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.