[Pete Svarrior]

So the OP is wrong about his/her own work?

No.

Stop trying to derail this thread. Final warning.

[troolon]

Physics is a model.
It's a good model if it produces the same results as we measure in reality.
The distance metric in my model gives the exact same answer as the globe model as the two are the same.
Shapes and distances only make sense compared to a basis.

[troolon]

Please find the sun_at_the_equator image hereby.

If you prefer, it's relatively easy to make these graphs yourself:
- generate the scene as the globe model describes (ie, an earth, a sun and straight lightrays from the sun to the earth)
- express every element in celestial coordinates (latitude, longitude, distance from the center of the earth)
- now draw (lat/long) as an AE projection (this creates a disc) and insert this disc at height `distance` in the cylinder.
Doing this for all elements in the scene will create the pictures i've made.

Or if you like hard math, transform the line-equation this way, and you'll have the equation for lightrays in the flat-earth universe.

[WTF_Seriously]

Please find the sun_at_the_equator image hereby.

If you prefer, it's relatively easy to make these graphs yourself:
- generate the scene as the globe model describes (ie, an earth, a sun and straight lightrays from the sun to the earth)
- express every element in celestial coordinates (latitude, longitude, distance from the center of the earth)
- now draw (lat/long) as an AE projection (this creates a disc) and insert this disc at height `distance` in the cylinder.
Doing this for all elements in the scene will create the pictures i've made.

Or if you like hard math, transform the line-equation this way, and you'll have the equation for lightrays in the flat-earth universe.

You posted another animation which showed this.  Thanks for the still shot though.

Still interested to know the maths and, more importantly, the mechanism which causes light to travel different distances and in different paths from a single light source in your model.

[Dr David Thork]

Please find the sun_at_the_equator image hereby.

If you prefer, it's relatively easy to make these graphs yourself:
- generate the scene as the globe model describes (ie, an earth, a sun and straight lightrays from the sun to the earth)
- express every element in celestial coordinates (latitude, longitude, distance from the center of the earth)
- now draw (lat/long) as an AE projection (this creates a disc) and insert this disc at height `distance` in the cylinder.
Doing this for all elements in the scene will create the pictures i've made.

Or if you like hard math, transform the line-equation this way, and you'll have the equation for lightrays in the flat-earth universe.

You would only expect those sunlight patterns on 20th March and 23rd September.

[troolon]

The mechanism by which light travels is the same as on the globe.
If you can tell me why light travels straight in the globe model, i'll give you the same answer for a flat earth
It just does, it matches observations.

for more maths you can check the site. https://troolon.com.
The computer code that generates the images has these functions:
      xyz_to_celest(), celest_to_ae() and ae_to_screen()
We then chain them together as: ae_to_screen(celest_to_ae(xyz_to_celest(coordinate)))
Or coloquially: express in celestial coords (lat, long, distance), draw the celestial coords as AE cylinder, plot on screen....

This model will work, and all of physics will work, but the maths will be a lot harder than the globe.
The only use of this model is to show that it's possible the earth is flat, (or square, or a globe or duckshaped),
However the true shape of the planet can never be deduced from within this. Only an outside observer can tell us if we truly live on a duck

[WTF_Seriously]

The mechanism by which light travels is the same as on the globe.

Well, not exactly.  Light travels the same distance in all directions from a single light source.   In your model, it does not.  In fact, not only does it not travel equal distance in all directions but the distance that it is able to travel in a given direction changes with position of the sun.  Why is that?

[troolon]

The mechanism by which light travels is the same as on the globe.

Well, not exactly.  Light travels the same distance in all directions from a single light source.   In your model, it does not.  In fact, not only does it not travel equal distance in all directions but the distance that it is able to travel in a given direction changes with position of the sun.  Why is that?

The distance metric is totally whacked and the speed of light is the same. So the light looks also pretty wacked.
But again it's a coordinate transformation. You're trying to debunk globe physics expressed in spherical coordinates...

[AATW]

This is nonsense.
A square has certain properties - 4 equal length sides, 90 degree corners. These are physical things which can be measured.
If the four sides are different lengths - compared to each other - then it's not a square.
If the angles are not all 90 degrees - which is a definition, and the angles can be compared against each other - then it's not a square.
You can define these things in different ways, but that doesn't mean the different things equivalent.

[WTF_Seriously]

The mechanism by which light travels is the same as on the globe.

Well, not exactly.  Light travels the same distance in all directions from a single light source.   In your model, it does not.  In fact, not only does it not travel equal distance in all directions but the distance that it is able to travel in a given direction changes with position of the sun.  Why is that?

The distance metric is totally whacked and the speed of light is the same. So the light looks also pretty wacked.
But again it's a coordinate transformation. You're trying to debunk globe physics expressed in spherical coordinates...

I'm not trying to debunk anything.  I'm talking about this animation.



It shows light changing how far it travels as well as how much it curves.  As others have said, distance is distance whether you're measuring it on a flat surface or curved.  Your 'distance metric' is nonsense.  A point source of light will emit light radially the same distance in all directions if not obstructed.  Your model doesn't show that to be the case.

Something else I noticed about your animation.  It appears the sun is changing elevation as it changes the shape of the pattern of light on the disc.  Is this the case in your model?

[GoldCashew]

Please find the sun_at_the_equator image hereby.

If you prefer, it's relatively easy to make these graphs yourself:
- generate the scene as the globe model describes (ie, an earth, a sun and straight lightrays from the sun to the earth)
- express every element in celestial coordinates (latitude, longitude, distance from the center of the earth)
- now draw (lat/long) as an AE projection (this creates a disc) and insert this disc at height `distance` in the cylinder.
Doing this for all elements in the scene will create the pictures i've made.

Or if you like hard math, transform the line-equation this way, and you'll have the equation for lightrays in the flat-earth universe.

In your above image at the left, the Sun's rays project light in only a "downward" direction from a single point.

Would not a spherical Sun emit rays in all directions though? If this be the case, than wouldn't you have rays projecting sideways  from the Sun and then bending downwards (due to bendy light) towards the flat Earth surface that you show as not lit?

[troolon]

This is nonsense.
A square has certain properties - 4 equal length sides, 90 degree corners. These are physical things which can be measured.
If the four sides are different lengths - compared to each other - then it's not a square.
If the angles are not all 90 degrees - which is a definition, and the angles can be compared against each other - then it's not a square.
You can define these things in different ways, but that doesn't mean the different things equivalent.

Math doesn't care how you draw things. It can't see. For math all these shapes are rectangles.
For an observer inside the coordinate system, all these shapes are also rectangles.
For an observer outside of the system, none of these are a rectangle.
This is the way we construct our universes. From the inside they're identical to globe physics, they're only different from the outside.
But as they're all identical on the inside, we can never know the true shape of the universe.

[scomato]

I have one question about what people see on the ground, vs. what the map describes. In the animated diagram below you have light from the Sun warping every which way, stretching and contracting as needed to illuminate the Earth as we observe it. Wouldn't the sun appear to be wildly contorted in the sky, as the light from one part of the Sun will go on a roller coaster ride around Antarctica, while another will fall straight down? The shape of the sun in the sky should appear to alternate between an elliptical and crescent-moon shape with the changing seasons.



Yet what we actually observe when we look at the Sun is a perfect sphere of unchanging size disappearing over the horizon. You can see dark spots on the Sun, which refutes the idea that sunlight takes different diverging paths to arrive at their special destinations, depending on where it is emitted from. From the observer's perspective, the light from the Sun has taken a straight-line path between its surface and your eyes. Regardless of whether the light was emitted from the top of the Sun or the bottom, straight-line path.



So the only possible explanation I can see is that spacetime itself is warped in between the Sun and the ground observer, creating the illusion that the light travelled in a straight line, configured precisely to simulate the appearance of a massive Sun-like object in distant space. But this introduces more problems than it answers - warping space to that degree would require a phenomenal source of mass or energy - that somehow only acts upon sunlight, without interfering with other forms of matter or energy on Earth.

[DuncanDoenitz]

This is nonsense.
A square has certain properties - 4 equal length sides, 90 degree corners. These are physical things which can be measured.
If the four sides are different lengths - compared to each other - then it's not a square.
If the angles are not all 90 degrees - which is a definition, and the angles can be compared against each other - then it's not a square.
You can define these things in different ways, but that doesn't mean the different things equivalent.

Math doesn't care how you draw things. It can't see. For math all these shapes are rectangles.
For an observer inside the coordinate system, all these shapes are also rectangles.
For an observer outside of the system, none of these are a rectangle.
This is the way we construct our universes. From the inside they're identical to globe physics, they're only different from the outside.
But as they're all identical on the inside, we can never know the true shape of the universe.

I assume you are not a carpet fitter by trade. You would experience some very surprised customers. 

If that sounds facetious, it isn't meant to be.  Forget the complex maths (and I'm not convinced that you're not trolling), but if you can't get your head around the practicality of measuring something you are physically experiencing (be it with a straight ruler, odometer or whatever), and transferring those dimensions onto a model with which you are interacting (be that an engineering drawing, map or mental concept), I don't think you are really in a position to postulate the shape of the cosmos. 

[troolon]

It shows light changing how far it travels as well as how much it curves.  As others have said, distance is distance whether you're measuring it on a flat surface or curved.  Your 'distance metric' is nonsense.  A point source of light will emit light radially the same distance in all directions if not obstructed.  Your model doesn't show that to be the case.

Something else I noticed about your animation.  It appears the sun is changing elevation as it changes the shape of the pattern of light on the disc.  Is this the case in your model?

Imagine i have 2D cartesian coordinates (0,0) and (4, 4).  Then the distance formula would be sqrt((x1-x2)² + (y1-y2)²
In polar coordinates these coordinates would be (0°, 0) en (45°, 4).  If i plug these numbers into the distance formula, i get total gibberish. When you do a coord transform, you must update all formulas. That is what i mean with a new distance metric. It's the old one with compensation for the coord transform.

The animations show both day and night  and seasons. In this animation a year is 3 days long (just to keep the filesize reasonable)
So in all the animations, the sun is both rotating and moving from -23° to +23°.
Because of the dome, yes the sun seems to move up and down, but only for an outside observer.
For someone inside the universe/coordinate system, the height change is not measurable. From the inside this still looks like the reality globe physics describes.

But to understand the grand picture, all you should understand is that
- Physics supports coordinate transforms. Physics does not break if i make my measurements in celestial coordinates.
- coordinate transformations can change any shape into any other shape
=> physics works in any shape universe

[troolon]

I have one question about what people see on the ground, vs. what the map describes. In the animated diagram below you have light from the Sun warping every which way, stretching and contracting as needed to illuminate the Earth as we observe it. Wouldn't the sun appear to be wildly contorted in the sky, as the light from one part of the Sun will go on a roller coaster ride around Antarctica, while another will fall straight down? The shape of the sun in the sky should appear to alternate between an elliptical and crescent-moon shape with the changing seasons.

When you draw a sun on an AE-map (or run a sphere through an AE-projection) it will become distorted. The sun is no longer a sphere.
The light curving compensates exactly for this distortion and an observer will see it as perfectly circular again.
This is because the flat-earth universe is nothing but a coordinate transformation of the old one.
If i make the universe 2 times bigger (you and your ruler included), would you notice?
From inside the coordinate system you don't notice the change, everything appears as before. It's only for an outside observer that the universe appears flat.
From inside the universe looks exactly identical to the globe physics model.

[GoldCashew]

Please find the sun_at_the_equator image hereby.

If you prefer, it's relatively easy to make these graphs yourself:
- generate the scene as the globe model describes (ie, an earth, a sun and straight lightrays from the sun to the earth)
- express every element in celestial coordinates (latitude, longitude, distance from the center of the earth)
- now draw (lat/long) as an AE projection (this creates a disc) and insert this disc at height `distance` in the cylinder.
Doing this for all elements in the scene will create the pictures i've made.

Or if you like hard math, transform the line-equation this way, and you'll have the equation for lightrays in the flat-earth universe.

In your above image at the left, the Sun's rays project light in only a "downward" direction from a single point.

Would not a spherical Sun emit rays in all directions though? If this be the case, than wouldn't you have rays projecting sideways  from the Sun and then bending downwards (due to bendy light) towards the flat Earth surface that you show as not lit?

Hi. I think you may have skipped over the above inquiry.

Was curious to get your take.

[troolon]

I assume you are not a carpet fitter by trade. You would experience some very surprised customers. 

If that sounds facetious, it isn't meant to be.  Forget the complex maths (and I'm not convinced that you're not trolling), but if you can't get your head around the practicality of measuring something you are physically experiencing (be it with a straight ruler, odometer or whatever), and transferring those dimensions onto a model with which you are interacting (be that an engineering drawing, map or mental concept), I don't think you are really in a position to postulate the shape of the cosmos.

Given the time i've spent on programming the animations, i would be a very determined troll
Also i've had this work verified by at least 10 physicists by now. It is correct, i just find it very difficult to explain.
I wholeheartedly agree that this model is not practical. The only uses found so far are:
- to show it is possible to construct a flat earth model
- From this it can be deduced a flat earth is possible, but so is a globe, a pyramid or any shape universe, however it will be impossible for us to ever test, as all models will always predict all the same values for every test
- maybe there could be some value in visualization. I would love to be able to make space tangible and hold it in my hand.

Maybe i should update the title of the post to "Flat earth possible, but true shape of the universe unprovable"
As i spent a lot of time on the animations, i do tend to get bogged down into the details which i can imagine would be confusing.

Maybe I can explain better in 2D:
Imagine a cartesian coordinate grid with a line in it, terminated by points p1 and p2.
Also imagine you have some formulas that work on lines (say a length-formula)
- Now, we will express p1 and p2 in polar coordinates. Mathematically there's nothing wrong, but all our formulas break.
  We can fix our formulas by reverting back to cartesian coords.  so length_in_polar(polar1, polar2) = length_in_cartesian(polar_to_cart(polar1), polar_to_cart(polar2))
  The details don't really matter, just that all formulas can be made to work when we switch to polar coords.
  All right we now have done a coordinate transform, and everything still works.
- Now we will draw these polar coords funnily. An angle is a value between [0° and 360°].
  What if we were to draw our coordinate (angle, dist) onto a orthonormal X/Y grid.  Ie X ranging from [0°, 360°] and Y just the distance.
  Suddenly our line, would become an arc.
  It might seem crazy to draw angles linearly on an X-axis, but remember that in radians, the angle represents also the arclength of a unitcircle. It's also a length.
  However this is just a representation. Mathematically nothing changed. The polar coordinates are still the same we just drew them differently.
Conclusion: We did a little magic trick whereby we changed a line into an arc, while all maths and formulas still treat it as a line.

That's what i'm doing in 3D and with all of physics:
- We take any point in x,y,z-coordinates and transform to celestial coordinates. (latitude, longitude, distance)
  We must also update all formulas in physics to support this change. For example the distance formula needs to be updated as it works on cartesian instead of celestial coords.
  This is not impossible, in fact physicists use celestial coords and coordinate transformation all the time.
- Now we draw our celestial coords funnily. We do the same trick as before where we draw latitude (an angle) on a straight axis instead of as an angle. This effectively creates and AE-projection like effect, and the result is the flat earth you've seen.

So we've taken the globe, and all of physics and transformed it into a flat earth, with all the physics intact.
That's it. Hope it makes more sense like this.

Philosophical implications:
The trick above generalizes to all coord transforms. You can practically turn any shape into any other shape.
And physics, well they use coord transforms all the time.
So all of physics basically works on any shape universe.

From this we can conclude that the universe can have pretty much every possible shape
As all the models always produce the same answer for every question, we'll never be able to test which model is correct and thus what shape the universe has.
Earth could be a globe, or flat or ..... Any shape is possible, none is provable. At least not from within.

[AATW]

Math doesn't care how you draw things. It can't see. For math all these shapes are rectangles.

But there is an underlying physical reality. One which can be observed and measured.
If I go 1m in a direction on a flat plane, turn 90 degrees to my right and go another 1m and repeat that 2 more times I will be back where I started and will have traced out a square with side length 1m.
You can change your units, redefine 1m as 1km. OK, so now you have a square with side length 1km.
You've redefined the units but that doesn't mean the square is now 1000 times bigger on each side.
Or you could change the units so that units in one direction are twice as long as the units in the other. OK, so now you have a square 1m by 2m. It's still a square, it hasn't changed shape.
You could redefine the angle units so now we have a square which has 45 degree internal angles. Again, that doesn't change the shape of the object.
So sure, you can do all kinds of transformations but there is an underlying physical reality which can be observed and measured.
Being "inside" a coordinate system is a meaningless concept.

[troolon]

There is indeed but one reality.
And if i measure/observe something in reality, and i check with the globe model, the model will say everything checks out.
And when i check in the flat model, everything will also check out.
So both models explain/predict reality correctly.

And what does reality really look like? We don't know. It could look like one of the models, or it could be a simulation, or it could not even have a shape.
There's no way to tell.

Being "inside" a coordinate system is a meaningless concept.

You are quite right, and yet you are assuming you're in an orthonormal one because that's the only basis where earth looks like a sphere.
Ultimately a model is just a representation of reality and you can represent it in infinitely many ways.