Questions about terminology

[Vector_Graphics]

1) me singlehandedly keeping this forum alive lol
2) the actual question.

a) "bulk" seems like the word was arbitrarily chosen, same with the "hole" series which I can't even bother to remember. i'd just make "volume" n-dimensional and "area" between 2 and n-1 dimensional.
b) "surcell"? seriously? what possible ambiguity could "surface" have in the context of 4d shapes.
c) "realm" may lead to misconceptions about 4d space. Just use "plane" in 4d, "edge" in 5d, and "3-plane" in higher dimensions.
d) does anyone actually say geochoron instead of tesseract? Not as much of a complaint about this one because I do like the elemental naming scheme, but it's like calling a cube a regular hexahedron.

[Hugh]

Hi Vector_Graphics, I’m still around too in the VRI thread. Feel free to add in any thoughts in there. Ever had a VRI?.

I asked ChatGPT4 about what you said in this post and here was its response:

a) The terminology in higher-dimensional geometry can indeed feel arbitrary or confusing, especially when trying to extend our 3D intuition. Your suggestion of using "volume" in an n-dimensional context and "area" for 2 to n-1 dimensions does offer clarity.

b) Terms like "surcell" are specialized and intended to distinguish higher-dimensional analogs from their lower-dimensional counterparts. However, you have a point; in a well-defined context, "surface" shouldn't be ambiguous.

c) "Realm" could indeed lead to misconceptions. Using "plane" for 4D, "edge" for 5D, and "3-plane" for higher dimensions is a good suggestion for avoiding any such confusion.

d) "Geochoron" is much less commonly used than "tesseract." While the term does align with a particular naming scheme, it's far less intuitive for most people compared to the more popular "tesseract," just as "regular hexahedron" is less commonly used than "cube."

[Vector_Graphics]

uhh, what is your response, though?

[Hugh]

I’m open to whatever is decided by the community regarding terminology. I just thought I’d help in whatever way I could to help keep the forum active. Cheers!

[gonegahgah]

I'm with you Vector. I prefer to make things more translatable and understandable for people and not so arcane.

[quickfur]

Terminology is a complicated topic, because there's a lot of historical baggage that clouds the waters.

Almost all of today's "standard" mathematical terminology for higher-dimensional geometry comes from generalizations from 3D. Unfortunately, there are a lot of things in higher dimensions that don't exist in 3D, so the result was that 3D terms got stretched and generalized in ways that often contradicted or are ambiguous with one another.

There are generally 3 approaches to dealing with this contradictory mess: (1) just accept that it's the way it is. This is obviously the least desirable option, but it does have the benefit that the maximum number of people would actually understand you. (2) Keep as much of existing terminology as possible, while eliminating the contradictory parts and/or standardising on one alternative when a single term has multiple, contradictory meanings. This is a compromise between (1) and (3), and generally is easier to understand, but nevertheless suffers from deficiencies, because inevitably a higher-dimensional concept will come up that can't be adequately mapped to an existing term, so you end up having to coin, ad hoc, new terms for it. But without an over-arching consistent approach to creating new terms, the result is equally as messy as (1). (3) Discard existing terminology and start over from a blank slate, construct a consistent scheme that can handle stuff from arbitrary dimensions.

Wendy's Polygloss is an example of (3), and generally I like it because it is completely consistent and does not suffer from 3D-specificity. However, it has one big disadvantage, that by discarding existing terminology, it comes across as obtuse and hard to understand (that's actually untrue, but you do need time to learn it, so there's the initial appearance that it's obtuse).

Now to answer your questions:

a) IIRC "bulk" is from Wendy's Polygloss. Why "bulk" instead of "volume"? Because "volume" in general refers to the measurement of the content of some n-dimensional object, the amount of space enclosed by the object, whereas "bulk" refers to the n-dimensional content itself. The difference is subtle but occasionally important. You could say "volume" is how you assign a number to the "bulk" of an n-dimensional object, whereas "bulk" refers to the content of the object itself. In some contexts, there can be multiple ways of assigning a number to the same object; in those cases it's important to distinguish between the two. Personally, though, I'd just use "n-volume" because more people understand "volume" than "bulk", and specifying the dimension clears up any ambiguities (it's obvious what "4-volume" refers to, whereas simply using the word "volume" without further clarification can be confusing, especially when you're talking about objects from multiple dimensions in the same discussion.)

Now, "holes": the reason there's a whole separate set of terms for holes is because once you go beyond 3D, there are actually different kinds of holes(!). This is one of the things that is hard to find the best terminology for, because there is no such thing in 3D, so in our 3D human language there's no native term for such a thing. In 4D, for example, there is a difference between a hole produced by puncturing an object with a long 4D spear (in the shape of a very long spherinder), and a hole produced by bending a long tube (say, a long spherinder) around so that its two ends meet, making a donut shape with a "hole" in the middle. The analogous actions in 3D produce the same type of hole, but in 4D two different kinds of holes are produced, with very different properties, so it's important to differentiate between them!

b) "surcell" refers specifically to a flat fragment of the surface of an object that's in the shape of an (n-1)-dimensional polytope. "Surface" is a more general term, that could refer to curved manifolds that form the boundary of the object. For example, we can speak of the surcell of a 600-cell, which is a tetrahedron. But a duocylinder does not have any surcells, because its two bounding surfaces are curved toroids. They form its surface. So both the 600-cell and the duocylinder have a "surface", but the surface of a 600-cell consists of tetrahedral surcells, whereas the surface of a duocylinder consists of two toroidal manifolds.

c) "realm", "plane", etc.: I'm also not a fan of "realm", but it does lead to nicer-sounding derived words like "two facets are co-realmar". Whereas if you use "plane", you definitely have to quantify it with the dimension, e.g. "3-plane" as opposed to "2-plane". Which leads to less pronunciable terms like "two facets are co-3-planar" (which may also be confusing since this is in 4D, but "co-3-planar" may sound like we're talking about 3D).

d) "geochoron", et al were coined by Keiji, this site's admin. I'm personally not a fan of that terminology, though I'm not opposed to it (it does have a nice ring to it, however it's very obscure and very few people would actually understand it unless they're from this forum!). I'd just stick with n-cube.