by **quickfur** » Thu Mar 07, 2019 8:32 pm

But anyway, coming back to the original topic, my current idea is to imitate IUPAC nomenclature in naming duoprism augments, just as I imitated IUPAC with the "m,n-duoprism" notation for duoprisms.

Here is a possible (sketch of a) scheme:

- For duoprisms with only one augmentable ring, the relevant parameters are just the number of augments, which would be named as (mono)augmented, biaugmented, triaugmented, etc.; the relative positions of the augments; and if the augments are orientable which augments are gyrated.

- For positioning the augments, we pick one of the augments as position 1, and then count the number of positions on the ring to reach another augment either clockwise or counterclockwise, and number the other augments this way. Furthermore, we choose the augment of position 1 and the direction of the counting such that the resulting series of numbers are minimized.

So for example, if we have a 5,20-duoprism with 4 augments at positions 1, 3, 5, 8, we'd write that as the 1,3,5,8-tetraaugmented 5,20-duoprism. (Adjacent augments are not possible with the 5,20-duoprism.) We could have started counting from 8 instead, which would have resulted in 1, 4, 6, 8, but we choose 1, 3, 5, 8 because the numbers are smaller (3 < 4; 5 < 6). Or we could have started counting from 3 instead, but then either 1 or 5 would become 19, which is much larger than the numbers we obtain by counting as 1, 3, 5, 8. Similarly for counting from 5.

- If there are gyratable augments, we choose the ortho orientation to be the majority orientation, and use gyro for the augments with the other orientation. The ortho augments would be listed first in the name, then "gyro-" followed by the gyro augments, then "x-augmented m,n-duoprism". For example, the 10,20-duoprism has orientable augments, so we might have something like 1,3,8-gyro-5,10-pentaaugmented 10,20-duoprism, meaning there are augments at positions 1, 3, 5, 8, 10, and augments 5 and 10 are gyrated relative to the other 3 augments. We could have chosen 5 and 10 as the ortho orientation instead, but that would lead to 3 gyro augments instead of 2, so the two gyro assignment is preferred. Similarly, we could have started counting from 10, which would give us 1, 3, 6, 8, 10 as the indices of the augments, with 1 and 6 as gyro augments. But since 6 > 5, we prefer the assignment 1, 3, 5, 8, 10 instead.

- If both rings are augmentable, things get a lot more hairy. My present idea is as follows: we again number augments on each ring as before, choosing the assignment that gives the smallest indices, then we list the augments for each ring grouped by parentheses in order of the m,n prefix of the duoprism name. So for example, the 4,5-duoprism can have augmentations on both rings; the 4-membered ring of pentagonal prisms allows adjacent augments, while the 5-membered ring of cubes only allows non-adjacent augments. So we might have a (1,2,3),(1,3)-pentaaugmented 4,5-duoprism, meaning 3 augments on the 4-membered ring in positions 1, 2, and 3; and 2 augments on the 5-membered ring in positions 1 and 3.

- If there are orientable augments, things become even hairier, because that also induces orientation across rings. For example, the 10,10-duoprism allows simultaneous augmentation of both rings with adjacent, orientable augments. Suppose on each ring all augments have the same orientation. We might think it suffices to choose ortho orientation for all augments on either ring and minimize augment indices like before. However, gyrating all augments on one ring has the effect of shifting the indices of augments on the other ring by 1. I.e., a (1,3,5),(gyro-1)-tetraaugmented 10,10-duoprism is equivalent to a (2,4,6),((ortho)-1)-tetraaugmented 10,10-duoprism. So the two sets of augment indices are not independent; so there needs to be some rule to choose between these two notations. Perhaps the rule can be to choose the name with the minimal indices, so that (1,3,5),(gyro-1)-tetraaugmented 10,10-duoprism is preferred over (2,4,6),(1)-tetraaugmented 10,10-duoprism even though the former means assigning gyro orientation to the only augment, as opposed to defaulting to ortho orientation. This also applies where one ring has orientable augments but the other doesn't. In this case, gyrating the orientable augments induces a renumbering of the non-orientable augments on the other ring.

- Furthermore, a rule needs to be established in the case of orientable augments how exactly the indices on the other ring map to the positions of the orientable augments. For example, in the 10,10-duoprism, the augments are 10-prism||5-gon, which have lacing cells that alternate between square pyramids and trigonal prisms. If there's an augment in the other ring, its lacing cells could either share the same ridge as a square pyramid, or a trigonal prism. So there are 3 cases to consider for a biaugmentation with 1 augment on each ring:

(1) 2 trigonal prisms meet at a ridge

(2) 2 square pyramids meet at a ridge

(3) a square pyramid meets with a trigonal prism at a ridge.

Let's arbitrarily assign the first case to the name (1),(1)-biaugmented 10,10-duoprism. Then the second case involves gyrating both augments, which is equivalent to gyrating one augment and shifting the other's index by 1. So we'd have (gyro-1),(gyro-1)-biaugmented 10,10-duoprism, which is equivalent to (1),(gyro-2)-biaugmented 10,10-duoprism. Which is also equivalent to (2),(2)-biaugmented 10,10-duoprism. The latter cannot simply be renumbered to (1),(1)-..., because shifting indices by 1 induces a gyration of the other ring. The third case involves gyrating a single augment, so we have either (1),(gyro-1)-biaugmented 10,10-duoprism, or (2),(1)-biaugmented 10,10-duoprism (which is equivalent to (1),(2)-biaugmented 10,10-duoprism).

So a rule needs to be made about which among these possible names should be canonical. Should we minimize the indices, at the expense of having one or more gyro- prefixes? Or should we minimize gyro- prefixes and permit indices that don't start from 1?

Incidentally, this complexity in dealing with orientable augments across rings is one of the factors that cause the combinatorial explosion of augmentations of the 10,10-duoprism to 11+ million distinct augmentations up to isomorphism.

//

Further thoughts: using parentheses for both-ring augmentations can become hard to parse and hard to read out loud; perhaps a better scheme might be to split the number of augmentations across rings, like thus: 1,3,5-tri-2,4-biaugmented 10,10-duoprism. This would require writing "mono" for the single-augment-per-ring case, to prevent ambiguity: 1-mono-2-monoaugmented 10,10-duoprism means 1 augment on each ring, whereas 1,2-biaugmented 10,10-duoprism means both augments are on the same ring. Probably it's best to always write "mono" when both rings are augmented, otherwise we end up with confusing names like 1,2-bi-1-augmented 10,10-duoprism: the "bi" seems to indicate only 2 augments, yet 3 indices are specified. 1,2-bi-1-monoaugmented 10,10-duoprism makes more sense.