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 > More Geometry Examples
Geometry Examples #3
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## More Geometry Examples

geom2-texture

These examples demonstrate the new custom texture feature. In 3D what you see is a block with an octagon painted on the front face, and in 4D what you see is a tesseract with a cuboctahedron painted on the front face. As you move around the side of either one you'll see the painted face flatten and pass out of sight.

geom2-wall

These examples let you play with blocks built up into a wall. The middle block is a good place to start. You can pull it out and make an arch, or you can pretend it's a door and pull it shut behind you after you've gone through. The walls are three units long in the "a" variants and five units long in the "b" variants. One lesson you can learn here is that it's expensive to build walls and fences in 4D since the number of cubes you need grows as r2 rather than r.

geom2-tile

These are the same kinds of walls except that the solid blocks have been replaced with thin tiles painted red on the back. The tiles are also scaled down a little so that you can rotate them in place.

geom2-clamp

The idea in these examples is that we start with a single block, red in front, floating in the air, and we apply clamps to it in different directions and see how that restricts its movement and rotation.

In "3a" the block is clamped in one direction. It can slide in the other two directions, and can turn in place in one way. Note that when the block is turning, the forward points obstruct your view of parts of the clamping blocks. This is easiest to see if you go out of align mode so that you can stop the turn partway through.

In "3b" the block is clamped in two directions. It can slide in one direction but can't turn at all.

In "3c" the block is clamped in all three directions so that you can't even see it without moving some of the clamping blocks.

That's all simple enough, but what about 4D?

In "4a" the block is clamped in one direction. It can slide in the other three directions and can turn in place in three ways. As in 3D, the forward points cover up parts of the clamping blocks during turns. That's why the block appears to get larger and not fit between the clamps.

In "4b" and "4c" the block is clamped in two directions. It can slide in two directions and can turn in place in one way. This is just the same as a singly clamped 3D block, and in fact that pattern holds for all the other examples as well. A clamp removes one degree of freedom, so a 4D block clamped N times can move in the same ways as a 3D block clamped N-1 times.

Then, in "4d" the block can slide in one direction and can't turn, and in "4e" it's totally clamped.

In all these examples, but especially "3c" and "4e", you might want to slide up one space so that you have a symmetrical view of the situation.

geom2-house

Here are some simple buildings I put together using modular room units.

In the "a" variant we have just a simple one-room hut, for reference.

In "b" four huts are arranged around a central courtyard (six huts in 4D). In "c" the central courtyard is replaced with a room with a skylight (through which you can see the sun, naturally). In "d" the skylight is removed and the huts are separated by long corridors to make … a moon base! I also put an "airlock" in the room behind you, a single cube that's easy to move and replace so that you can get outside.

In "e" we have a 2x2 layout of rooms (2x2x2 in 4D). You can look around for yourself, or you can switch to "f" and let a train give you a guided tour. (Or you can just follow the rails.)

In all of these, you can fly up and out to get an overhead view, and you can move or delete the roofs if you want to peek inside.

Finally, in "g" we have a three-story building with an open floor plan and an elevator shaft down the middle. Be sure to look up at the front wall from the outside before you go in! When you get to the top floor you can get a nice panoramic view by pushing the roof up or folding the walls out. Also, if you've ever wanted to jump off a four-dimensional building, now's your chance.

geom2-arch

Here I just wanted to show off a more compact way to make arches that I thought of. In "a" you can see the new arches, and in "b" you can see how they fit together with the old arches from Geometry Examples.

geom2-ring, geom2-sphere, geom2-3sphere

In just the same way, there are more compact versions of rings, spheres, and 3-spheres. If you're interested, here's a three-dimensional table that shows how the pieces correspond.

`block3  diag3  ceil3block4  diag4  ceil4   extra4block3  half3  rtetrablock4  half4  wedge4  rpenta`