In my answer to my own question, Physical laws for a matter-filled universe, I detail the gross physical laws that would shape my four-spatial-dimensional matter filled universe.

Given a slowly expanding four-dimensional universe, where the bulk of it is filled with solid matter, and nexi and streams of energy interpenetrate the bulk and carve out hyperspherical and hypertubular voids that may be filled with liquids, gases and vacuum, what would the geography of this world/universe be like?

In particular, how does having four spatial dimensions change the nature of this world's geography in comparison to three-spatial-dimensional worlds? Are there any geographical features that occur only in 3D space, or only in 4D space?

How would the dimensional stresses of the expanding universe be relieved in the solid bulk, and what effects would that have on the universe's geography?

This is part of the related series of questions, What would the periodic table of a 4-Dimensional universe look like? and What would organisms be like in four physical dimensions?.

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    $\begingroup$ @PyRulez, A hypercube has eight three-dimensional sides. $\endgroup$ – Monty Wild Dec 9 '15 at 1:55
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    $\begingroup$ @PyRulez, a 3d room doesn't necessarily have four walls, and a 4d room doesn't necessarily have 6. Talking about hypercubes cuts out any potential confusion. $\endgroup$ – Monty Wild Dec 9 '15 at 2:41
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    $\begingroup$ Not geology related but close: the concept of "curl" only appears in 3 dimensions (as does cross product). This is interesting because vortices are intrinsically tied to curl, so you would see no 4d vortex equivalent in your geography. $\endgroup$ – Cort Ammon Dec 9 '15 at 3:02
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    $\begingroup$ I doubt your universe would form solid bodies large enough to make the question relevant. Minute Physics has a video about 4D space topology. $\endgroup$ – Draco18s no longer trusts SE Dec 9 '15 at 15:42
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    $\begingroup$ As @CortAmmon mentioned, there are topological features of 3D space that are lost in 4D. Low-dimension spaces are "special" and have a lot of mathematically "nice" properties. 4D and higher have, in a sense, too much flexibility, and are less structured. $\endgroup$ – Era Dec 10 '15 at 21:08

There are many physical laws that would be changed simply from going from a 3D to a 4D universe. The first one that comes to mind is that the does not exist a vector cross product in four-dimensional space.

If one requires only three basic properties of cross-product, properties which >are explained in practically all undergraduate textbooks that discuss vector >analysis, it turns out that a cross product of vectors exist only in 3->dimensional and 7-dimensional Euclidean space.

"Cross products of vectors in higher dimensional Euclidean spaces"

Just by the nature of this not being usable in the physics of your 4D world, all the rules would change. Vector cross products appear in torque and angular momentum (both of which would be very important in your swirling universe). Because of the strangeness that would ensue, it is practically impossible to guess at how certain geographies would form in this mostly solid universe. The geographies would be complex and extremely different from that of our 3D world.

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    $\begingroup$ "Just by the nature of this not being usable in the physics of your 4D world, all the rules would change. Vector cross products appear in torque and angular momentum (both of which would be very important in your swirling universe)." - it could be defined in term of tensors or bivectors, derived from rotational symmetry. Witness the definition of angular momentum in 2D, despite the fact that there is no cross product defined for 2D vectors. $\endgroup$ – Radovan Garabík Dec 12 '15 at 13:55
  • $\begingroup$ @RadovanGarabík Could it? Maybe, I'd would look that up before taking it as fact. However, either way, it would not be a vector cross product that has the same properties or follows the same rules as our cross product. This makes the rest of my answer unchanged; the physics of torque and angular momentum would still be drastically different in 4D. $\endgroup$ – Ethan Dec 12 '15 at 14:01
  • $\begingroup$ This doesn't answer the question. Saying what the geography isn't doesn't say what it is. $\endgroup$ – Monty Wild Dec 13 '15 at 22:00
  • $\begingroup$ @MontyWild: I'm trying to say that it'll be practically impossible to say what it is. Therefore, you should take your best guess, and no one will really say that you're wrong. $\endgroup$ – Ethan Dec 13 '15 at 22:02
  • $\begingroup$ I'd be interested to read that article without having to pay for it or register. Do you have another link to it? $\endgroup$ – Monty Wild Dec 13 '15 at 22:07

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