Troolon,

If you have oroved that the earth could be any shape, then the earth could be round. Interesting. The only thing I know all FEs to agree on is that the earth is not round. Could be north pole centric, could be south pole centric, could be bi-polar, but they all agree couldn't possibly be round. FEs will like you better if you prove it isn't round rather than that it could be round. To do that, I think you would have to prove that light rays in vacuum can't be straight and measurement is definitely broken. You will be the hero of FE.

I have a degree in math, my son is a math major, and I have reviewed the math behind your claims. Known mathematicians and discussed math all my life. Your claim might be restated as: The earth appears round in Euclidean 3-space but reality is secretly non-Euclidean (measurement is broken, as you say, and rays of light that seem to travel straight are actually bent, perhaps in different ways depending on the position of the observer). Once you assume measurement is broken and light curves in unknown ways, an infinite number of possibilities with no way to know which one.

You also did not incorporate the astronomical transform. When you apply your transform to earth, what about stars millions of light years away? Somehow they moved into a dome, but you have no math for that. Once you say the light bends and you don't know the equation, those stars could be anywhere, sun, moon, etc. The earth could be a cylinder a million miles long and an inch diameter. Meanwhile, Euclidean 3 space with RE gives us GPS, ICBM, airliner finds the airport, sextant north star latitude makes perfect sense, and the solution is singular. If light doesn't bend and measurement isn't broken, the earth is round.

Your ideas re coordinate transforms, basis, and "mathematically equivalent", and your physics claim that it follows a shape change through coordinate transformation are wrong. If you are right, you can go to the math and physics community and explain, for example, the power that coordinate transforms have to bend light. You will be famous, but you will need experiments and equations. You found a reasonable (to you, although not to a professor) of how the earth's shape is unknown. I imagine you will not share your discoveries with them, perhaps posting on TFES is the thing to do with it. To what end?

So the choice of what to do with your discoveries is up to you. You can post on TFES and get some agreement and some explanation of why you are wrong from RE. If I knew some part of science was wrong, I would go to scientists and explain in an attempt to set them straight. But I suppose if they did not agree, it would be because of conspiracy, or perhaps scientists and mathematicians are stupid. Conspiracy can really explain a lot, and you never have to have details or evidence, because its secret, so how could you know the details and the evidence is they all say the wrong RE stuff.

Still, just the idea that changing coordinates changes the shape of a geometric figure would be a starting point. Go to some mathematicians and show them how you changed a sphere into a disk by coordinate conversion. If they say "It doesn't work that way, you don't understand coordinate conversion, the figure remains the same size and shape." Perhaps all mathematicians are wrong, just the ones at UCLA, maybe you can explain, but this is their definition, so don't they get to decide?

You started with a globe where Australia's size matched real world measurements and Sigma Octantus made sense. Then you transformed the surface into a disk, where Australia was wrong size and Sigma Octanus made no sense. Then you claimed this as mathematically equivalent, and then that the physics is then equivalent. Forget which is true, which is useful? If it pleasures you to think that under the RE appearance, non-Euclidean math means it is flat, well, I can't stop you. But the fact that you can map the points on a sphere to a disk if you don't care that the size is wrong and the light bends, but that doesn't prove the earth could be any shape. If anything, you have proved that it is round. In the infinite possibilties of shape agnostic earth, only one works with measurement and light waves traveling straight in a vacuum, Polaris and SIgma Octantus, etc, etc etc.

Does that mean anything? As I said, forget true, what is useful?