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Geometry Examples #3
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These scenes are designed to let you practice the very useful technique of looking down on trains from above. What you want to do is fly to the center of the mat, turn to face straight down, then move back until the whole track comes into view.
In both cases, the only railcar is a flatcar. In 3D you can see from the start that the car is flat, so it's no surprise that when you look down from above, even at an angle, it looks pretty much like a rectangle right on the tracks. In 4D it's not so obvious that the car is flat, but it is, so when you look down from above, it looks like a rectangular cube right on the tracks.
So, if you're looking at some other 4D scene from above and you just want to admire the 3D track structure without being distracted by the 4D nature of the railcars, you can make all the cars flat. All you have to do is edit the scene file and change "boxcar4" to "flatcar4".
Here you can see most of the different kinds of railcars I've set up. If you want to play with any or all of them in another scene, you can edit the scene files and cut and paste them over. I've used single boxcars in most of the other scenes so that the peculiar shapes of the cars don't distract you from the other points I'm trying to illustrate.
Another set of railcars that you can cut and paste.
Now that you're used to the cars we can start looking at some tracks.
"4a" is the four-dimensional loop I used in the previous 4D train scenes. It's the exact analogue of the loop in "3". It extends in the forward, backward, left, and right directions, and it looks bigger where it's closer to you. Also note how the left and right sides of the loop recede toward the vanishing point straight ahead.
"4b" is a loop that extends in the forward, backward, in, and out directions. It's nearly equivalent to "4a"—all you have to do to make them match up is spin 90 degrees about the vertical axis, and that's nothing, the 4D equivalent of tilting your head. (Actually it's one of three 4D equivalents of tilting your head, the weakest one that doesn't even change the vertical axis.)
"4c" is the third possible loop. It looks a lot like the 3D scene we started with, but that's a 3D-person mistake, because in fact it's significantly different from the other two. See how the track is the same size all around? That's because it's more or less equidistant from you! The loop extends in the left, right, in, and out directions, not forward or backward at all. This will become much clearer in the "hub" scenes below.
All three are simple 2x2 loops, so they're equivalent in that sense. The difference with "4c" is that you can't make it look like "4a" or "4b" without moving to a different location.
Here's a good exercise for you. In 3D, move to the center of the track, but don't look down and move backward like you've been doing, instead just turn left or right to follow the train as it moves around. Easy, right? Now do the same thing in 4D, in both "4a" and "4b". In the first case it's natural to turn left or right, in the second in or out.
"4d" contains all three loops together. (I recommend hitting "Q" to make the track display as a monorail.) You can see that together they outline something sort of like a sphere. But, it's a flat sphere, since it sits flat on the 3D floor. If you want a clear view of that, switch out of align mode and slide down until you hit the floor. The mat and sphere will collapse into a plane, but the trains won't, since they're not flat. I don't have good intuition for flat spheres myself, it's just something to think about.
"4e" is the same as "4d" except that I've connected the loops together so that it's more fun to watch the trains zoom around. Again, the "Q" key is useful.
These scenes are the same as the loop scenes except that a building has been placed in the center of the loop(s). In "4a" and "4b" you can see the train going behind the building, just as in 3D, but in "4c" you can't, because it's going around the building, not behind it. This is a good place to stop and think about the four dimensions: the train is moving in a two-dimensional loop, but it's not moving up or down or forward or backward. Can you move to a different location and make "4c" look like "4a" or "4b"?
I said that "loop4a" was the exact analogue of "loop3", and it is, but there's a different analogy we can use. Instead of thinking of the loop as going around the center point, we can think of it as following the edges of the floor mat. The result is what's shown here.
I like these examples a lot, I think there's a lot to be learned here. In the "a" versions you can see the street grid before construction, then in the "b" versions you can see all the skyscrapers. Welcome to the big city!
If you take two steps forward, you can see that you have significantly better visibility in 4D. One step forward and one step up is also good. If you move all the way into the center of the mat, turn to face down, and move back two or three steps to get the news chopper view, you can see why the visibility is better: the buildings are off in fully diagonal directions, so they only obscure the outer corners. Note that in 4D you can see three sides of every building, plus the roof if you're at height 3.
Actually you can also see the roof from height 2, you just have to turn and look for it. If in 4D you stay in align mode and look sideways from height 2, you get a nice effect.
Anyway, once you're used to the buildings you can watch the little green streetcar move around! The view from ground level is especially nice. If you stand in the central intersection, you can turn to watch where the car goes, and then sometimes it will come straight at you. That's a very counterintuitive picture to us 3D folks. To get the full effect you'll need to switch out of align mode and get down near the ground.
The problem with this scenario is that I'm not sure news choppers and streetcars have ever coexisted.
The idea here is self-explanatory, but the 4D tunnel is pretty mind-boggling. It will help if you go back and read about / look at "geom-arch" and "geom-tunnel" (in Geometry Examples).
For a train in 4D, even after the vertical and forward directions are fixed there are still two dimensions free, so the train can still rotate. It's the same as for an airplane in 3D, which can roll without changing its forward direction.
As a result, there are two special kinds of track tile that exist only in 4D: helical tiles that make the train roll left or right. When you open up "4a" you can see a train in the process of rolling right, and after you move the train out of the way you can see the helical tile. It's visible even in monorail view, but the multi-rail view is better since then you can trace the rails all the way through.
The special boxcars I'm using here have two of their four sides painted, one magenta and one yellow. The other two sides are still green, as are the front, back, top, and bottom, though you'll have to turn on some texture other than 0 to see that. Actually I made the back gray instead.
The only difference between "4a" and "4b" is that "4a" uses flatcars.
"4c" is a simple variant that shows that trains can accumulate roll even without crossing helical tiles. If you let the train go around the track, it will come back rolled by 180 degrees. "4d" is a more complex track that causes a 90-degree roll. If you look at it from the right perspective you can see that it actually has a larger helical structure. "4e" causes a 90-degree roll, and is one of the shortest tracks that cause roll. There are others of length 8, including some inequivalent ones, but none shorter.
This accumulation of roll isn't some weird 4D effect, it happens in 3D too. To see it, load a 3D scene, get in align mode, and face some horizontal direction. If you turn left, up, and then right, you'll end up facing the original direction but rolled by 90 degrees. You can get the same effect on a smaller scale if you turn off align mode and make small turns left, up, right, and down in that order over and over. I think the mathematical name for this effect is "parallel transport"; unfortunately the Wikipedia page for it is super technical. Basically you're carrying your local "up" direction around with you, and where it ends up depends on your path.
The weird 4D effect is, if you fly around freely in 4D instead of being stuck on the ground, there are more kinds of roll that you can accumulate.
These are just some miscellaneous 4D examples I made.
4a - the center is a three-way cross tile
I'm not particularly proud of these. I still have a long way to go as a designer of 4D train tracks.
This is the test layout I used during development.