How does a 3D space fit inside the 4th D?

[Dolokun]

I'm not clear on how infinite 3d worlds can exist. It seems like if one 3d world is infinite, then there isn't room for any others? Do 3d worlds have to be bounded for the 4th dimension to exist? 3d space can't just overlap other 3d space..

Also, is this true: in order for independent 3d worlds to exist, there must be a 4th dimension?

[Keiji]

Think about 2D worlds in 3D - given an infinite 3D universe, you can stack as many 2D objects (paper, for example) next to each other as you want, because they appear "flat" in one dimension. The exact same applies for 3D in 4D, it's just that we can't see that flat dimension from our view.

[Dolokun]

It seems to me, though, that if we were living in a true 3d universe, solid mass would be impenetrable. Nothing physical can exist on a 2d plane, so the 2d is still a concept. There are inherent differences in the 2 and 3 dimensions besides an additional axis. In order for what we perceive as 3d to lay 'flat', we would have to be living in a higher dimensional reality, it seems. Am I off here?

[Keiji]

Well, we could be living in a higher dimensionality without knowing it. It'd be impossible to find out.

[Eric B]

The whole point of the fourth dimension is that it si a dimension that doesn't exist in lower spaces. So the lower spaces can extend infinitely in their own dimensions, but there will be this whole new pair of "directions" perpendicular to every point of the spaces. "Solid" impenetrable objects in those dimensions will appear as "flat" projections (with every point of their "insides" exposed) from the higher dimension.

Each dimension is a totally different kind of existence from the other. The fourth is much more complex than the third, we can't visialize it, and matter as we know it would not hold together. But the second is actually just as hard to imagine, for how can you have vision within a line (and not in the 2D "screen" we see as we look out into 3D)? In 1D, objects cannot even pass each other. wherever they are ins where they forever remain.