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I'm curious about whether or not it would be useful to build 3D models out of pipecleaners or plastic or even 3D modeled in blender would assist in projecting 4D into 3D space effectively. I'm currently learning to understand how a hypercube "folds" from an unfolded model of one in my minds eye, starting to barely grasp how a new direction exists. But I think building some models would help. Any info on this?

- solarjackal
- Nullonian
**Posts:**2**Joined:**Fri Sep 04, 2020 7:18 pm

What I recommend is working with 3D things from the 4D world. The floor plan of a 4D house is 3D. The up-down dimension has been left out. A 4D house has six walls. A 3D cube has six sides. It works very well, the spaces relate to one another correctly, you just have to get used to replacing up-down with ana-kata. A door going ana-kata appears to be in the ceiling. But it isn't. I've considered actually building a model. Some of the furniture would appear to be floating in air. But they aren't. A floor plan is 100% floor.

The full 4D cube is hard to deal with. The worst of all -- or most fun, if you look at it that way -- might be the sphere. That's REALLY weird, and it's so symmetrical that simple tricks won't work. You have to decompose it into 4D tori(!). But you CAN make a map of the surface of a 4D planet. That's 3D, so it can exist in our Universe. Maybe some day I'll hire a wood carver to make one.

The full 4D cube is hard to deal with. The worst of all -- or most fun, if you look at it that way -- might be the sphere. That's REALLY weird, and it's so symmetrical that simple tricks won't work. You have to decompose it into 4D tori(!). But you CAN make a map of the surface of a 4D planet. That's 3D, so it can exist in our Universe. Maybe some day I'll hire a wood carver to make one.

- PatrickPowers
- Tetronian
**Posts:**394**Joined:**Wed Dec 02, 2015 1:36 am

The 4D sphere isn't all that hard. Here's how you get a mental model of it:

1) Take a 2D circle. But don't imagine the overhead view of the circle that we usually think of. Instead, think of it as seen edge-on. Or, if you like, look at a cylinder sideways. Notice the "bulge" in the middle, where it curves "outwards" at you? Now imagine the cylinder becoming flatter and flatter. Keep that "bulge" in the middle in mind. Imagine the cylinder as flat as a razor-thin strip. Now you have a line segment, but it's not really a line segment; it's an edge-on circle.

2) Take the 3D sphere. As seen on your computer screen, it's actually nothing more than a shaded circle. But actually, it's not a circle; the middle "bulges out", as conveyed by the way it's shaded. Notice how the envelope of the sphere is just a circle on your 2D screen, but your mind "sees" it as a sphere with a bulge in the middle, not a flat circle.

3) Now take the 4D sphere. In projection it's just a 3D sphere. However, it isn't a "flat" 3D sphere; it has a "bulge" in the middle, i.e., inside the core of the spherical projection image. The part of the 4D sphere's surface that projects to the inside is "bulging out" at you; it's nearer to you than its spherical boundary. Now you've "seen" the 4D sphere.

Easy, wasn't it?

Of course, that was only half of it. The "bulging out" part is the half of the 4D sphere facing you. Just like when you look at a 3D sphere, the "bulging out" is the near half. There's another half on the far side, that, if you cut away the near half, also occupies the space of a circular disk, but it "caves in" rather than bulges out. Glue these two halves together in your mind, and you've a complete picture of the 3D sphere.

Now do the same thing with the 4D sphere: as described above, it's just a 3D sphere, or rather ball (filled sphere), where the insides are "bulging out" at you. Well this is actually only half of the 4D sphere. The other half lies on the "far side", also spherical in shape, but ths insides are "caving in", i.e., curved away from you in the 4th direction. Glue these two halves together, and there you have it: you've successfully visualized the entire 4D sphere.

1) Take a 2D circle. But don't imagine the overhead view of the circle that we usually think of. Instead, think of it as seen edge-on. Or, if you like, look at a cylinder sideways. Notice the "bulge" in the middle, where it curves "outwards" at you? Now imagine the cylinder becoming flatter and flatter. Keep that "bulge" in the middle in mind. Imagine the cylinder as flat as a razor-thin strip. Now you have a line segment, but it's not really a line segment; it's an edge-on circle.

2) Take the 3D sphere. As seen on your computer screen, it's actually nothing more than a shaded circle. But actually, it's not a circle; the middle "bulges out", as conveyed by the way it's shaded. Notice how the envelope of the sphere is just a circle on your 2D screen, but your mind "sees" it as a sphere with a bulge in the middle, not a flat circle.

3) Now take the 4D sphere. In projection it's just a 3D sphere. However, it isn't a "flat" 3D sphere; it has a "bulge" in the middle, i.e., inside the core of the spherical projection image. The part of the 4D sphere's surface that projects to the inside is "bulging out" at you; it's nearer to you than its spherical boundary. Now you've "seen" the 4D sphere.

Easy, wasn't it?

Of course, that was only half of it. The "bulging out" part is the half of the 4D sphere facing you. Just like when you look at a 3D sphere, the "bulging out" is the near half. There's another half on the far side, that, if you cut away the near half, also occupies the space of a circular disk, but it "caves in" rather than bulges out. Glue these two halves together in your mind, and you've a complete picture of the 3D sphere.

Now do the same thing with the 4D sphere: as described above, it's just a 3D sphere, or rather ball (filled sphere), where the insides are "bulging out" at you. Well this is actually only half of the 4D sphere. The other half lies on the "far side", also spherical in shape, but ths insides are "caving in", i.e., curved away from you in the 4th direction. Glue these two halves together, and there you have it: you've successfully visualized the entire 4D sphere.

- quickfur
- Pentonian
**Posts:**2873**Joined:**Thu Sep 02, 2004 11:20 pm**Location:**The Great White North

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