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- Thread starter Rosen
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In summary, the statement is evidenced by the fact that the sum of the interior angles of the triangle formed by the Prime Meridian, the 90° line of longitude, and the equator...is 180°.f

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Off the top of my head, if there are indeed curled up extra dimensions as string theorists say, one could conceive at least of somehow a dimension curling or unfurling over time, but this is just idle speculation about a topic I know precious little.

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AFAIK there is no mathematical description for a dimensionless void.

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also, i don't think there is such a thing as a "dimensionless void." by definition, a void is an empty space, or a space completely absent of matter and/or energy. the only dimensionless entity in our universe would be a point. and i think the simplest definition of a void would be an empty space between two points. in this case, the simplest "void" would be the absence of a line (or infinite set of points) between two endpoints. in 2 dimensions, a void would a surface (another infinite set of points) absent of any matter/energy, contained by 2 or more lines (you might be thinking that it takes at least 3 lines to constitute a "closed" area, but think outside of Euclidean geometry, i.e. think curved surfaces). in 3 dimensions, a void would be a volume absent of any matter/energy, contained by 2 or more surfaces. what I'm getting at here is that a void implies dimensions, and is not dimensionless.

finally, it would be impossible to tell if the uncertainty principle applies in such a void if it existed b/c a void is by definition empty of matter/energy. there would be no particles' positions or measure, and no particles' momentums to be uncertain about, or vice versa.

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By appealing to differential geometry, it is known that surfaces possesses certain properties (like curvature) independent of any embedding. For example, take the 2D surface of a sphere embedded in 3D space. We can readily visualize this manifold as the surface of a ball. Interestingly, the curvature of a sphere of radius r (K = 1/r^2) is independent of this embedding (i.e. does not depend on the nature of the higher-dimensional space). In fact, the embedding is not even necessary -- the curvature isI was inquiring as to what models may have described such an area, and wondering if the uncertainty principle applied there. Great forum by the way. Thank you.

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In fact, the embedding is not even necessary -- the curvature isintrinsicto the surface!.

is this statement evidenced by the fact that the sum of the interior angles of the triangle formed by the Prime Meridian, the 90° line of longitude, and the equator = 270° (as opposed to 180° on a truly flat surface)?

TIA,

Eric

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Right. In particular, that you can make this determination by measuring the surface itself, irrespective of the space the Earth happens to be floating in.is this statement evidenced by the fact that the sum of the interior angles of the triangle formed by the Prime Meridian, the 90° line of longitude, and the equator = 270° (as opposed to 180° on a truly flat surface)?

TIA,

Eric

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