Keiji wrote:quickfur wrote:For that very reason, I fully agree that a single, consistent terminology accepted by everyone involved will be most beneficial. Which is why I wanted to update the wiki to rename some of the terms used there. But I have been too busy with other things for the past while, so I haven't gotten to anything on that front yet (I haven't even found enough time to update my own website, which is already a few months behind schedule on the Polytope of the Month series!). Plus, I don't run the wiki -- Keiji does -- and I don't want to step on his toes either by introducing or renaming terms that contradict the ones he used in other places on the wiki.
Let's look at the (small)
rhombicuboctahedron. Firstly it doesn't help that there's already confusion over whether you include the word "small" or not, and as for the great rhombicuboctahedron, even Wikipedia prefers to call it the less ambiguous name of
truncated cuboctahedron, even though
"the name truncated cuboctahedron, given originally by Johannes Kepler, is a little misleading. If you truncate a cuboctahedron by cutting the corners off, you do not get this uniform figure: some of the faces will be rectangles." Now for the main bit, rhombi-cub-octa- is one big portmanteau, cube and octa referring to the cuboctahedron which is half way between the two duals (a mesotruncate) while rhombi was chosen simply because
"twelve of the square faces lie in the same planes as the twelve faces of the rhombic dodecahedron which is dual to the cuboctahedron", and rhombi- takes merely from the rhombus which really has nothing to do with the rhombicuboctahedron itself.
I think we can all agree that the "standard" terminology sucks. There's nothing rhombic about the "rhombicuboctahedron", and "truncated cuboctahedron" is outright wrong (I've always argued that x4x3x is better understood as the Stott expansion of the hexagonal faces of the truncated octahedron, or the Stott expansion of the octagonal faces of the truncated cube, or the Stott expansion of the non-axial squares of the "rhombicuboctahedron").
I wasn't suggesting that the wiki should use the "standard" terminology.
My name for this is the
stauroperihedron. Now, there was a big fuss back when with Tamfang decided I'd misused his notation idea when I decided to extend it to everything under the sun (and used it a dimension higher or lower than intended), but there is no need to dwell on that. The concepts I latched onto were simple: pyro- refers to 3, stauro- refers to 4, rhodo- refers to 5, and geo,aero- and cosmo,hydro- refer to the two "ends" of the stauro- and rhodo- spectra. Mesotruncates are in the middle, pantomotruncates are all (=pan), and peritruncates are outside (=peri, like perimeter). And even though -hedron really means "faces" which are 2 dimensional, they are what bound a 3 dimensional shape and is already widely accepted as being a suffix for one. So what do we call a 3 dimensional shape which is the outside truncate (xx, xox, xoox, xooox etc.) in the 4 series? A stauro(4)peri(xox)hedron(3D).
I think the core idea (having different stems for different symmetry groups) is sound. As far as the regular polytopes go, such a system is perfect. I'm on the fence about whether we should adopt "hedron" for 3D shapes, or Tamfang's preferred "morph". But that's a question of implementation; the core idea is sound.
I must admit I've been very pushy with my names for things, however I am sure if you looked at my system against the widely accepted names for convex uniform polyhedra, if you hypothetically ignored the fact that the current system has been around for pretty much all relevant history you'd agree mine makes more sense, even if you might use different words or parts of words for it. It is just a question of sticking to what has evolved, versus designing something new from scratch. Sure, it doesn't properly cover all of the 2^n possible truncations as you extend to higher dimensions, but it's a lot easier to say "cosmochoron" than "hecatonicosachoron", and I'd argue even if not complete, it is an improvement over the existing system.
Well, just about
anything beats the existing system, I'm sure we all can agree.
What keeps me on the fence about the whole issue, though, is the issue of communication vs. representation. For the sake of communication, it is best if one uses pre-existing, well-known terminology so that newcomers can understand what we're talking about. To give a contrived example, imagine if we all agreed on this forum that what we called "cube" before should now be called, for whatever reason, "lumafyqonomus" (the actual term is irrelevant, you can substitute any arbitrary name here). As long as we all agree on this, there's no contention at all. But a newcomer coming to this forum, or an anonymous web surfer encountering the wiki, say, will have no idea what "lumafyqonomus" means, even though he may be very familiar with exactly what a "cube" is. We have only re-lexified our terminology, but the essence of the thing has not changed. Yet in doing so, we've alienated potential newcomers (not to mention made any materials we publish online unreachable via typical search engine terms). We may also potentially alienate other researchers, who may regard our neologisms as unnecessary.
On the side of representation, though, one has to concede that a consistently-derived name like "geohedron" is far better than an arbitrary, unanalysable term like "cube". In this case the benefit may not be obvious, but once we start talking about "truncated cuboctahedron" then the consistently-derived scheme will definitely be shown to be superior. However, consider this: from the side of representation, we are interested not in the pronuncibility of the label, but a compact, mathematically-accurate representation that unambiguously designates a specific shape. But we already have such a representation: the Coxeter-Dynkin symbol. When we write x4o3o (using Wendy's notation), for example, it unambiguously designates the cube, and furthermore provides full information about its symmetry group, the shapes of its surface elements, its construction, etc..
So it seems that we're still stuck with the sucky traditional terminology for communication purposes, and Coxeter-Dynkin symbols for representation purposes (which, really, doesn't have any significant drawbacks save being unpronunciable).
Coming back to Klitzing's original comment, his point was that we seem to be too trigger-happy with coining new terms and reinventing existing names, when doing so doesn't really help clarify the essentials of the subject, but only adds more noise to an already noisy, inconsistent, traditional terminology. I'm guilty of this too -- it
is rather fun to invent new names, after all: it's the thrill of being the first discoverer of some shape and having bragging rights to name it -- but let's take the name "rotunda", for example. Back when Mrrl first discovered a non-monostratic shape containing pentagonal rotunda cells, we didn't know of the existence of other non-trivial polystratic shapes (non-trivial meaning not made by simply gluing two segmentochora together). At the time, we only knew of Klitzing's monostratic segmentochora, so this was an exciting discovery for us. The fact that it was bowl-shaped (or cap-shaped, depending on which way you look at it), and had pentagonal rotunda cells, inspired us to name it a 4D "rotunda".
In retrospect, however, this choice of name was premature. As it turns out, subsequent research has revealed
lots of other polystratic bowl-like (cap-like) shapes. Should they all be called rotundae now, just because they aren't monostratic? Should the term "rotunda" merely mean "not monostratic"? Seems like a waste of such a good term for something that could've been called, oh, a "polystratic cap"?
Not to mention that Wendy has already anticipated them by defining the notion of lace prism and lace city long before Mrrl made his discovery. Klitzing's definition of "rotunda" makes much more sense: a hemispherical CRF, generalizing the 3D pentagonal rotunda in a slightly different, and arguably more useful manner. It lets us conveniently designate the pseudo-bisected 600-cell as an augmented icosahedral/icosidodecahedral rotunda, for example, without needing to count just how many layers of vertices it has (in order to know whether it's a bistratic, tristratic, or tetrastratic polychoron, e.g.).
Furthermore, there's the issue of, do we really need to invent a new set of terminology to cover
every possible variation of every possible class of shapes that we can find? There's only so many short, pronunciable names available in English, no matter how cleverly you try to derive them from Greek/Latin/whatever-else, we should reserve them for the important things. Like, shapes with special properties that we may want to refer to frequently. Obscure shapes that are just another entry in the big list of CRF polychora (and now we know that there are
lots and lots and lots of them) are OK to be named something less convenient to pronounce. Like 1,(1,2,1',2'),0,1-hexadiminished 24-cell, one of the 19 diminishings of the 24-cell that we
probably won't be referring to over and over again.
Perhaps the way to proceed is for all of us to lay all currently existing terminology / naming schemes on the table, and see if we can distill it into an IUPAC-style consistent set of terms. I don't think we can ever completely get rid of different naming schemes -- in any sufficiently interesting subject, there's always more than one way of looking at the same thing, which leads to different ways of generalizing / fitting things into patterns. But there should at least be a "lingua franca" of terminology that lets us communicate without being completely frustrated by inconsistent use of terms.