## 4D tectonics

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 tectonics

This morning it occurred to me that if a 4D planet had plate tectonics, it would produce mountain ranges of significantly different characteristics than in 3D.

In 3D, mountain ranges produced by converging tectonic plates are linear, because the boundary between plates is 1D. In 4D, however, converging plates would interact at a 2D boundary, so they would tend to produce 2D regions of mountains, not linear ranges! A range produced by two directly-converging plates would have mountains with 2D peaks. Volcanic mountains would still have 0D peaks, and would have sphericonical geometry, as opposed to mountains produced by converging plates, which would have wedge geometry with 2D ridges.

Furthermore, if the two plates are also shearing w.r.t. each other, then it would produce 1D columns of rotational stress (e.g., what you might get if you rub two erasers against each other -- you get coils of rubber peeling off because of the shearing motion; in 4D the stress would get pushed into the 4th dimension as 4D height), which would produce 1D sub-ranges with spiralling geometry within the 2D range of mountains. The mountains would tend to have 1D peaks instead. So just by looking at the rotational configuration of the mountains you'd be able to tell the direction the plates are moving w.r.t. each other.
quickfur
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### Re: 4D tectonics

Hi,

Thank you for all your engaging thoughts on here. I'm still acclimating to the context (here and by reading at your site) so I hope you'll forgive the newbie questions. You made reference to wedge geometry, sphericonical geometry, and spiralling geometry - is there some online reference where I can read about these (and maybe other relevant) geometries? (assuming some general mathematical maturity is ok.)

Thanks!
Auden
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### Re: 4D tectonics

I'm mainly reasoning by dimensional analogy here. For an overview of my usual methods, you may read the 4D visualization document on my website. (But sounds like perhaps you have already read that.)

More specifically, the spherical cone I mentioned goes by "spherone" on my website (it dates from a time when I was infatuated with portamenteaus, though in retrospect it sounds a bit silly today). The general construction is to begin with a sphere in 3D, and drag it along the 4th direction while shrinking it linearly until it has shrunk to a point. The shape traced out is the spherone, or spherical cone. Now consider a hypothetical volcano which is erupting from a crater in the ground. As it erupts, it throws lava and pumice which would fall in all directions around it; since the surface of the ground on a hypothetical 4D planet would be a 3-manifold, which is locally approximated by a 3D hyperplane, this debris would collect in a spherical region around the crater. As the eruption continues, the debris would collect around the crater in spherical symmetry; and as it rises in height, the loose debris on the top would roll down the sides in a roughly spherical shape. The resulting shape of the new mountain would be a spherical cone.

With respect to converging tectonic plates: if we approximate the converging plates to be subsets of 3D hyperplanes, it follows that the boundary between them would occupy the space of a 2D plane. Consequently, any mountains that result would be parallel to this plane; with the plane itself roughly where the highest peaks would be pushed, sloping down on either side to the respective plate, forming a 4D wedge. This shape would be roughly like that of the 3,4-duoprism, if you imagine the latter placed on a flat surface on one of its cubical facets; its other two cubical facets would form a wedge-like shape approximately like the resulting mountain range.

The preceding assumes that the tectonic plates are converging directly at each other. This is unlikely to happen often, since one would expect that generally the tectonic plates would have at least some lateral movement relative to each other at the boundary. Without loss of generality, we may fix a line on the 2D boundary between plates along which the two plates are moving in opposite directions relative to each other in the horizontal component (besides their mutual converging motion). The converging motion would uplift the surface on either side of this boundary into the wedge-like shape; however, there is also lateral shearing motion happening. This shearing produces a curling motion along the 2D boundary; so like the strings of rubber which rub off when you drag a rubber eraser across the table, the uplifted mountains would shear past each other, causing the rocks to twist past each other consistently clockwise or anticlockwise, thus producing spiralling or vortex-like folds, almost like rolls of toilet paper. There would be a lot of erosion by friction around the axis of these rolls; so one might expect that along the axis of each roll the debris would accumulate and so be more uplifted than the surrounding spiralling folds. So you'd have 1D peaks surrounded by spiralling formations, which would clearly show the direction of the shearing motion, and thus the relative directions of the tectonic plates, and moreover would also be clearly distinguishible from volcanic peaks which would be 0D peaks that converge to a point.

Thus just by looking at the shapes of the peaks of the mountain range, one would be able to tell whether it's volcanic in origin, or resulting from the convergence of tectonic plates; and of the latter, whether the convergence is head-on or there is lateral motion as well, and which direction this lateral motion is.
quickfur
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### Re: 4D tectonics

Wow, thanks for all this. I wasn't expecting such a thorough response. And I did read (and stare into space thinking about parts of) your 4D visualization document. I'm just starting on the following sections though.

The spherical cone formation makes sense - having 3D volcanoes to compare with (with their growing, layered, circular accumulations) helped a lot in trying to conceive of what you're describing.

I'm having a harder time conceiving of 4D wedges. (The general idea of what's going on in that scenario makes sense though.) I'm hoping I can try to describe my attempt at thinking of them, and maybe get some corrections on my thoughts.

We have a 2D plane that is the boundary between converging plates, and on which we should (approximately) find the highest peaks of the range. I'm going to think of that plane as y-z, with z being height. y seems to have the least relevant role in the geometry of the range (I mean my very simplified visualization of it), other than variation in the peaks, so I'm going to look at what happens when we hold y constant. In the x-z plane, we should be seeing peaks on the z axis, sloping down as the magnitude of x increases, and in the w-z plane we'd see the same kinds of shapes as the magnitude of w increases. Probably vastly oversimplifying the possibilities here, but this makes me want to think of the 3D slice (in xzw space) at a fixed value of y as a conical shape, but then having incidences of this type of shape at (potentially) every value of y. (But I've probably gone astray here, given how different this image seems from the figures on the page you linked to. Or maybe I'm just not seeing it.)

The shearing scenario feels sufficiently complicated that I don't expect to get a solid handle on every detail, but the spiral pattern makes sense, and in general that you'd have these different types of patterns depending on a range's origin.
Auden
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### Re: 4D tectonics

quickfur wrote:Furthermore, if the two plates are also shearing w.r.t. each other, then it would produce 1D columns of rotational stress (e.g., what you might get if you rub two erasers against each other -- you get coils of rubber peeling off because of the shearing motion; in 4D the stress would get pushed into the 4th dimension as 4D height), which would produce 1D sub-ranges with spiralling geometry within the 2D range of mountains. The mountains would tend to have 1D peaks instead. So just by looking at the rotational configuration of the mountains you'd be able to tell the direction the plates are moving w.r.t. each other.

But couldn't we reduce the dimensions by 1 to get the same effect in 3D? Do we see 0D mountains with spiralling geometry within the 1D range where two plates shear past each other?
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mr_e_man
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### Re: 4D tectonics

Auden wrote:[...]
I'm having a harder time conceiving of 4D wedges. (The general idea of what's going on in that scenario makes sense though.) I'm hoping I can try to describe my attempt at thinking of them, and maybe get some corrections on my thoughts.

We have a 2D plane that is the boundary between converging plates, and on which we should (approximately) find the highest peaks of the range. I'm going to think of that plane as y-z, with z being height. y seems to have the least relevant role in the geometry of the range (I mean my very simplified visualization of it), other than variation in the peaks, so I'm going to look at what happens when we hold y constant. In the x-z plane, we should be seeing peaks on the z axis, sloping down as the magnitude of x increases, and in the w-z plane we'd see the same kinds of shapes as the magnitude of w increases. Probably vastly oversimplifying the possibilities here, but this makes me want to think of the 3D slice (in xzw space) at a fixed value of y as a conical shape, but then having incidences of this type of shape at (potentially) every value of y. (But I've probably gone astray here, given how different this image seems from the figures on the page you linked to. Or maybe I'm just not seeing it.)
[...]

There's a simple way to visualize the 4D wedge we're talking about here. Let's take two 3D cubes, side by side, and ram them together such that they collide at a common square face. Now add a 4th direction that's perpendicular to the 3 axes the cubes lie in. For convenience, imagine this as temperature or color. Let's say color. So imagine the two cubes as being white, white being w=0, and let's say as w increases the color changes to yellow then orange then red. As these two cubes squish together, the square face where they meet gets forced into the 4th dimension -- because there's nowhere else for it to go to relieve the pressure from the colliding cubes. So this common square face turns slightly yellow (i.e., it gets displaced into w>0). As the cubes squish together even more, the square face turns orange, and now the parallel slices of the cubes next to this common face will begin to acquire a yellow tinge (i.e., they also get displaced into w>0, albeit slightly lower than what's now the orange face). As the cubes continue to converge, the w-displacement of the common square increases, so that it's now red (i.e., the w-coordinate is now at a large value), and as you move away from this square, the parallel slices of the cubes gradually diminish in w-coordinate: they fade from red to orange to yellow then eventually to white at the opposite far ends of the two cubes.

What you get is a gradient of a large w-coordinate at the common face where the cubes meet, gradually fading away as you move away to either opposite end of the cubes. If we say that this gradient is approximately linear, then what you get is a 4D wedge shape. If the cubes are pushed together so much that the distance between the far face of one cube to the far face of the other cube becomes equal to the edge length of either cube, then the resulting shape is the 3,4-duoprism I referred to. The "edge" of the wedge is the square face between our two starting cubes, and the two cubes form the slopes of the mountain range -- noting that in 4D, surfaces are 3-manifolds, so the slopes occupy a 3D surface (hyper)area.

In general, when visualizing 4D terrains, it's useful to draw an analogy with height maps in 3D. In 3D, maps occupy a 2D area, and we imagine that the mountain ranges drawn on the map are "protruding" upwards from the 2D plane of the map. Similarly, in 4D, maps would occupy a 3D volume (or hyper-area), and mountain ranges drawn therein have 4D height, and can be imagined as "protruding upwards" into the 4th direction. Just as in 3D, our 2D maps use shades of color to represent height in the 3rd direction, so we may also imagine our 4D maps as 3D volumes wherein shades of color represent height in the 4th direction.

So a long mountain range in 3D produced by converging plates would be drawn as a roughly linear line of peaks across the map, fading away on either side to level ground; in 4D, a mountain range produced by converging plates would be drawn as a 2D region of peaks cutting across the 3D volume of the map, fading away on either side to level ground. A 4D volcano would be drawn as a spherical feature, with the peak in the center of the sphere, and fading radially to level ground. The two would be very distinctive-looking.

There's another possibility in 4D that I haven't mentioned so far: that is, where three tectonic plates are converge to a common pole. On our 4D map, this would look like 3 planes converging at a common line across the map. The 3 planes would be mountain ranges produced by each pair of converging plates, and the line would be where all 3 plates meet, and one can surmise that the peaks would be the highest along this line, being forced upwards into the 4th direction in 3 directions at once.
quickfur
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### Re: 4D tectonics

mr_e_man wrote:
quickfur wrote:Furthermore, if the two plates are also shearing w.r.t. each other, then it would produce 1D columns of rotational stress (e.g., what you might get if you rub two erasers against each other -- you get coils of rubber peeling off because of the shearing motion; in 4D the stress would get pushed into the 4th dimension as 4D height), which would produce 1D sub-ranges with spiralling geometry within the 2D range of mountains. The mountains would tend to have 1D peaks instead. So just by looking at the rotational configuration of the mountains you'd be able to tell the direction the plates are moving w.r.t. each other.

But couldn't we reduce the dimensions by 1 to get the same effect in 3D? Do we see 0D mountains with spiralling geometry within the 1D range where two plates shear past each other?

Hmm, very good point! Apparently areas of shearing don't produce peaks, but instead faults and fissures. I guess the same would apply in 4D then: shearing plates would produce 2D regions of faults and fissures. No spiralling geometry then.

Though in 4D, you do get something you don't get in 3D: if the plates are rotating relative to each other at the boundary of convergence. Like two cubes squishing together and also twisting relative to each other at the same time. What would happen then? There'd be some kind of feature with radial symmetry. That could well produce spiralling geometry...? Not sure.
quickfur
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