Structure of Planck volumes in String theory This question (as the previous one) is mostly arose from such pictures:

As explained by Brian Greene, this is something what our Universe should look like at a Planck scales in superstring theories.
Now, from an article on Wikipedia:

The central idea is that the visible 3D Universe is restricted to a brane inside a higher-dimensional space, called the "bulk". If the additional dimensions are compact, then no reference to the bulk is appropriate.

So here goes my questions (all for the compact extra dimensions):

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*If we imagine our whole Universe as a giant 3D Volume, am I correct that this compactified "extra space" is a "complete subset" of that 3D Volume? I.e. (since I am terrible with math and it's terms), there're no "points" inside them (like coordinates) that do not belong to (shared with) our 3D Volume, right?..

*Are these shapes (called Calabi-Yau manifolds) meant to be connected with each other? Like, if you were small enough to place yourself inside 1 Planck volume of our Universe (or even smaller if it needed), could you use 1 (or any) of these compactified dimensions to travel from 1 Planck volume to another Planck volume?.. Or prof. Greene did picture them separately on purpose, to mean specifically that they're unconnected? And if he did, how a String can get from 1 Planck (or any) volume to another if there's no "bulk" - ?..

*If there's no higher-dimensional space (as suggested by any recent experiments, including this one), how compactified dimensions could be even considered "dimensions", if 3D (4D) is all we need to specify any point inside them? May be "solutions" (of some equation, like $\text{Lagrangian}$, or other) is the better term then?..

 A: I cannot really answer this from a string theory point of view, just from the mathematical perspective of extra dimensions:

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*I am not sure your intuition is correct, or if I just misunderstood you. The points in the compact dimensions are not a subset of our large 3D space, but what we call 'points' in the large dimensions are actually a subset of all of space, where the latter includes the compactified dimensions. Mathematically spoken, there are NO points that are either in
compactified dimensions OR in the large ones. Every point is just
a point in 9D space (ignoring the time-dimension for the whole answer, but taking the 6 compact dimensions from superstring theory) and if you
want to give a coordinate of such a point in local coordinates,
every point always has 9 coordinates, which you can write as
$x=(x^1,x^2,x^3,...,x^9)^T$, where, if we have 6 small dimensions,
w.l.o.g. we can say the first three coordinates can have arbitrary real values, while the other 6 can only have 'small' values,
or just loop very fast (meaning larger values corresponds to the
same point as smaller values), because the dimensions are so small.
Since we are so large compared to the compactified dimensions, it
seems we can never distinguish for any point with the same first 3
coordinates any of the coordinates from $x^4$ to $x^9$, so we only
use the first three to specify points in 3D space. So the points in
the compactified dimensions are not a subset of the 3D volume, and
also not the other way around, but the 3D world we experience is
actually a subset of the whole 9D universe, while it not relevant in
our three dimensional macroscopic world at which coordinates $x^4$
to $x^9$ any object is, if that is even defined, since even what
we call elementary particles seems to be larger than the
compactified dimensions, at least that is how I understand it.


*All the small dimensions are connected, and the picture you posted is misleading in that sense. If you would be small enough to move around in compactified dimensions, you would need to move along a large dimension to get to 'another' Calabi-Yau manifold (at another 3D point) (which is quite obvious if you think about it). Think about the example with the straw, which is a small cylinder: The length of the straw is the large dimension (let's assume the straw is infinitely long), while the circle around the 'hole' of the straw is the small dimension. You can specify a point on a cylinder by two coordinates, one that can have arbitrary real values (since the straw/cylinder is infinite), and one that loops as soon as you circled around the straw. Mathematically this is equivalent to gluing a circle to every point of the number line (remember we said the straw is infinite), which could be written as $\mathbb{R}\times S^1$. All these circles, $S^1$, are definitely connected. How? No matter at which point on the circle you are, you can always get to 'another circle' just by moving along the 'large' dimension, which does not change your coordinate on the circle, i.e. in the small dimension, but nevertheless could be pictured as moving you to 'a ('different') circle glued to another point on the straw'. It is analogous with 6D-CY-manifolds and 3 large dimensions, just harder, or impossible, to picture.


*I do not understand this question. If there are no additional dimensions additional to our large space-time dimensions and time, then there cannot be any compactified dimensions. But I do not think that was your question. :-D
