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Geometry Examples #3
Elevated Train Examples
Toys and Puzzles
Round Platform ExamplesThere's not a lot of educational value here. You can learn about flat objects, I guess, and get some practice visualizing complicated 3D shapes, but mostly I just wanted to show off the 4D round platforms.
Let's warm up by looking at the square platform in "3a". There are several things that I've done differently here.
The platform has width 3 and is located half a unit above the ground, so if you select the platform and rotate it, it will hit the ground right away and become unaligned. That's fine, no harm in that! But, if you back off a ways, and lift the platform up two units, you can rotate it through a full 360 degrees. (One unit isn't quite enough because the platform has some thickness.) What does that rotation look like? Well, first the square shrinks to a line (line segment), then it expands back into a square, a square that we know represents the back side of the platform. And, if we keep going, of course we return to the front side.
The platform in "4a" is exactly analogous. What does it look like when we rotate it? First the cube shrinks to a square, then it expands back into a cube, a cube that we know represents the back side!
Another thing you can do with these platforms is fly down and look underneath them. The procedure is the same in any number of dimensions: get out of align mode, aim at a point a bit outside the platform, move forward until you're near the ground, and then turn toward the platform.
How many faces does a 4D platform have? There are two big flat ones, the visible top face and the hidden bottom face. Then, for every 2D "subface" the top face has, there's a short vertical face that connects that subface to the corresponding subface on the bottom. If you have good intuition for that, you're way ahead of me, but that's basically what's going on.
Now we can look at some round platforms!
The 3D case isn't very interesting. We start with a square, and round off a corner, and that's all that happens. The other corners don't interact with it at all.
In 4D we round off edges, not corners. So, we start with a cube, and round off an edge. Then we can round off an adjacent edge and see an interesting join between the two, and after that we can round off one more edge and see a three-way join.
Note that if you take the projection of the 3D top face along any of the coordinate axes, you get a shape that could be the 2D top face of a 3D platform. Actually, let's turn that around: a 3D top face is exactly the intersection of three 2D top faces … no more, no less.
Except, sometimes it's a bit less, because when there's a three-way join, I like to clip off a bit of the corner, as in "4e". That's optional, and can be turned on and off in the scene file.
These examples show platforms of different sizes that have been rounded in all possible directions.
That's the last of the 3D platforms, by the way.
These examples show all the possible 4D corner shapes. "4a" through "4e" are just for reference, since already we've seen them in the "round" examples above.
In the general case, that's all the possible shapes. But, if a platform has width 1 in one direction, there are two more possibilities. One is the reverse join shown in "4f". The planes only match up nicely because sin 30 is exactly 1/2! The other possibility is shown in "4g", where the reverse join has been rounded off (to make two normal joins).
If a platform has width 1 in two directions, we can get the double reverse join shown in "4h". Or, just as with the three-way join earlier, we can clip off a bit to get the more pleasing shape shown in "4i". However, there's some trickery here. The shape generation code can't handle double reverse joins, so I made these models by hand. Fortunately, the code that decides what shapes to generate never produces double reverse joins!
Finally, if a platform has width 1 in all three directions, we can get the peculiar shape shown in "4j". The code that decides what shapes to generate never produces it, but by an accident of good design the shape generation code can handle it. (The shape consists of three reverse joins.)
These remind me of Japanese paper lanterns.
I like these dome shapes, especially the smallest one. Note that they can be derived from the "all" examples by cutting off the bottom parts of the platforms.
I named this set of examples "shell" because the final stages "4c" and "4f" remind me of seashells. They also remind me of potato chips. "4d" is sort of like those magnetic children's toys. "4e" is my favorite.
I named this set of examples "clamp" because the flat faces in "4a" look like C-clamps. The shapes here are the same as in "shell" except that the vertical edges have been rounded off. However, there's a more interesting way to think about the relationship between the two sets of examples. Given any platform, we can produce a dual platform by unrounding all the round edges and rounding all the unround edges. "clamp" is the dual of "shell". (Of course this operation is not the same as the geometric dual that e.g. swaps cubes and octahedra.)
I named this set of examples "clamp" too because the flat faces in "4a" look like … L-shaped angle brackets? Well, close enough. The interesting new feature here is that if you turn the platform over you can see that the bottom face is the mirror image of the top face. That's always been true, actually, we just couldn't see it until now because all the earlier top faces had reflection symmetry (except a few of the "corner" ones). An object with no reflection symmetry is called "chiral", did you know that?
Although the top face is a chiral 3D object, the platform as a whole is not a chiral 4D object, because of course it has reflection symmetry in the vertical direction. It's easy to find chiral 4D objects, though. A platform with tracks on it could be chiral. Most puzzle pieces are chiral. I don't think I've made any shapes that are intrinsically chiral, but that's easy too. For example, you could construct a pyramid with a cubic base and an asymmetrical apex.
This isn't some 4D effect, by the way. Even in 3D, the bottom face of a platform is the mirror image of the top face.
"leaf" is the dual of "3clamp", and so is chiral as well.
Here the six rounded edges form a ring or loop.
"antiring" is the dual of "ring". The border of the top face is also nice and ring-like in its initial orientation.
Finally, here are the "platform" examples from Elevated Train Examples with the platform style changed to PS_ROUND. All the platforms are generated from the train layout automatically! I like both "4a" and "4b" a lot. Unfortunately, the other two examples are a bit too complicated for my taste.
This is the same thing but with platform style PS_ROUND_MORE. "4a" is the same as before, which shows that the code can't always find more rounding to do, but "4b" is quite pleasing.