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## Geometry Examples

geom-tower

These are just cute constructions. You might like to fly above the towers and look down.

geom-obstruct

These show how one object can be behind another. You can slide left and right around the obstruction to peek behind it and see the second cube, and in 4D you can also slide in the in and out directions.

In the "a" files, the obstructed cube is the same size as the one in front, so when you peek around the sides you can see that the diagonal perspective lines match up.

In the "b" files, the obstructed cube is half the size of the one in front, so you have to slide further to see it, and the side face of the front cube becomes visible before the obstructed cube does.

In the "c" files, the object in front is a cap rather than a cube, so you have to go much further around it to see any of the side faces. This is how you can begin to understand angles in 4D! You can also see the obstructed cube that much sooner.

geom-clock

This has little educational value, it's just something I put together when I was trying out the rotate command.

geom-arch

Besides being pretty to look at, these do have educational value. In 3D an arch with two sides encloses your path, but in 4D it doesn't. You need to revolve the two sides to get something more like a dome. Examples "4a" through "4d" show a possible construction sequence.

4a - two-legged arch
4b - four-legged arch
4d - add ceiling pieces to make a complete dome

If you're not seeing the difference between 4c and 4d, be sure to turn on an interior texture (1-9).

Although to us 3D folks it looks like the dome is completely enclosed and impossible to get into, in fact from the starting point you can fly straight through it. What you have to remember is that the focus of your vision is supposed to be the crosshairs at the center of the retina. There are no blocks in the way there!

To look at it another way, turn on align mode and then take five steps right and three steps forward from the starting position. Look to your left and you'll see that the arch-dome is flat, as it should be. Or you can fly into the middle and look all around, then you'll see that only four of the six horizontal directions are blocked.

Now that you can move the blocks around, these are fun to explode and reassemble.

geom-tunnel

It's only a small step from arches to tunnels. The "a" files are arches in a blocky style. The advantage over the curvy style is, it's easier to make a complete ceiling for the 4D arch. The "b" files are exactly the same except that the arch has been extended to length 3 to make a tunnel. From the front, the only difference you can see is that the exit hole is further away (smaller).

geom-tree

The "a" forms of these are trees with trunks, while the "b" forms are just the green part, lifted up into the air so that you can fly right underneath and look up at the base. For your viewing convenience, the base has been colored red. The 3D trees are cones based on 12-sided polygons (dodecagons), while the 4D trees are cones based on dodecahedra. You can see the dodecahedron clearly in "4b", just fly underneath and look up. You can also see a dodecahedron if you fly above the tree and look down, but in that case you're not really seeing the base, just the outline of it. If you switch from texture 0 to any other texture, you'll see green, not red. The red parts are just a shared two-dimensional boundary.

The "4c" tree has is based on a pentagonal prism.

geom-nav

These examples are some experiments I did regarding navigation, compass points, and absolute directions.

Let's start with "3a". At the start you're face to face with a cube. If you fly around you'll see there are a bunch of colored cubes, but if you fly to the center of the mat you'll be back to seeing just single faces, one in each direction.

The idea here was to use lighter colors for positive directions along the axes and darker colors for negative directions. So, we have a yellow sun overhead (or a big ugly yellow fluorescent light) and brown dirt underfoot (since brown is like dark yellow), and we have cyan in front of us and dark cyan behind us, and so on.

The clipping between the brown cube and the mat is wrong, sorry about that. If it was easy to fix, I would have done it.

Now, imagine you're at the center of the mat, and I move the cubes away from you but at the same time make them larger and larger. The result is "3b". You might think the cubes are off at infinity, but no, the distance is only about 150. Because I wanted to make the cubes larger while keeping them level with your eyes, they now extend below y=0 and have the same clipping problem as the brown cube did before. Just ignore that and everything should make sense.

Now is a good time to look at "4a" and "4b", which are the same thing but with another dimension and another two compass directions. A good exercise in "4b" is to spin all around like crazy and then restore yourself to your original orientation. The page How to Orient Yourself from the maze game documentation might be of some help.

If you want to get to the center of the mat, the steps are the same in 3D and 4D. In align mode, slide left, take three steps forward, then slide right.

The "c" and "d" scenes go through the same two-step process as "a" and "b" but with better markers that show both what direction you're looking and what colors are in adjacent directions. You'll need to turn on some interior textures to get the full effect, and you might even want to hit "A" to get rid of the mixed-up boundaries.

In "c" it's interesting to back away from the markers and look at them from a distance. From some points of view you can see all of them at once in a nice triangular / tetrahedral arrangement. Note that the colors are almost exactly the same on all the markers; only the inward-pointing faces are different.

In the "e" files I've gone back to single-colored markers, but I've moved them closer together so that there are almost no gaps. If you turn on all the textures, the effect is something like a particularly colorful sunset. I plan to experiment more with patterns at infinity some time, both skies and horizons. (I did that; see Scenery Examples.)

If you like any of these navigation schemes, you could try cutting and pasting them into other files.

geom-sub

These useful examples show what subspaces look like. "3a" and "3b" show a line and a plane (one- and two-dimensional subspaces) while the four-dimensional ones show a line, a plane, and a three-dimensional plane. All these objects are flat along the forward-backward direction, so you can always scoot around to the side and look at them edge on, even "3b" and "4c".

geom-ring

Now I want to return to the arch examples and go off in another direction. What if we take the arch (two-legged arch) and pull it out of the ground so we can see a complete ring? If you fly to the side you can see the ring is flat, in either 3D or 4D, and if you fly around to the back you can look through the ring at a red cube over your start location.

But! In 3D the ring encloses the line between you and the cube. There's no way to move the cube outside the ring without having your line of sight to the cube be blocked. In 4D, however, the ring doesn't enclose anything. If you've got the ring in the natural orientation (in the up-down-left-right plane), you can slide in or out, then slide left or right until the cube is outside the ring, and the cube will stay visible the whole time.

Another point of interest is, in 3D you can spin so that the ring turns with the cube in the center, but in 4D you can do that and more. There's one spin command that makes the ring turn just as in 3D, but there are also two other spin commands that make it turn and trace out a sphere. And, the cube is still always visible in the center.

So, another way to get the cube out of the ring in 4D is to spin the ring into the up-down-in-out plane and then just slide left or right. The cube is at the center of the ring but not at all trapped inside it.

geom-sphere

So, as I said in the arch section without really explaining it, if we want to enclose things in 4D we need a sphere, not a ring. That's what you can see in these examples. I've done the construction in two stages, "a" and "b", with the final ceiling pieces being added in "b". In 3D the result is what you'd expect, a sphere that you can't get inside of. In 4D, as with the arch-dome, the result is something that encloses your line of sight but that you can still see through and fly through.

Seeing is believing, so fly around to the other side of the sphere and line up the red cube in the center. Now there's no way to get the cube out without having your line of sight blocked somewhere along the way. Also, if you stop halfway around, you can look to the side and see that the sphere is flat.

Instead of lines of sight we could equally well be talking about threads, so in 4D you can string spheres along a thread and they won't fall off.

Now that you can move the blocks around, you can fly to the other side of the sphere and pull the red cube through. You can do the same with the ring in 3D, of course.

The "ring", "sphere", and "3sphere" examples are also fun to explode and reassemble. The spheres have enough pieces that when they're scrambled they create natural maze-like structures. I particularly enjoy imagining that "sphere3b" is a nuclear reactor. You open it up, put the red fuel cube inside, and then, oh no, something's gone wrong! Kaboom!

geomx-3sphere

To complete the analogy, we need to construct a 4D object that we can't get inside of, and that object is the 3-sphere. Again, I've done the construction in two stages. And, just as "sphere3a" had holes that you could fly through to get to the center of the sphere, so does "3sphere4a". It's a good challenge. You'll definitely know when you're there—the view from the center is very symmetrical, especially if you turn off all textures except 0.

Another thing you can do in both "sphere3a" and "3sphere4a" is fly around to the other side and catch a glimpse of the red cube through the cracks. Once you've done that, if you line up the crosshairs with care, you can fly straight through to the cube.

I don't know much of anything to do with "3sphere4b", but it's certainly very colorful. I guess you can fly to one of the diagonal directions and get a nice view of a tetrahedral outside face.

geom-claw

When I was putting the ring examples together, the first step was to take the arch examples and lift them up, and that gave me these amusing claw shapes.

geomx-line

These show a line orthogonal to a plane (3D) and a three-dimensional plane (4D).

geomx-plane

These show two planes (two-dimensional planes) orthogonal to one another in 4D.

geomx-ring

A ring can't enclose a line in 4D, but there's a different analogy we can look at: a ring can enclose a point in 2D, a line in 3D, and a plane in 4D. If you want to think about the 4D case mathematically, what's going on is that the ring x2 + y2 = 1 is enclosing the zw plane x = y = 0 without intersecting it.

Now that you can move the blocks around, you can wiggle the enclosed line or plane and see that it's genuinely stuck inside the ring.