How do y'all distinguish 2d and 3d stuff in 4d?

[Vector_Graphics]

A 3d object is flat in 4d, and a 3d space is analogous to a plane, but we also often talk about planes of rotation, and 2d surfaces as boundaries on the 3d surface of a 4d object (i.e. boundary lines on the ground). How do you guys talk about 2d objects in 4d space? Like, what analog to "line/length" (for 1d), "plane/area/surface" (for n-1 dimensions) is there?

[Hugh]

I think that 3D is full 3D in 4D, not really “flat” like a 2D plane. The way I see it a 4D being sees a 3D slice of its 4D surroundings all around itself in full 3D.

[gonegahgah]

@Vector,

The general thinking here is to call 4-Volume "bulk". My preference is to call it 4-Volume. All other objects are objectively flat.
We consider a line and a plane to be flat. However, it would need a new identifier for 3D objects or use the generic 3-Volume.
So I myself would also refer to 3-Volume as 4-Plane (as to their eyes a 3D object would be a plane flat object like a piece of paper).
To 4Der's a 2D object would be an edge so 4-Edge. So as a table:

Us / 4Der
3-dot / 4-dot
3-line / 4-line
3-plane / 4-edge
3-volume / 4-plane
3-hypervolume / 4-volume

That is how I like to think of it; but that is not the convention, I believe.

[PatrickPowers]

A 3d object is flat in 4d, and a 3d space is analogous to a plane, but we also often talk about planes of rotation, and 2d surfaces as boundaries on the 3d surface of a 4d object (i.e. boundary lines on the ground). How do you guys talk about 2d objects in 4d space? Like, what analog to "line/length" (for 1d), "plane/area/surface" (for n-1 dimensions) is there?

My goal is/was to see a 4D world as though I were 4D myself and had grown up in it, seeing everything as natural. So things are named according to their function. What to us is a solid sphere is to them a disc. What to us is a hollow sphere is to them a ring. And so forth.

Sometimes this isn't enough. Let's say we've got a line to measure distance. That's 1D in both worlds. But a line to partition a surface is 2D in the 4D world. So there are both 1D and 2D lines. In 4D a shoreline is 2D. Weird, eh?

Planes of rotation are 2D no matter how many dimensions you have. You could quite reasonably say that they aren't planes in 4D but I don't like to invent new words. So I just write "3D plane" and "2D plane" if there's any chance of confusion.

[Vector_Graphics]

@Vector,

The general thinking here is to call 4-Volume "bulk". My preference is to call it 4-Volume. All other objects are objectively flat.
We consider a line and a plane to be flat. However, it would need a new identifier for 3D objects or use the generic 3-Volume.
So I myself would also refer to 3-Volume as 4-Plane (as to their eyes a 3D object would be a plane flat object like a piece of paper).
To 4Der's a 2D object would be an edge so 4-Edge. So as a table:

Us / 4Der
3-dot / 4-dot
3-line / 4-line
3-plane / 4-edge
3-volume / 4-plane
3-hypervolume / 4-volume

That is how I like to think of it; but that is not the convention, I believe.

Yeah, I specifically don't like "bulk". It feels just like pulling out a new word where it isn't necessary, especially when 'bulk' already has other meanings. I probably end up using "volume" for 4d when describing a 4d world. And apparently people use words like 'surcell' or smth here? Like, since when was 'surface' 2d specific? I feel like the 1d boundary of a 2d object has been called the surface. If anything, "surface" should be n-1 the same way "plane" is.
Most of the properties of 2d planes in 4d have to do with rotation, so I will probably end up deriving the terms for 2d stuff from words relating to rotating.

[gonegahgah]

If anything, "surface" should be n-1 the same way "plane" is.

Yes. Absolutely.

Most of the properties of 2d planes in 4d have to do with rotation, so I will probably end up deriving the terms for 2d stuff from words relating to rotating.

That sounds good.

But a line to partition a surface is 2D in the 4D world. So there are both 1D and 2D lines. In 4D a shoreline is 2D. Weird, eh?

I would be more inclined to call it a "shoreedge" than a shoreline (like water's edge) as it is more of a boundary than a line.
To my thinking a line always travels from and to somewhere whereas a 4-edge extends out in the full 360° of directions.

[PatrickPowers]

I would be more inclined to call it a "shoreedge" than a shoreline (like water's edge) as it is more of a boundary than a line.
To my thinking a line always travels from and to somewhere whereas a 4-edge extends out in the full 360° of directions.

I like that.

[Vector_Graphics]

2 candidates so far for naming 2d "plane"
"rotad" or other "rot..." or "gyr..." something - from the fact that rotation is a 2d operation and can be described using a 2d space.
"edge" (which seems to be gaining more traction than I thought) due to the fact that:
- the boundary of a region of the ground is a 2d surface in 4d
- the edges of 4d shapes are 2d
- a bladed edge in 4d would be 2d

[mr_e_man]

Wendy uses "hedrix" for the 2D thing and "chorix" for the 3D thing. https://os2fan2.com/gloss/pglossh.html#PGHEDRIX

[Embi]

Talking about 2D objects in 4D space can get a bit mind-boggling, but it's an interesting concept. In a way, you can think of a 2D object in 4D space as a "slice" or a "cross-section." It's like taking a snapshot of a 2D shape as it intersects with the 4D space, just like how a 3D space intersects with a 2D plane to create a 2D shape.