle through these two points.

That's assuming Round Earth Theory. Gauss's point is that the shortest distance can be known in a world where the ground is made of steel, and no possibility of tunnelling, and where there is a constant cloud covering, so no clues from the stars. He claims that if we know just the distances by themselves and nothing else, we can work out whether the surface is curved or flat.

Perhaps this is obvious. I think not, otherwise he would have not called it the 'Remarkable Theorem', but the 'Obvious Theorem'.

I also detected a presumption elsewhere that the assumption of roundness or flatness depends on the projection used. My point is that projections are irrelevant. Just look at the actual (shortest) distances measured, and that tells you the kind of surface you are living on.

[rabinoz]

the shortest distance/connection between two points on a sphere is following the great circle through these two points.

That's assuming Round Earth Theory. Gauss's point is that the shortest distance can be known in a world where the ground is made of steel, and no possibility of tunnelling, and where there is a constant cloud covering, so no clues from the stars. He claims that if we know just the distances by themselves and nothing else, we can work out whether the surface is curved or flat.

Perhaps this is obvious. I think not, otherwise he would have not called it the 'Remarkable Theorem', but the 'Obvious Theorem'.

I also detected a presumption elsewhere that the assumption of roundness or flatness depends on the projection used. My point is that projections are irrelevant. Just look at the actual (shortest) distances measured, and that tells you the kind of surface you are living on.

Gauss's Theorema Egregium

Gauss's theorema egregium states that the Gaussian curvature of a surface embedded in three-space may be understood intrinsically to that surface. "Residents" of the surface may observe the Gaussian curvature of the surface without ever venturing into full three-dimensional space; they can observe the curvature of the surface they live in without even knowing about the three-dimensional space in which they are embedded.

In particular, Gaussian curvature can be measured by checking how closely the arc lengths of circles of small radii correspond to what they should be in Euclidean space, 2 π r. If the arc length of circles tends to be smaller than what is expected in Euclidean space, then the space is positively curved; if larger, negatively; if the same, 0 Gaussian curvature.

Wolfram MathWorld, Gauss's Theorema Egregium

So, it sounds fine in principle, but how does any ordinary person measure the radius (along the surface) and circumference of a circle large enough to measure the exceedingly small Gaussian Curvature of the earth.

The Gaussian Curvature at a point is defined as K = 1/(R₁R₂) where R₁  and R₂ are the principle scalar radii of curvature (just the earth's radius) at the point.

The Spherical Excess Angle can also be used but, for a triangle that is 720 × ((area of triangle)/(surface area of earth)) degrees.
This is so small that it needs a Geodetic Surveyor to get any reliable measurement. And from this flat-earther on the other site

Also, geodetic surveying has been refuted as a credible source of info. No doubt they used telescopes for centuries and would have quickly seen that there is no curvature with the equation given at 8 inches × distance squared.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Geodetic surveying is also a Masonic created. They lied about the poles and curvature. No curve means no ball

So can we trust these surveyors ?

PS I hope you know more about Differential Geometry that I do.

[mitsyara]

I've actively been researching the flat earth idea for a while now. I've found many compelling flat earth points regarding everything from sunsets to seismic waves.
I can honestly say that there are many good points, or at the very least things worth discussing, brought up by many of the flat earth models that I've encountered.
I have not yet found a flat earth response to how we are able to navigate and travel on this earth. I'm hoping I can get a response.

1. If no accurate map of the earth exists how am I able to accurately travel long distances on a consistent basis using a map?

The maps are accurate enough and the maps are flat the same as the earth you travel on.

2. If i'm able to use a map to accurately travel long distances all over the world would that not make my map accurate?

It would make the map accurate enough and the map is flat the same as the earth you travel on.

3. If we don't have an accurate map of the earth (and don't know the distances between far cities) how are ships and planes able to navigate long distances?

We have maps that are accurate enough and those maps are flat the same as the earth you travel on.

These questions really?

Do your own research, as flatter always said that.

Look, the greenland island on a 2D earth map, pick it up then compare to the Africa continel. It's about the same size, right? Actually, no. Africa is much much bigger than Greenland. It's because the Google map 2d model trade off the size to correct the direction. You cant have size and direction always correct when trasfer 3d to 2d. You have to trade off. Either size or direction. Keep learning.

[iamcpc]

We have maps that are accurate enough and those maps are flat the same as the earth you travel on.

Then why do so many flat earthers like Tom Bishop and Max_Almond say that there is no accurate map?

[AATW]

We have maps that are accurate enough and those maps are flat the same as the earth you travel on.

Then why do so many flat earthers like Tom Bishop and Max_Almond say that there is no accurate map?

Because if the distances shown on those maps are correct then the earth cannot be flat. So they have to claim the map is not accurate.
If they produced a flat earth map then it would be immediately shown wrong because it wouldn't match distances or other observations like sunrise/sunset times.
So they don't.

It's a bit like me saying "I'm thinking of a prime number which is also a square number" and you saying "that's not possible".
So long as I don't tell you what number I'm thinking of, you can spend all day going "4 is a square number but it isn't prime, it divides by 2, 9 is a square number but it isn't a prime number, it divides by 3" or "7 is a prime number, but it isn't the square of any whole number"
And I can just say "I'm not thinking of 4, 7 or 9..."

The fact is the globe has been mapped very accurately. The proof of this is the global airline and shipping industry. They get people and goods around reliably, they don't drop out of the sky because Paris was a lot further from New York than they thought it was. They don't get lost. They know where they are accurately, they know how fast they're travelling accurately.
I was on a plane back from Dubai recently, on these flights they have maps so you know where you are at any given point, they have cameras if you're not near a window. Now, I didn't spend 7 hours on a plane checking out of the window to see if I could identify landmarks and see if I was where they're claiming, but someone paranoid enough is free to do so. The claim that airlines don't know how fast they're going, that cable laying ships don't know how much cable they're laying across oceans, that the whole history of cartography has got it wrong is crazy.

[iamcpc]

We have maps that are accurate enough and those maps are flat the same as the earth you travel on.

Then why do so many flat earthers like Tom Bishop and Max_Almond say that there is no accurate map?

Because if the distances shown on those maps are correct then the earth cannot be flat. So they have to claim the map is not accurate.
If they produced a flat earth map then it would be immediately shown wrong because it wouldn't match distances or other observations like sunrise/sunset times.
So they don't.

I disagree. Why can't the distances be correct and the earth be flat? Google maps is not a map of a sphere. It's a 2 dimensional map. I believe that the distances on that 2 dimensional map are accurate. Let's just leave the shape of the earth out of it.

[ICanScienceThat]

We have maps that are accurate enough and those maps are flat the same as the earth you travel on.

Then why do so many flat earthers like Tom Bishop and Max_Almond say that there is no accurate map?

Because if the distances shown on those maps are correct then the earth cannot be flat. So they have to claim the map is not accurate.
If they produced a flat earth map then it would be immediately shown wrong because it wouldn't match distances or other observations like sunrise/sunset times.
So they don't.

I disagree. Why can't the distances be correct and the earth be flat? Google maps is not a map of a sphere. It's a 2 dimensional map. I believe that the distances on that 2 dimensional map are accurate. Let's just leave the shape of the earth out of it.

Google maps IS a map of a sphere. They project a portion of the sphere onto a flat projection for display on your flat screen.

Perhaps you've never seen this presentation before, but I remember doing this in school when I was a kid... here is a video of all sorts of ways to chop up a globe to get it to lie flat(er). No matter how you slice it, you HAVE to slice it:


Try this with a paper cube for example. Most of us have made one of these before. https://www.template.net/business/paper-templates/paper-cube/
When it's folded up into a 3D shape, draw something across several of the faces. Now unfold it. Some of the lines you drew got separated. There are now huge gaps between areas that were touching when it was a 3D cube. There is no way to lay a 3D cube out into a 2D flat sheet without creating gaps like these. The same goes for spheres.

You can't turn a sphere into a flat sheet without gaps. You'll sometimes see these gaps called "pac-manning". That just means a spot where you go off the edge of the map in one area and suddenly pop back onto it in another area.