## Naming of duoprism augmentations

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

### Naming of duoprism augmentations

I'm working on projections of duoprism CRF augmentations for my website, and I'm not sure how to go about naming these things.

In particular, I'm looking at augmentations of the 10,10-duoprism, of which there are 11.9 million up to isomorphism. Each decagonal prism can be augmented with a pentagonal magnabicupolic ring (aka 10-prism || pentagon). Adjacent augmentations are possible, and both rings can be augmented simultaneously. Furthermore, augments can be gyrated relative to each other, and gyrating all augments in one ring is equivalent to shifting all augments by 1 position around the other ring.

The most interesting among these myriad augmentations are the omniaugmentations (icosaaugmentations), where all decagonal prisms are augmented. This produces two rings of pentagonal bicupolae, which can be orthobicupolae or gyrobicupolae, interspersed with square pyramids and triangular prisms. All decagonal prisms become internal to the polytope. Either or both rings can have all augments in ortho orientation, or have some augments in gyro orientation.

The most interesting augmentations, then, would appear to be the ortho-augmented case, where all augments line up with each other, and the gyro-augmented case, where every other augment is gyrated. Each of these cases have two subcases, depending on other one ring is ortho to the other, or gyrated relative to the other.

How would we name these cases? Is there a feasible scheme for naming all 11.9 million augmentations? How to denote the various combination of ortho and gyro augmentations, and the positions of the augments? Any ideas?
quickfur
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### Re: Naming of duoprism augmentations

This is not directly related to your question (I am definitely not qualified to answer it), but I just noticed that your countings of augmented duoprisms did not include using the digonal magnabicupolic rings on the n,4 duoprisms. Are there no CRFs from those? There are definitely tesseract augmentations using those.
ndl
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### Re: Naming of duoprism augmentations

Hmm you're right!

The digonal magnabicupolic ring is just the square pyramid prism, and the intra-ring angle is 90°. So it can only augment 4,n-duoprisms only up to n=4, because past that point it becomes concave relative to the adjacent cubes. So besides the augmentations of the tesseract, the only other candidate for augmentation is really just the 3,4-duoprism (= 4,3-duoprism). And for the same reason of the intra-ring angle being 90°, only non-adjacent augments are possible, which leaves us with a maximum of one augment in the ring of cubes.

I don't know off the top of my head what the inter-ring angle is... If it's small enough, we might be able to augment both the cubes ring and the trigonal prisms ring simultaneously. Well, wait. Since the square pyramid prism is just the extrusion of the square pyramid, the inter-ring angle should just be the angle between a triangle and the square in the square pyramid, which is 125.264°. But the inter-ring angle of the trigonal prism pyramid is atan(√5) = 65.905°. Adding these two would be > 180°, so unfortunately, these augments cannot coexist on the same 3,4-duoprism. So we're only adding one more augmentation to the 3,4-duoprism.

The tesseract (4,4-duoprism) case is a bit more interesting: the square pyramid prism (spp) has orientation, so we can have the orthobiaugmented tesseract and the gyrobiaugmented tesseract with spp augments. No adjacent augments are possible (90° tesseract dichoral angle + 2*65.905° inter-ring augment angle of two augments > 180°) in any orientation. So these two are the only pure augmentations possible. However, the concave gap between adjacent augments can be filled with square pyramid pyramids and become CRF again; if we do this with 4 adjacent augments around a ring of cubes, we obtain something I found before, which can be described as the convex hull of a tesseract and an octagon. It can also be regarded as a partial Stott expansion of the 16-cell, the square pyramid pyramids being quadrants of a 16-cell.

There remain also other possibilities: the 3,4-duoprism itself could act as an augment to itself in gyro orientation, thus producing one of the several 4D analogues of the gyrobifastigium. The spp can also augment 4,n-duoprisms up to n=6 by putting it in an orientation where the square pyramid is corealmar with the cells of the orthogonal ring, thus producing augmentations that are equivalent to prisms of the Johnson augmented n-gonal prisms. E.g., augmenting a 5,4-duoprism with an spp with square pyramids on the inter-ring ridges would produce the prism of the augmented pentagonal prism, and so on. Unfortunately, we can't use the 3,4-duoprism in place of spp here, because we would end up with augmented pentagons (the triangles and pentagons become coplanar) which are not CRF.
quickfur
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### Re: Naming of duoprism augmentations

And it just occurred to me: on the tesseract, both spp and the cubical pyramid can exist simultaneously as augments. This gives us a few more combinations (I didn't count how many though).
quickfur
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### Re: Naming of duoprism augmentations

There remain also other possibilities: the 3,4-duoprism itself could act as an augment to itself in gyro orientation, thus producing one of the several 4D analogues of the gyrobifastigium.

In fact that one is nothing less as the gybef prism! This is because of being just oxx xxx xxo&#xt.
--- rk
Klitzing
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### Re: Naming of duoprism augmentations

@klitzing: Haha, you're right, of course.

Though I'd point out that there are other analogues of gybef in 4D, for example, stacking a 3,4-duoprism with a square pyramid prism, or stacking two square pyramid prisms together in gyro orientation.

On a different line of development, you can analyze gybef as a digonal gyrobicupola, which in 4D can be generalized to a wider class of shapes, e.g., xNo3o||xNo3x||oNo3x for various values of N.
quickfur
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### Re: Naming of duoprism augmentations

wider class of shapes, e.g., xNo3o||xNo3x||oNo3x for various values of N

with N=2,3,4 only as ike||srid does not exist (except in hyperbolic environment).
And none of these feature gybefs, as the joins between trip and gyrated trip there never are corealmic.
--- rk
Klitzing
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### Re: Naming of duoprism augmentations

Klitzing wrote:
wider class of shapes, e.g., xNo3o||xNo3x||oNo3x for various values of N

with N=2,3,4 only as ike||srid does not exist (except in hyperbolic environment).
And none of these feature gybefs, as the joins between trip and gyrated trip there never are corealmic.
--- rk

True. The N=2 case would be the most likely candidate to be named among the 4D gybef generalizations; the others are more properly considered bicupolae instead. And yeah, I wasn't looking for gybef cells here, and it's obvious from the global curvature of the constitutent cupolae that there wouldn't be corealmar trigonal prisms that might form gybefs.
quickfur
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### Re: Naming of duoprism augmentations

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.
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