wendy wrote:The PG seeks to set a way of constructing names for which the meaning is apparent from the stems of the word. For example, tetra+hedr+on gives 4 + 2d + patches, say a figure so bounded. Much research was undertaken in the linguistics of other fields of science, that made the transition to practical technology, a lot of the faults and triumphs of these endeavours form the basis of the polygloss.
That it is so dispart from the current termonology is more implication of the way that the current termonology is set.
Current terminology is one big self-contradictory tangle of inconsistencies. That is why I was only half-joking when I referred to the IUPAC--I think the study of polytopes would greatly
benefit from a standardization of terminology.
From three dimensions, 2d can appear as either 2d or N-1 d. A word like "face" or "hedral" can be freely applied to 2d or N-1 d, which match in three dimensions.
Ah, yes, the classic example of the mess of self-contradicting terminology: the word "face". Traditionally, it refers to the 2D face of a polyhedron, of course, but people have extended it in completely inconsistent ways. "Facet" used to be a synonym for "face" (as in, facets of a diamond), but now it has come to mean (N-1)-face, whereas "face" itself could mean 2-face, (N-1)-face, or i-face for any i, as the abstract polytopists would have it (at least they're consistent). Coxeter, however, uses "cell" for (N-1)-face, whereas amongst 4D-centric folk, "cell" refers to a choron (a 3-face). I've seen "hypercell" being used for 4-face as well as (N-1)-face.
In six dimensions, N-1 is 5, one speaks of a 'dihedral angle' meaning the angle that five-dimensional surtopes meet.
Yes, "dihedral" is another one of those inconsistent terms: strictly speaking, a hedron should be a 2-face, so "dihedral" should be relating to two 2-faces. However, a dihedral angle thus defined has no meaning except in 3D, so it makes more sense to define it as "angle between two (N-1)-faces". Except that now "hedron" is taken to mean (N-1)-face rather than 2-face.
I have yet to see a widely-accepted term for this concept that does not contradict other uses of "hedron".
Likewise, faces are not facing things, but facets are.
Yes, so really, "face" shouldn't really apply except as an (N-1)-face (i.e., facet). Unfortunately you have the entire society of abstract polytopists contradicting such a proposition.
So nice is it to divide at the outset 2d (hedron) vs N-1 d (face), so were one to encounter hedron, one is talking invariably of something of 2d, or made of 2d things.
Would that we could apply this to "dihedral angle".
The problem of apposition (placing two numbers side by side) is always present. One has several solutions to this: eg
twenty of four-pound guns (ie 20 in number, of guns that shoot 4 lb shots)
twenty-four-pound guns ( ie guns that shoot 24-lb shots)
One method is to use 'cerimonial numbers', such as latin or greek: eg 1992 XII 31 (christmas 1992). [month in latin numbers]
The problem with polytope count eg 24ch, is that one might end with three strung numbers, eg
eighteen 24tn is 18-24-4s, ie count of eighteen + sides at 24 + patch of 4d.
The crux of the problem is that the three numbers come from three distinct domains: the first from the cardinality of the set of polytopes, the second from the cardinality of the set of facets, and the third from the ordinality of the dimension.
A straightforward solution suggests itself as using three distinct sets of number words for forming each of the three parts. A phrase like "24 icositetrachora" have this property, in that "24" comes from the Arabic numbers, "icositetra-" from the Greek prefix construction for 24, and "choron/-a" from a separate sequence -gon, -hedron, -choron, -teron, etc.. My only complaint here is the multiplicity of syllables in the second component; otherwise the system is impeccable.
Some work is done on this.
In weights and measures, one might hear references to things like "metre-tenths" or metre-tens, being E-10 m and E10 m. The idea of postfix number of this kind has been looked on favourably. This would allow the use of a greater number of compound roots, subscripted by the dimension. Some terms, like 'margin' are descending, but the ordinals serve here, eg margin-sixth = M-6 = N-8 dimensions. margin-sixths dual into edge-sixes (E+6 = 6). The terminology is set that one and first refer to the unadorned word: a margin is a margin-first (or margin-prime), while an edge-one is an edge.
The matter awaits formal consideration though.
Hmm. Is there a reason for using "margin" where others have used "ridge", for (N-2)-faces? The reciprocity of this system is nice, though. It makes comparisons of a polytope and its dual much less verbose.