4D Water Pressure

Ideas about how a world with more than three spatial dimensions would work - what laws of physics would be needed, how things would be built, how people would do things and so on.

4D Water Pressure

Postby ICN5D » Fri Apr 25, 2014 5:52 am

In my adventures of SCUBA diving, it's impossible not to notice the increase in pressure when descending to the bottom. In 3D, we have a cube of water in a swimming pool, where a 2D flat sheet of water has the same pressure gradient throughout. Pressure only increases when traveling along the 3D height. So, in 4D, I'm imagining a 4D ocean, and two possible pressure gradients. I suppose it's a mater of how many directions gravity would be pulling, or which 4D toratope is the planetary structure is shaped like.

4D Ocean # 1:
--------------------
Surface is an entire 3D oceanic volume of water. Depth is deeper into 4D, stacking an infinite number of 3D water volumes together of increasing pressure. Each 4D depth level contains an entire 3D volume of water at the same pressure.

4D Ocean # 2:
--------------------
Surface is some other kind, where 3D depth in addition to 4D depth will increase pressure. Each 3D water volume is an entire functional 3D ocean, where going deeper into 4D increases the overall base pressure of each 3D volume.
It is by will alone, I set my donuts in motion
ICN5D
Pentonian
 
Posts: 1135
Joined: Mon Jul 28, 2008 4:25 am
Location: the Land of Flowers

Re: 4D Water Pressure

Postby Secret » Fri Apr 25, 2014 7:11 am

I believe 4D Ocean #2 is the most common model for a 3-spherical planet
Capture2.PNG
3-ball case
(159.58 KiB) Not downloaded yet


If my 1st year physics did not go too rusty

Assuming our planet is 3-spherical and isotropic, that is, each piece of the planet has the same mass and the masses are uniformly distributed, thus by Newton's Law of Gravitation, the same magnitude of gravitational force Fg is exerted onto each mass towards the centre

Also assuming our planet has its rotational energy equalized, thus it is in isoclinic rotation (i.e. Clifford rotation with the orthogonal planes rotating at the same rate, so that circles were traced out instead of spirals

Fluid pressure is caused when the fluid particles is pushing against another, if particles in the bottom layer get stacked by those on the top layer, then the bottom layer experience a larger force per unit area, or pressure

Now, zooming in a small patch of the 4D ocean, you will see that in general Fg is contributed by two components, the centripetal force in the xy plane and the centripetal force in the zw plane. What we feel as weight is the reaction force we felt exerted by the object we are standing on, thus will feel heavier if there is less contribution to make Fg since gravity contributed most of it in order to give the same Fg and this gravity pushes us against the ground more (because the ground is in our way of falling towards the centre), thus the reaction force we felt increases.

As we dive deeper towards the centre in the xy or zw plane, the xy or zw contribution becomes smaller thus we felt a greater reaction force, thus the water pressure increases as we move towards one of the circular 'poles' (xy or zw) so that we only felt one of the two rotations from the isoclinic rotation.

So in general, the pressure increases as we diver deeper towards the centre of each rotational plane. Thus a depth which is closer to the centre of the planet definitely will experienced higher water pressure, but in addition, just diving towards one of the poles while maintaining your altitude will also increases water pressure, thus 4D oceans in general works with two pressure gradients, despite gravity only acts along a single direction (down)

Capture2.PNG
4-ball case
(53.27 KiB) Not downloaded yet


For 5D, since the isoclinic rotation axis becomes a line (latrix), the pressure behavior is sort of a fusion between 3D and the 4D case. Not only you have the double pressure gradients like in 4D, but the 'poles' fatten up into clifford tori since one of these circles don't rotate, this they cannot create an extra component of the centripetal force

And I suspect in 5-balls (6D) you have triple pressure gradients, given you can set up an isoclinic rotation with 3 orthogonal planes
Last edited by Secret on Fri Apr 25, 2014 7:48 am, edited 4 times in total.
Secret
Trionian
 
Posts: 162
Joined: Tue Jul 06, 2010 12:03 pm

Re: 4D Water Pressure

Postby wendy » Fri Apr 25, 2014 7:28 am

Here 'surface' is being confused with 'hedrix', which is what i tried to avoid in the polygloss.

It really is quite simple. Water flows everywhere, but is held by gravity. Gravity is one axis of 4d (height), so 4d divided by height is surface (chorix). So the surface of a 4d ocean is 3d, and its depth is 1d.
The dream you dream alone is only a dream
the dream we dream together is reality.

\ ( \(\LaTeX\ \) \ ) [no spaces] at https://greasyfork.org/en/users/188714-wendy-krieger
User avatar
wendy
Pentonian
 
Posts: 2014
Joined: Tue Jan 18, 2005 12:42 pm
Location: Brisbane, Australia

Re: 4D Water Pressure

Postby anderscolingustafson » Fri Apr 25, 2014 3:39 pm

The water on top is only affected by the pressure of the air while the water further down is affected by the pressure of the water on top of it and the water below it as well as the water beside it. The water at the bottom is affected by the pressure of the water above it and the pressure from the Earth underneath. The further down you go the more water there is above and so in any number of dimensions the pressure will tend to increase with depth.
AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA
anderscolingustafson
Tetronian
 
Posts: 316
Joined: Mon Mar 22, 2010 6:39 pm

Re: 4D Water Pressure

Postby ICN5D » Fri Apr 25, 2014 11:41 pm

Hey, thanks for the input!

I felt it could be a toss-up between two pressure gradient models. One is a glome-shaped ocean mass, with 3D chorix volumes stacked at each 4D depth level. Or, perhaps some other 4D toratopic design, which could allow two depth directions. Like some sort of ocean-prism, where we have simply extruded/rotated a 3D normal ocean into a 4D equivalent. Once again, I always fall back on the idea of 4D toratope-shaped planetary structures, where a tiger-world could have two gravity vectors, and thus, two pressure gradients. It's the allowance of such things as the two ortho circles in 4D that are a curiosity. And, how it's manifested out in the open. Still not sure what to make of it, both ideas sound legit.
It is by will alone, I set my donuts in motion
ICN5D
Pentonian
 
Posts: 1135
Joined: Mon Jul 28, 2008 4:25 am
Location: the Land of Flowers

Re: 4D Water Pressure

Postby quickfur » Thu May 01, 2014 5:23 pm

Assuming by "water" you mean any fluid that passively flows in the direction of lowest potential, wendy's model seems most accurate. The global shape of the planet ultimately doesn't matter, because water molecules (i.e. fluid constituents) don't "know" about global shapes, they only sense the local, effective force acting on them, and move accordingly. So it doesn't matter how many different forces are acting on it; what matters is only the combined, effective force in a particular localized region, which can only act in a single direction.

So it makes sense to consider a small neighbourhood of water molecules that all experience (approximately) the same (effective) force in the same direction. The natural response would simply be to move in the direction of this force. Now supposing that we're talking about a still body of water (i.e. oceans and lakes as opposed to rivers and waterfalls), obviously the water molecules cannot actually flow anywhere, but they simply exert pressure in the direction of the force. Assuming that the swimmer also undergoes the same force, the effective pressure he feels would be proportional to the amount of water above him in the direction of the effective force. Thus, water pressure increases in a single direction in proportion to the cube of the depth (because the cross-section of the water perpendicular to the direction of the effective force is a 3D manifold).

Note that this analysis applies regardless of the global shape of the planet -- be it a glomic planet or a toroidal planet or some other such exotic shape. While the global properties of the forces that shape the planet may differ, at the local level they are all approximately the same because forces, being vectors, sum to a single vector (the effective force); they do not sum to bivectors or other such exotic things. A water molecule can only move in a single direction that corresponds with the sum of all the forces acting upon it.
quickfur
Pentonian
 
Posts: 2935
Joined: Thu Sep 02, 2004 11:20 pm
Location: The Great White North

Re: 4D Water Pressure

Postby Prashantkrishnan » Thu Dec 25, 2014 12:52 pm

ICN5D's idea of gravity pulling in many directions is interesting, though difficult to imagine. I suppose if gravity were to pull us in two directions, we might be compelled to use complex numbers to indicate the gravitational potential at any given point ... and then it would be necessary to define comparison of complex numbers, as a mass always tends to move from higher potential to lower potential. Objects would behave in weird manners.

Also, being 3d creatures, we can visualise a purely horizontal plane (i.e. a plane in which all lines are horizontal) but not a purely vertical plane. Note that our notion of "vertical" is actually induced by gravity. So if gravity were to pull in two directions, we would be able to visualise a purely vertical plane.
People may consider as God the beings of finite higher dimensions,
though in truth, God has infinite dimensions
User avatar
Prashantkrishnan
Trionian
 
Posts: 114
Joined: Mon Jan 13, 2014 5:37 pm
Location: Kochi, Kerala, India

Re: 4D Water Pressure

Postby Prashantkrishnan » Thu Dec 25, 2014 12:53 pm

By the way, I find the first model of the 4D ocean more believable
People may consider as God the beings of finite higher dimensions,
though in truth, God has infinite dimensions
User avatar
Prashantkrishnan
Trionian
 
Posts: 114
Joined: Mon Jan 13, 2014 5:37 pm
Location: Kochi, Kerala, India


Return to Higher Spatial Dimensions

Who is online

Users browsing this forum: No registered users and 12 guests