Building 4D objects and worlds

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.

Re: Building 4D objects and worlds

Postby gonegahgah » Sat Dec 17, 2011 9:46 pm

hi mrrl. It does look lovely. What is the transparent stuff surrounding everything?
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Re: Building 4D objects and worlds

Postby Mrrl » Sun Dec 18, 2011 5:50 am

Empty land. Plains or desert or something like that. Nothing special...
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Re: Building 4D objects and worlds

Postby gonegahgah » Sun Dec 18, 2011 2:24 pm

hi mrrl. What I mean is: is it air or rock?
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Re: Building 4D objects and worlds

Postby Mrrl » Sun Dec 18, 2011 3:44 pm

Rock, sand, grass... Whatever, but not air. You can walk by it.
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Re: Building 4D objects and worlds

Postby quickfur » Sun Dec 18, 2011 10:45 pm

Mrrl wrote:Rock, sand, grass... Whatever, but not air. You can walk by it.

Or rather, walk on it. "On" as in, in the 4th direction perpendicular to the 3D model.
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Re: Building 4D objects and worlds

Postby gonegahgah » Sun Dec 18, 2011 11:03 pm

Cool. The transparent area would have to be something solid that is connected to hyperbelow or else be denser than water.
Otherwise the water above would move down to displace it and push it up.
This is because the entire volume shown is all in the same hyperhorizontal slice.
What you are representing is a 'hyperhorizontal' slice because the unseen dimension is in the down direction.
It is a cool way of thinking about it so thanks for introducing me to this perspective.

There are some important things to note about the image; and about all 4D rivers.
One is that the very bottom slice of the hyper river will not have continuous streams like shown.

If we take a bottom thin slice of any of our 3D rivers you will see a speckled map of blue dots.
You may also see some misshapen lines spread far apart but following a general direction.
A river tends to travel towards a body of water which is lower than the start point.
It is unlikely to have too many stretches of bottom that have dots in the very bottom hyperhorizontal slice.
Instead the bottom of a river covers a whole range of hyperhorizontal slices.

If we take any slices above this it will be like taking slices above from any of our 3D rivers.
For our 3D rivers the dots in the horizontal slices will turn into 2D misshapen shapes spread apart.
The further you go up the more these coalesce and connect to form bigger 2D misshapen shapes until to reach some nice continuous stretches.
At all times the layer puddles are bounded by solid earth in the same horizontal slice.
Otherwise water from above moves into those unbounded regions.
Get away from the very bottom layers and you hit continuous horizontal stream slices of just water bounded on two rough sides.
Eventually you get to slices where air begins to appear.
On a calm river this may appear as a sudden change from water to air.
On less calm rivers pockets of air will begin to appear until the air dominates.

The same goes for our 4D rivers.
The hyperhorizontal slices will begin with mostly continuous solid.
This will be followed by ever increasing presences of water which begin as hyperdots.
These coalesce to form hyperbubbles of water which then coalesce to form hyperlines and/or hyperregions of water.
The constant is that at any given hyperhorizontal level that the water must be fully surrounded by solid.

If any level were to have air pockets they would quickly be dispelled by water from hyperlayers above.
This is until again we start to reach the air layer.

There are a few problems with the diagram because of this.
The bottom of our 3D rivers can have valleys and peaks.
Taken at non-very bottom horizontal slices these are seen as mishappen 2D slices of ridges and peaks.
The bottom of a 4D river can have hypervalleys and hyperpeaks.
Taken at non-very bottom hyperhorizontal slices these are seen as misshapen 3D blobs and other solids appearing mid-water.
The overall trend will be for these misshapen solids to all align in the same direction as the hyperstream.
This is just like the valleys of the bottom of our faster moving river all tend to align with the direction of the stream.
Slower rivers and hyperrivers will have fatter bottom patterns.

The overall overriding feature is they are always bound by solid on the sides.
For our 3D river it is bound on both sides.
For our 4D river it is bound on all four hypersides at any place in the river.

The only way to achieve something like what you have shown is via tunnelling.
Could I also ask what hyperhorizontal level of the river you are attempting to depict in your diagram?
Is it a bottom slice, a slightly off bottom slice, a middle level slice, a top slice?
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Re: Building 4D objects and worlds

Postby quickfur » Mon Dec 19, 2011 12:20 am

gonegahgah wrote:[...]
This is because the entire volume shown is all in the same hyperhorizontal slice.

I can't speak for mrrl, but why does it have to be a slice?

In 3D, we use 2D maps to represent terrain, and not every point on the map is at exactly the same elevation. To only include points that lie on exactly the same elevation would limit the usefulness of maps, because if there's a road that's sloping downhill, for example, only a tiny section of it will appear on the map. Which defeats the purpose of a map in the first place.

Rather, we include points from different elevations so that we can see the interconnectivity of roads and rivers and such. If we wish, we can include contour lines to give some idea of which parts of the map represent higher elevations and which represent lower elevations, but on the whole, this kind of information is not necessary (unless you're a geometer or land survey scientist).

[...]
There are a few problems with the diagram because of this.
The bottom of our 3D rivers can have valleys and peaks.
Taken at non-very bottom horizontal slices these are seen as mishappen 2D slices of ridges and peaks. [...]

Again, this is only a problem if you're including only points at exactly the same elevation. There is no need to do this. Just think of this as a map of a section of a 4D planet's surface. The purpose of the map is to show the connectivity of various geographical features. It doesn't have to correspond with a perfectly horizontal slice. The important thing is that it has to represent the entire surface. In 3D, the surface is 2D, so we can draw maps on pieces of 2D paper. In 4D, the surface is 3D, so we can draw maps on "3D paper", which is represented here as a 3D model of the interconnectivity of various geographical features (cities, roads, rivers, etc.).
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Re: Building 4D objects and worlds

Postby gonegahgah » Mon Dec 19, 2011 1:12 am

I'm guessing now that you must be representing the hyperhorizontal layer directly below the 4D atmosphere. ie the hypertop of the river.

Just as our 3D rivers tend to follow one preferred direction for most of their lengths with occasional splits - instead of splitting off into several different north, south, east, west or between directions all the time - I tend to believe a 4D river will tend to follow one preferred direction as well.

Generally splits occur to form forks and you would get something similar with 4D rivers.

Just as rivers widen and narrow at the top along a meandering path; so would 4D rivers fatten and thin along a meandering pattern at the hypertop.
An important constant is that they are bounded on their constraining sides by the river bank or hyperbank.

Banks by their very nature rise into layers above the river's top layer.

One of the cool things with 4D that you can actually touch the inside of 3D shapes.
You can't touch the inside of hypersolids but you can touch the inside of any of their hyperfaces.
So if you have a tesseract you can touch one inside of each of the 8 cube faces from the one hyperside.

But our 4D river by necessity requires a 4D river hyperbank.
You would see this in the next slice above as the water being replaced by air.
As the bank hyperrises it would recede giving more hyperspace to air creating a funny shaped chimney of air in a hyperhorizontal slice.
Soon you would see 'floating' solids separating away from the bank in the air until it gives way to full atmosphere.

I must concede that the chimney hyper-riverbank provides a means to encircle the river without needing to cross it.
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Re: Building 4D objects and worlds

Postby quickfur » Mon Dec 19, 2011 2:32 am

gonegahgah wrote:[...] Just as our 3D rivers tend to follow one preferred direction for most of their lengths with occasional splits - instead of splitting off into several different north, south, east, west or between directions all the time - I tend to believe a 4D river will tend to follow one preferred direction as well.

Generally splits occur to form forks and you would get something similar with 4D rivers.

Yes, one would expect that would happen.

This topic of river formation is actually quite interesting. I experimented with automated terrain generation algorithms some time ago, where you start with an essentially random elevation map (which looks nothing like a realistic terrain, with unlikely spikes and precipitous pits sprinkled everywhere with almost no flat surface) divided into tiles. Then you simulate rainfall by assuming X amount of rain falls on the terrain, and then trace the direction of water flow by following the path that leads to the lowest altitude from each map tile. This in itself doesn't produce anything realistic either, since the water just collects in randomly-scattered dots where the pits are. But then you add water erosion to the simulation: every time water flows from tile X to tile Y, it carries some "topsoil" along with it, so that tile X reduces its elevation and tile Y increases its elevation. You can also optionally add wind erosion to the simulation, by causing very high tiles surrounded by very low tiles to lose part of its height and distribute them into surrounding tiles (simulating rocks falling from steep cliffs into valleys below). After running this simulation for some cycles, all the strange jutting spikes and precipitous pits have evened out into a more realistic-looking terrain, with many more-or-less "smooth" areas that one can actually walk on.

Now what's most interesting is this: as you run the simulation, store a "water saturation" value in each tile, which basically represents how much rain falls on it plus how many times water flows over it. At the end of N cycles of the simulation, color the tiles whose water saturation is below a certain threshold T brown (the idea being that soil can soak up a certain amount of water), and color the tiles whose water saturation is above this threshold blue (soil can only soak up water up to a certain point, past that point it can't hold anymore and the water level rises above the soil). What's really amazing is that when you adjust the value of T appropriately, you get an amazingly realistic terrain complete with rivers that begin at mountainsides and fork every now and then until water collects at the lowest elevations as little pools. You can even get oceans if you fill up tiles below a certain elevation (the "sea level") with water.

So basically, I started out with a completely random elevation map, and ended up with beautiful mountains, valleys, and plains, complete with rivers with forks and lakes, just by simulating the action of rain and water erosion. Of course, the terrain is still not 100% realistic-looking, but it's pretty darn good for such a simplistic algorithm.

So assuming in 4D we also have rainfall and water erosion, both of which are very reasonable assumptions, we certainly expect that 4D terrain would also have mostly-linear rivers with the occasional fork, with water collecting in lakes and seas.

[...] One of the cool things with 4D that you can actually touch the inside of 3D shapes.
You can't touch the inside of hypersolids but you can touch the inside of any of their hyperfaces.
So if you have a tesseract you can touch one inside of each of the 8 cube faces from the one hyperside.

And in fact, most of the time 4D beings would touch the inside of the cube facets--just like most of the time, we don't handle a cube by its corners or its edges, but by its faces.

[...] I must concede that the chimney hyper-riverbank provides a means to encircle the river without needing to cross it.

Bingo. Now you know how to walk around a 4D river. :)
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Re: Building 4D objects and worlds

Postby Mrrl » Mon Dec 19, 2011 4:03 am

gonegahgah wrote:Cool. The transparent area would have to be something solid that is connected to hyperbelow or else be denser than water.
Otherwise the water above would move down to displace it and push it up.
This is because the entire volume shown is all in the same hyperhorizontal slice.
What you are representing is a 'hyperhorizontal' slice because the unseen dimension is in the down direction.
It is a cool way of thinking about it so thanks for introducing me to this perspective.
[...]
The only way to achieve something like what you have shown is via tunnelling.
Could I also ask what hyperhorizontal level of the river you are attempting to depict in your diagram?
Is it a bottom slice, a slightly off bottom slice, a middle level slice, a top slice?


It is not a slice - it's hypertop view of the surface, projection of it to horizontal hyperplane. If you want to think about map in terms of slices, consider it as a slice just below the water level, but assume that basements of town areas and deepest parts of road constructions are deep enough to be seen on this level. And, again, water hypersurface should be horizonal, there is no currents in channels (because land has no points with different elevation). This is why I call them "channels", not "rivers".
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