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Geometry Examples #3
Toys and Puzzles
Elevated Train Examplestrain2-window
Let me start with the examples that are not about elevated trains but rather about the new custom texture feature. These are just simple passenger cars with windows on the sides. In both 3D and 4D, there are no windows on the front, back, top, and bottom sides, leaving two sides with windows in 3D and four sides with windows in 4D. You can see all four sides in "4a" by sliding in the in and out directions, or you can switch to "4b" and just watch as the train goes by. There the sides of the cars are painted different colors and there's a helical track segment that makes them roll and display different sides.
These are simple monorail trains that run in circles. The 3D one is straightforward (but still very cute). The 4D one is instructive in many ways. If you slide a few units upward from the start position, you'll think you're seeing a normal 3D perspective view, but if you then spin about the vertical axis (shift-J or shift-L) you'll see that the whole thing still lies in a plane. You can see all four dimensions here: the vertical dimension that the pylons exist in, the two dimensions that the train travels in (forward, back, left, and right), and the dimension along which the scene is flat.
You can get another nice view by moving to the center of the track, flying upward, and looking down. From that perspective, downward is forward, so all the pylons extend toward the center. And, once again you can see all four dimensions.
Another thing you can do from overhead is see that the cars cover up the rail as they go past. You'll probably want to move away from the center and hover directly over the rail for that, because the rail extends downward a bit and so is visible underneath the train unless you're pretty high up. The elevated platforms are wider in all the other examples, so this is the only place you can get that effect.
Ramps in 4D are hard to understand, but let's work through the analogy and see where that gets us.
In "3a" we're looking at a ramp leading upward that's really just half of a block. Imagine that it started out as a full block and that we removed the top face by pushing the top front edge backward. The front face got stretched out to become the ramp; the back face was unchanged because it connected to the top along an unchanged edge; and the bottom face was unchanged because it didn't connect to the top at all. That leaves two sides that got cut in half. If you slide left and right from the starting position you can see them, and if you move around to the sides (or pick the ramp up and turn it) you can see that the faces are right triangles.
In "4a" we have the same thing. In the original full block, the front face cube connected to the top face cube through a square boundary. That's what got pushed to the back, and in fact that's what we can see at the top of our view. The front face that we're looking at got stretched out, and the back and bottom are unchanged as before. That leaves four sides that got cut in half. If you slide left, right, in, and out from the starting position you can see them, and if you move around to any of the sides (or pick the ramp up and turn it) you can see that the faces are right triangular prisms.
For extra credit, visualize the ramp and the original block at the same time and figure out where the other half block fits.
The ramps for elevated trains are similiar. In the "b" variants I've made the ramps into full triangles like the ones in "a" (and also like the children's blocks that I'm always thinking of), while in "c" the ramps have spaces underneath like the real ramps that we'll see in all the other examples. I left the large pylon in place for reference, although it won't be there in the real examples either.
Normally the elevated platforms will be that dark green color you saw in the ramp examples, but it's possible to change that, as you can see here. The colors go from cheerful cyan, magenta, and yellow to industrial gray. The layout is just a simple loop with a branch. Note the small pylons in back that hold it all up.
Elevated platforms also come in different styles. Example "a" is PS_SQUARE, "b" is PS_ROUND, "c" is PS_THIN, and "d" is the default PS_THIN_ROUND. Thin platforms in 4D can have some interesting shapes (see the "platform" examples below), but round platforms in 4D are truly amazing. See Round Platform Examples for details.
Now at last we come to the canonical figure-eight layout. In "3a" we have a simple crossing in the center, in "3b" we have a bridge or overpass, and in "3c" just for fun I elevated the track in both directions and put a building underneath.
The 4D case is much more interesting. It starts off the same way: in "4a" we have a simple crossing and in "4b" we have an overpass. But, remember the loop examples in Train Examples. In 3D there was just one loop, but in 4D there were three, in three different orientations. In the same way, we can have figure eights in three different orientations. In "4c" the track runs in-out instead of forward-back, so the whole thing is roughly equidistant from the viewer, while in "4d" the track runs in-out instead of left-right. These are all perhaps best viewed from overhead rather than the starting point.
Then, since we have so much room to work in, we can combine "4b" and "4c" to make the double underpass "4e", and "4b" and "4d" to make the double overpass "4f". Finally, in "4g" we have a triple overpass with a building underneath. This is another scene in which you can see all four dimensions. There are the three dimensions that the bridges run in, plus the vertical dimension that separates the bridges from the building.
There are actually a lot more options here than I showed you. It's true that the basic figure eight exists in a plane like a loop, and that there are three possible planes in 4D, but for a figure eight you also get to choose which track is elevated and in which direction it loops around to connect to the lower track. And which direction the train runs! That makes 12 possible figure eights (not counting the train direction) which you can then combine in various ways to make double and triple overpasses.
Another thing you can do here is watch the train go under the bridge. Or, better, pause it while it's underneath and take a long look. If you do that in "3b", and look at the train from above, then no matter how you move around, the bridge always blocks your view of part of the train. But, in "4b" you can simply slide a little inward or outward and get a clear view of both bridge and train. This is the kind of thing people are talking about when they say that a four-dimensional being would be able to see inside things. If you slide further inward or outward you reach a viewpoint that looks just like "4c", which is the real X-ray vision view. Watch the train go under the bridge there and you'll see what I mean. It's not really X-ray vision, though, as you can see with the double overpass in "4f". The platform there extends fairly far in both the left-right and in-out directions, so there's no easy way to look around it. To put it another way, that train layout doesn't fit in a plane, so there's no way to get around to the side and get an unobstructed view of everything at once.
Now let's go back to the basic figure-eight layout and follow a different analogy. In "4a" we got rid of a crossing by building a bridge, but why bother? We could simply have gone around in one direction ("4b") or another ("4c"). Or both at the same time! ("4d") If we want an obstacle that requires a bridge, we need a plane instead of a line … like, say, a whole railyard ("4e"). That's probably the most genuinely 4D of all the examples here.
Elevated trains aren't limited to one level of elevation, and to show that, here's a nice spiral layout. In 3D you can watch from the side or fly up to the top to get a spiral staircase view like in Vertigo. In 4D you can do the same things, or you can move around into the in or out direction and get the X-ray view, since although the spiral is complicated, it's still flat in that direction. You can see that clearly from the starting point (or the top) by spinning about the vertical axis (shift-J or shift-L). Again, it's not true X-ray vision. There are two clear indications of that. One, the train gets more and more obscured by the platforms as it ascends, because you're looking up at it from below. Two, you can't see the orange track on the upper platforms.
Detailed instructions for how to get the X-ray view:
Sorry there are no pylons. I know it's physically unrealistic.
To show off the different platforms shapes in 4D, here are the "misc" examples from Train Examples lifted up onto elevated platforms. There are eight different tile shapes (versus four in 3D). Can you find examples of all of them? It's definitely easiest to change to an overhead view and slide sideways to move around. I didn't bother with pylons in this case either, sorry.