This discussion centers on the complexities of determining the global size of a manifold, particularly in the context of Euclidean 3D space. Participants highlight that while manifolds can be locally Euclidean, their global properties, such as size and volume, are not inherently defined without embedding them in a Euclidean space. The conversation emphasizes that size measurements require external geometric definitions, and approximations can be misleading due to topological distortions. Ultimately, the discussion concludes that understanding the interconnections of manifold charts can provide insights into global measurements, such as geodesic distances.
PREREQUISITESMathematicians, physicists, and students of geometry interested in advanced concepts of manifolds, topology, and their applications in understanding spatial dimensions and measurements.
But since the Earth is locally 2D , topological properties will be distorted in 3d charts. Unless maybe you make a copy/replica of the Earth in 3-d. There is such a think as area forms and volume forms but these are more geometric than topological, i.e., they are not invariant under topological transformations.sqljunkey said:Taking the example of the spherical Earth what if instead of constructing atlases and charts in 2d space we construct these in 3d space? So we would have charts of cities, rivers, oceans and the Earth's core etc all in 3d.
If i pieced this all together again would i get what i started with? So if i knew the dimensions of each 3d chart let's say wouldn't i be able to know the dimensions of the whole earth. ( if Earth can be considered a manifold)
What is the dimension of the manifold itself? This is an invariant. And what other dimensions are you referring to?As an example, A circle is 1-dimensional but it may sit in 2,3 or higher dimensions.sqljunkey said:I was wondering if it was possible to determine the size of a manifold globally. Suppose I had a manifold that sits in 3 dimensions. I could construct a Euclidean space around in the same space and be able to say things of the dimensions right?
I assumed 3-D.WWGD said:What is the dimension of the manifold itself?
WWGD said:But since the Earth is locally 2D , topological properties will be distorted in 3d charts.
This doesn't work in the manifold picture. Locally flat means there is a certain kind of bijection, but it doesn't mean no deformation takes place. So in case of a three dimensional manifold when you cut out small pieces, transform them into a Euclidean space where size makes sense, you can add up those volumes, but it doesn't reflect the original size. And there is a technical difficulty: the local pieces of manifolds overlap. You cannot chose a closed part of them. Anyway, you can get an approximation, but of what? Volume isn't properly defined, it cannot be done. Except we have an embedding in a Euclidean space and we use its definition of volume. And this is geometry. The language of manifolds isn't necessary in this case.sqljunkey said:... wouldn't I be able to at least approximate the volume of the whole Earth by adding the number of scoops I dug?
I was thinking of the surface of the Earth which is locally 2D. The Earth with its interior is locally 3D.sqljunkey said:I assumed 3-D.
I didn't understand when you said locally flat.
If I brought a spoon today and started digging scoops of Earth one at a time, careful not to deform it's shape and I counted each scoop, one by one until I dug up the whole earth, wouldn't I be able to at least approximate the volume of the whole Earth by adding the number of scoops I dug?
You are somehow saying that even though I piece these 3-d scoops back together into the manifold I wouldn't have learned anything of the geometry because it could have had a higher dimension.
So, if I had a manifold with n dimension my charts should at least be n -1 ?
note my spoon measures volume in cubic cm.
fresh_42 said:You may still call those objects manifolds, but that would be like telling kids at school, that their fractions they calculate with aren't rational numbers but complex numbers instead. You can do this, but does it make sense?
That sounds eminently reasonable.sqljunkey said:By knowing how the pieces are interconnected and the size of each piece I can add all possible combinations and find the one that has the least "distance" from south pole and north pole. Wouldn't this be a global-like measurement of the manifold?