Johnsonian Polytopes

[quickfur]

Had the word "cube" been written "x4o3o" since day one, we would have no problem whatsoever in recognizing it on sight.

Actually, yes we would: from x4o3o, you can make very small changes to the notation, to get a completely different valid meaning. x3o3o, x4x3o, what have you. You can't do the same to cube... cybe, cupe, cuba, whatever, all just look like typos, and aren't confusing - the overall shape of the word, and the sound of it when spoken, are so different to the nearest thing you could misinterpret it as that you know what it is easily.

"cuba" is a valid name, and "cupe" is a valid acronym. We tell the difference based on context. A typo in "x4o3o" would also be caught by context, though I do agree that it is far more prone to typos than "cube". That's the price you pay for non-redundant, compact representation. Natural languages all have some degree of built-in redundancy ("cube" is a valid word, "cibe" isn't, etc.), but at the price of messiness (no obvious relation between "cube" and "octahedron").

In any case, like I said, I'm still somewhat on the fence. I'm not totally sold on the terminology derived from Tamfang's original idea, but neither am I opposed to it. I quite prefer not to use "standard" traditional terminology, but I feel compelled because that's what everyone understands. It's a hard sell to get the casual web surfer to accept terms like "geochoron" when they expect "tesseract" or perhaps "4D cube". To understand "geochoron" they'd have to amble over to the Terminology page, look up "geo", then look up "choron", then put the two together, then relate that to what they already know; whereas they already know what "tesseract" means (and if not, google reveals the answer pretty quickly, whereas googling for "geochoron" only turns up pages with more unfamiliar terminology), and "4D cube" consists of already-familiar terms "4D" and "cube", so they can skip the looking up part and just put two and two together immediately.

Another example: Klitzing habitually uses Bowers-style acronyms for various polytopes; I have a lot of trouble reading some of his posts because of that. I have to keep looking up what each acronym means, and it really makes communication more difficult than necessary, one might argue. If we had standardized on, say, CD diagram notation, then even if there are typos at least we'd be in the same ballpark. As things stand, if I have enough trouble reading the Bowers-style acronyms, I imagine most first-timers interested in the subject would give up before they get to some of the marvelous things we've found here.

One avenue to explore, perhaps, is to see if we can take Wendy's polygloss and use it as a basis for deriving more pronunciable names for things like x4o3o, x4x3x3x, etc.. Not all of these symbols need names, of course, but I think it's safe to say the regular polytopes are special enough to deserve their own names. I'm not sure which of the uniforms would deserve individual names -- it's not as though we go around every other day talking about the n'th truncate between the 6D cube and the 6D cross. I'm willing to just use IUPAC-style numerical suffixes to indicate the CD diagram's configuration. Dedicating a set of roots just for naming (some subset of) them seems a bit arbitrary -- we could argue all day about which to include/exclude with no real consensus.

[wendy]

One of the problems i do notice with my notation is that it is supprisingly dense. What does one do, though?

The material is likewise as dense. For example, there are 15 valid polytopes on 3,3,5, which correspond to all fifteen combos of 0,1 except 0,0,0,0. It is a coordinate system with position polytopes. So in one sense, it really can't help but be dense.

On the same line, i do see what Bowers and Klitzing are up to. You may well need different handles for different things, like we give suburbs different names. But if you don't work locally, it can be hard to work out if balmoral is north or south of belmont, or where banyo fits into the mix.

[Klitzing]

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..

The important problem here is that it's easy to look at "cube" and know what it is. But if you look at x4o3o, it's not immediately obvious what it is - you have to work it out, and you might take a glance and think you read x3o4o or o4o3x which are the octahedron instead. This problem gets worse the more complex the symbol gets.

Right to the contrary! "Cube" is just better known than "x4o3o" because its non-mathematical everyday prehistory. If I'd ask you what would be better known (or could be easier dechiffered), a term like "dischiliahectohexaconta-myriaheptachiliadiacosioctaconta-zetton" (and even its translation from greek number names to arabic digits: "2160-17280-zetton") or one like "o3o3o3o *c3o3o3o3x", then the answer would be quite clear: The dynkin symbol does provide all the informations needed, that classical way of naming (providing arrays of facet counts) not even provides a clue what those facets themselves would be.

In general Dynkin symbols even are shorter than Bowers acronyms, as those are based on those longish classical names. Esp. when lots of adjectives are contained within the name, each one will have to be abbreviated to at least one additional character, often more (like introduced vowels for a better pronouncabillity). It is only for some base polytopes (of either dimension) that those acronyms get even shorter than the Dynkin symbols. (E.g. that above mentioned thingy he does represent just as "fy".)

The load of information of that dynkin symbol, and its shortness clearly imply directly that any single character thereof is essential. I.e. any typo clearly has to be excluded, there has to be taken lots of care, both while typing and while "reading".

Getting used to either way of naming is a prerequisite. I'd go with Wendy od Quickfur in saying, that the best way to provide the most load of information always is the Dynkin symbol. Whether those are given as classical graphs or as Wendy's (or mine, which clearly is based on hers) inline notation is a subsequent topic. Inline notations clearly have the advantage of being printable inline, while the additional disadvantage that those need for a further legend translating that inline representation into the being meant graph. But in the great run that one is quite obvious and thus the legend can be dropped.

On the other hand, when dealing with rather few special polytopes, refering to them multiply within textual contexts, one clearly is tempted to use some sort of (readable) names instead. Here the shortest ones clearly are abbreviations. And nothing is more misleading than having individual abbreviations. That is why I stick to the standardized ones, i.e. to the OBSAs (official Bowers style abbreviations).

The only disadvantage to them, I agree, is that those are not too wide spread. Wiki one by one adopts them. Jonathan's and my website naturaly use them a lot. But neither his nor mine are a complete reference. Jonathan owns lots of handwritten lists of such abbreviated names. I've taken over the task to list at least all those he has mentioned sometimes somewhere into a spreadsheet, which now is online accessible. (Thus being at least part of an official reference.)

--- rk

[quickfur]

One of the problems i do notice with my notation is that it is supprisingly dense. What does one do, though?

The material is likewise as dense. For example, there are 15 valid polytopes on 3,3,5, which correspond to all fifteen combos of 0,1 except 0,0,0,0. It is a coordinate system with position polytopes. So in one sense, it really can't help but be dense.

And the CD symbol in this case really is the best possible representation, because not only can you tell what symmetry group it belongs to (*3*3*5*: immediately you know it's the 600-cell group), you can also tell what shapes its surtopes make: e.g., x3o3x5o: the shapes of the surtopes are: o3x5o, x.x5o (a prism, due to the disconnected node), x3o.o (a polygon, so not a full-fledged cell), and x3o3x. Furthermore, the symbol also tells you how you may go about constructing such a shape: each ringed node corresponds with a Stott expansion along that element of the symmetry group. By starting from the degenerate o3o3o5o (a point) and applying Stott expansions for each ringed node according to the underlying symmetry group, you can construct the whole shape, and accurately at that (none of that "truncated cuboctahedron" nonsense).

None of the traditional names AFAIK give full information about the shapes of facets and the symmetry group in such a compact way AND tell you how to construct the thing to boot. A name like "600-cell" only tells you there are 600 of something, but doesn't tell you what shape that something is. A name like "cantitruncated 600-cell", at the most, only tells you how to construct the object from a 600-cell, but has no information about what kind of cells it has. About the only advantage a traditional name has is memorability (and even that is questionable!) and the appearance of being less arcane than a symbol like x3o3x5o.

On the same line, i do see what Bowers and Klitzing are up to. You may well need different handles for different things, like we give suburbs different names. But if you don't work locally, it can be hard to work out if balmoral is north or south of belmont, or where banyo fits into the mix.

I really like this analogy. Locals would find these handles much more convenient than, say, reciting GPS coordinates each time they want to refer to a place. The solution for outsiders unfamiliar with the handles is a map that tells one the association between handle and location.

So in our case, a good solution is to have a "map" of sorts, specifying the correspondence between CD symbol and handle; that is, a gloss of sorts, like the Polygloss or Klitzing's collection of Bowers acronyms, etc.. The mappings can be between more than one naming convention -- just like a map of physical terrain not only has location and placenames, but also terrain and other kinds of landmarks.

In fact, we could make a spreadsheet that lists correspondences between whatever naming scheme is out there. Each row of the table corresponds with a single object, concept, item, etc., and there would be a column with the CD diagram, another column with Wendy's inline notation of the CD diagram, a column with the Olshevsky/Johnson-style "traditional" name, a column for the Bowers acronym, a column for Keiji's Tamfang-style name, and whatever else anyone cares to include, like nicknames, pet names, etc.. Such a spreadsheet could be made accessible in an online interface that allows selection of columns to display/suppress, as well as sorting by any chosen column to permit easy lookup by any notation one may choose. Many of the entries can probably be programmatically generated if a particular column has a mechanical mapping scheme from a primary key, let's say Wendy's inline CD symbol (for ease of programming).

If nothing else, this would be a fun project to work on.

[wendy]

I do like the idea. I'm pretty sure that Richard has this in hand, since he maintains a largish list in a table, complete with my form of the decorated dynkin graph.

You can get a limit to the size of the thing, by noting that if you know h for the group (ie the "Petrie polygon by reflection"), then the size of the polytope is limited by a polygon hx, where x is the number of marked nodes. So x3x3o5o is smaller than a {60} = 30*2. That should help somewhat.

h = n+1 for simplexes, h = 2n for the cross and cube, h = 12 for the {3,4,3}, h = 30 for {3,3,5}, and for the gosset groups 2_21 (12), 3_21 (18), and 4_21 (30).

So that's a start.

On the other hand, the notation that i describe can illistrate figures of different sizes. There are letters like q and f, which are used to show different sizes of edges. The golden rectangle is f2x, and a number of other figures that elsewise have no name get names like x3f3o vs f3x3o (both truncated tetrahedra, but with different size triangles). The oxqxo8ooooo&qt, has no obvious name elsewise (i call it the octagonal ball), but this nicely describes it.

[Keiji]

I've re-read through this thread and added (to CRF_polychora_discovery_project#Monostratic_stacks) three CRFs that quickfur rendered a long time ago, which I call the transstaurosemicupola, gyrogeocupola and bigyrogeocupola. I believe the transstaurosemicupola is part of a fairly large family (of 20) listed below, though I'd like someone to check these are all CRF before I add them to the wiki:

xox4oxo3ooo - bigeosemicupola
xoox4oxxo3oooo - elongated bigeosemicupola
xoxo4oxoo3oooo - augmented bigeosemicupola
xooxo4oxxoo3ooooo - augmented elongated bigeosemicupola
oxoxo4ooxoo3ooooo - biaugmented bigeosemicupola
oxooxo4ooxxoo3oooooo - biaugmented elongated bigeosemicupola

xox3oxo4ooo - biaerosemicupola
xoox3oxxo4oooo - elongated biaerosemicupola
xoxo3oxoo4oooo - augmented biaerosemicupola
xooxo3oxxoo4ooooo - augmented elongated biaerosemicupola
oxoxo3ooxoo4ooooo - biaugmented biaerosemicupola
oxooxo3ooxxoo4oooooo - biaugmented elongated biaerosemicupola

xoo4oxo3oox - transstaurosemicupola
xooo4oxxo3ooox - elongated transstaurosemicupola
xooo4oxoo3ooxo - aeroaugmented transstaurosemicupola
xoooo4oxxoo3oooxo - aeroaugmented elongated transstaurosemicupola
xooo3oxoo4ooxo - geoaugmented transstaurosemicupola
xoooo3oxxoo4oooxo - geoaugmented elongated transstaurosemicupola
oxooo4ooxoo3oooxo - biaugmented transstaurosemicupola
oxoooo4ooxxoo3ooooxo - biaugmented elongated transstaurosemicupola

It's also worth testing if these exist for the other cupolae:

pyroquasicupolae - 6 total - pattern is: ((bi)augmented) (elongated)
staurosemicupolae - 20 total, listed above
rhodosemicupolae - 20 total, same as above but with 5,3 instead of 4,3
pyropericupolae - 6 total
stauropericupolae - 20 total
cosmopericupolae - 6 total (hydropericupolae do not exist)
pyrosemimesocupolae - 6 total
staurosemimesocupolae - 14 total (trans forms do not exist because you'd have to join on oxo, which is smaller than xxo so it'd be concave)
hydrosemimesocupolae - 6 total (cosmosemimesocupolae do not exist)
stauromesocupolae - 6 total (cis forms only due to mesotopic bases)
rhodomesocupolae - 6 total (cis forms only due to mesotopic bases)
pyrosemipericupolae - 6 total
staurosemipericupolae - 20 total
rhodosemipericupolae - 20 total
pyrocanticupolae - 6 total
staurocanticupolae - 20 total
cosmocanticupolae - 6 total (hydrocanticupolae do not exist)

That's an upper bound of 194 polystratic cupolic forms that we can check. Also, I'm not sure if you'd be able to gyroelongate any of these in place of the prismic elongations. I'm thinking you probably couldn't because it'd make it concave, but it's something worth looking into!

Of course, these would have to be checked for coincidences with already counted polychora... for example, the elongated bigeopericupola is probably a uniform polychoron analogous to the stauroperihedron.

[wintersolstice]

Removed fully quoted preceding post. ~Keiji

I've been meaning to do search of monstratic stacks for a while, but never got round to it, I'll sort these out,

(though I'll probably have my own naming principle, one that I've had a few ideas for already)

I actully did ask this early in the thread, I wonder if I could put my names for the cupola forms on the wiki (aswell that it:D) I did after all spend 11 weeks sorting and catogrising the segementotopes second there are 3 more cupola forms not listed on table which are vertex transitive so I wonder if they should be added for completeness

[quickfur]

[...] That's an upper bound of 194 polystratic cupolic forms that we can check. Also, I'm not sure if you'd be able to gyroelongate any of these in place of the prismic elongations. I'm thinking you probably couldn't because it'd make it concave, but it's something worth looking into!

I wonder if a computer search can be made for all these stacked polystratic forms.

As for gyroelongations, the main problem is how to define a gyration in a 4D elongated object. There are at least two distinct possible interpretations: replacement with the dual of the base shape (perhaps more properly called a dualization? I suggested the term syncopation to prevent confusion with taking the dual of the entire polytope, but it didn't catch on), and an actual gyration as in rotate by some amount in a given plane. The latter has the further issue of which plane the rotation should happen in, since in 4D there may be more than one possibility.

Of course, these would have to be checked for coincidences with already counted polychora... for example, the elongated bigeopericupola is probably a uniform polychoron analogous to the stauroperihedron.

I'm finding that these "readable" names are getting into the way of comprehensibility. Can we pretty please use CD symbols? It's pretty obvious to me that the stack x4oo||x4ox||x4ox||x4oo is the same thing as x4oxo, but I've no idea what a bigeopericupola is -- at least, not without stopping to think about what it might mean, then translate it into CD symbols, then interpret it. Sorry, I don't mean to snub anyone here, but I'm finding that CD diagrams are far superior at communicating precise shapes in this case.

[quickfur]

P.S. and on that note, I find that wendy's polystratic shorthand of the form xox4oxo3xxo3xxo quite difficult to read as well, even though it is more compact than Klitzing's barred notation x4o3o||x4o3o||x4o3o. In the latter, the individual layers are immediately obvious, but in the former you have to squint really hard or rewrite it in layers to figure out which x lines up with which o. I vote for combining the monostratic CD symbol x4o3o with Klitzing's barred notation for maximum readability.

[Keiji]

P.S. and on that note, I find that wendy's polystratic shorthand of the form xox4oxo3xxo3xxo quite difficult to read as well

Aww, I was just getting into it, after finally figuring out how it worked!

It's pretty obvious to me that the stack x4oo||x4ox||x4ox||x4oo is the same thing as x4oxo

I thought it wouldn't be x4o3x3o, but x4o3o3x..?

[wendy]

The polystrate notation was intended to bring all of the elements onto the same symmetry, which is useful when dealing with groups like 2_22 (4B1) or the simplex. The temptation could be to take 4/B1 or 4B1/, and convert them into the identical, but differently orientated /4B1.

It should be noted that at first, the polystrate notation was intended for lace-prisms and lace-tegums. Here the various layers are at the vertices of a simplex, not several points on a tower. That came later. It's the function that comes after the & that tells you what to do with the layers.

Richard's || notation is clearer, but more limited. You can't have lacings other than 'x', (which is what segmentotopes have), whereas many of the polytopes i use the strata form might have 'q' or 'h' heights (or even the octagon 'k' shortchord. For example, the form oxqxo8ooooo&qt correctly describes the octagon-ball, the vertex-figure of one of my non-wythoff polytopes, but Richard's || has no form for it. You could write |q|, i suppose.

The real reason is that i wanted to keep the symbol as one word, ie without gaps. It looks awkwid, but i suppose that you're neally getting coordinates from the array of symbols, which is not bad.

[Klitzing]

... As for gyroelongations, the main problem is how to define a gyration in a 4D elongated object. There are at least two distinct possible interpretations: replacement with the dual of the base shape (perhaps more properly called a dualization? I suggested the term syncopation to prevent confusion with taking the dual of the entire polytope, but it didn't catch on), and an actual gyration as in rotate by some amount in a given plane. The latter has the further issue of which plane the rotation should happen in, since in 4D there may be more than one possibility. ...

There even is that difference of usage within 3D (Norman's own terms): Norman used

• "elongated ... gyro-..." for something having caps, which are gyrated (i.e. relatively rotated) mirror images, being spread apart by a medial prism; resp.
• "gyroelongated ..." for 2 caps, which are spread apart by an antiprism.

Thus the (in that sense) correctly to be used extrapolation to 4D then would be in each case quite obvious. - Just that a true gyration in 4D is much less often encountered, as those there require caps of some 3D axial symmetry with a base polytope (the one for join) having a larger symmetry. (In 2D this was found rather often: just using an axial xNo symmetry and a xNx base.)

Of course, these would have to be checked for coincidences with already counted polychora... for example, the elongated bigeopericupola is probably a uniform polychoron analogous to the stauroperihedron.

I'm finding that these "readable" names are getting into the way of comprehensibility. Can we pretty please use CD symbols? ...

I'd second that.

... It's pretty obvious to me that the stack x4oo||x4ox||x4ox||x4oo is the same thing as x4oxo, but I've no idea what a bigeopericupola is -- at least, not without stopping to think about what it might mean, then translate it into CD symbols, then interpret it. ...

Wrong quickfur - and yes, Keiji is right in pointing this out -
x4oo||x4ox||x4ox||x4oo surely is sidpith = x4o3o3x (not srit= x4o3x3o). - This already can be seen from the facets: x4o3o3x has for facets x4o3o . (those cube, which are used in that stacking as top layer), x4o . x (further cubes, used rather as tetragonal prisms), x . o3x (trips), and . o3o3x (tets). Whereas x4o3x3o has for facets x4o3x . (sircoes), x . x3o (trips), and . o3x3o (octs) - i.e. no cubes at all.

But none the less srit = x4o3x3o can be given as multistratic stack: x4o3x || x4x3o || x4x3o || x4o3x.

P.S. and on that note, I find that wendy's polystratic shorthand of the form xox4oxo3xxo3xxo quite difficult to read as well, even though it is more compact than Klitzing's barred notation x4o3o||x4o3o||x4o3o. In the latter, the individual layers are immediately obvious, but in the former you have to squint really hard or rewrite it in layers to figure out which x lines up with which o. I vote for combining the monostratic CD symbol x4o3o with Klitzing's barred notation for maximum readability.

Well, xxxx4oxxo3xoox&#xt has layers x...4o...3x..., .x..4.x..3.o.., ..x.4..x.3..o., and ...x4...o3...x; therefore this is completely equivalent to x4o3x || x4x3o || x4x3o || x4o3x. Wendy's lace tower description has the advantage of being shorter and allowing for different edge-qualifiers (e.g. Wendy's q- or f-edges). - My layer description is longer but allows for non-Dynkin-symbol usages as well, esp. stacking things atop each other, which lack a useful common symmetry, like "cube || ike". I.e. Wendy designed her symbol esp. for usage of non-unit edges too, while mine was designed e.g. for usage of diminished bases too. - And yep, quickfur, I agree, mine (even when using Dynkin symbols for layer polytopes) is slightly easier to read, as here no paralaxes might be encountered (which usually are the main reading problem in Wendy's towers).

Btw., even so Wendy herself often drops that tail "&#xt" it is an essential part of her symbol! Telling that those layers have additionally (&) lacings (#) of length "x" and that those layers form a linear tower (t) - rather than forming a mere simplex within orthogonal space (no further extension beyond "&#x", i.e. all layers are pairwise to be laced accordingly) nor would form a circuit or ring (r). This is because Wendy aimed that staggered symbol without any extension for layers which are NOT shifted apart, i.e. for according compounds.

--- rk

[Klitzing]

... Richard's || notation is clearer, but more limited. You can't have lacings other than 'x', (which is what segmentotopes have), whereas many of the polytopes i use the strata form might have 'q' or 'h' heights (or even the octagon 'k' shortchord. ...

Hey, super, Wendy, I was missing that symbol so far in all your writings (in either list). So let us use that 'k' from now on.

... For example, the form oxqxo8ooooo&qt correctly describes the octagon-ball, the vertex-figure of one of my non-wythoff polytopes, but Richard's || has no form for it. You could write |q|, i suppose. ...

Haha, great idea! - I used || just as a true parallel-sign so far, but to be read as 'atop' instead. And indeed, those layers are to be stacked parallely. - But in cases, where such easy lacing lengths are to be used, this might be usable indeed. (It should be pointed out then, that those are not to be misread as the to be used heights.)

... The real reason is that i wanted to keep the symbol as one word, ie without gaps. It looks awkwid, but i suppose that you're neally getting coordinates from the array of symbols, which is not bad.

In fact (except to the total number of to be typed characters) "oxqxo8ooooo&qt" is completely equivalent to "q-laced tower of o8o || x8o || q8o || x8o || o8o" or even, using your new addition, to "o8o |q| x8o |q| q8o |q| x8o |q| o8o".

The polystrate notation was intended to bring all of the elements onto the same symmetry, which is useful when dealing with groups like 2_22 (4B1) or the simplex. The temptation could be to take 4/B1 or 4B1/, and convert them into the identical, but differently orientated /4B1. ...

Yep, the orientation matters. This I had to do wordingly: hex = tet || dual tet, but you simply would write xo3oo3ox&#x. OTOH that then clearly could be rewritten as x3o3o || o3o3x.

But there is a further issue. My || notation allows only for multi-layered symbols, i.e. for linear sequences of stacks, that is: your lace prisms or lace towers. Never for lace rings (at least without any further looping symbol to be added at the end), and esp. not for lace simplexes. Neither for compounds.

Or we'd opt for |l=q| instead for just |q| (i.e. "l", the lacing edge length is to be set accordingly), and add further |h=0| (the height is to be set accordingly). Then the compounds could be included now: xo3oo4ox = x3o4o |h=0| o3o4x.

--- rk

[quickfur]

[...]... It's pretty obvious to me that the stack x4oo||x4ox||x4ox||x4oo is the same thing as x4oxo, but I've no idea what a bigeopericupola is -- at least, not without stopping to think about what it might mean, then translate it into CD symbols, then interpret it. ...

Wrong quickfur - and yes, Keiji is right in pointing this out -
x4oo||x4ox||x4ox||x4oo surely is sidpith = x4o3o3x (not srit= x4o3x3o). - This already can be seen from the facets: x4o3o3x has for facets x4o3o . (those cube, which are used in that stacking as top layer), x4o . x (further cubes, used rather as tetragonal prisms), x . o3x (trips), and . o3o3x (tets). Whereas x4o3x3o has for facets x4o3x . (sircoes), x . x3o (trips), and . o3x3o (octs) - i.e. no cubes at all.

Oh, right, I was confusing it with x4ox||x4xo||x4xo||x4ox. Sorry!

[...]

P.S. and on that note, I find that wendy's polystratic shorthand of the form xox4oxo3xxo3xxo quite difficult to read as well, even though it is more compact than Klitzing's barred notation x4o3o||x4o3o||x4o3o. In the latter, the individual layers are immediately obvious, but in the former you have to squint really hard or rewrite it in layers to figure out which x lines up with which o. I vote for combining the monostratic CD symbol x4o3o with Klitzing's barred notation for maximum readability.

Well, xxxx4oxxo3xoox&#xt has layers x...4o...3x..., .x..4.x..3.o.., ..x.4..x.3..o., and ...x4...o3...x; therefore this is completely equivalent to x4o3x || x4x3o || x4x3o || x4o3x. Wendy's lace tower description has the advantage of being shorter and allowing for different edge-qualifiers (e.g. Wendy's q- or f-edges). - My layer description is longer but allows for non-Dynkin-symbol usages as well, esp. stacking things atop each other, which lack a useful common symmetry, like "cube || ike". I.e. Wendy designed her symbol esp. for usage of non-unit edges too, while mine was designed e.g. for usage of diminished bases too. - And yep, quickfur, I agree, mine (even when using Dynkin symbols for layer polytopes) is slightly easier to read, as here no paralaxes might be encountered (which usually are the main reading problem in Wendy's towers).

Well, we could just adopt the notation for different edge lengths into the barred notation, like o4ox||o4of||o4xo for the 4D teddy. And for orientation-sensitive constructions, we could write things like x3oo||x3oo = tetrahedral prism, x3oo||o3ox = hex (i.e. tetrahedral antiprism), that is, the symmetries of each element lines up with what comes before.

You're right, though, that things like cube||icosahedron would be difficult to write in the same way. One may get away with writing x4oo||o5ox, but then it would be unclear exactly how the .4.. and .5.. symmetries line up.

Btw., even so Wendy herself often drops that tail "&#xt" it is an essential part of her symbol! Telling that those layers have additionally (&) lacings (#) of length "x" and that those layers form a linear tower (t) - rather than forming a mere simplex within orthogonal space (no further extension beyond "&#x", i.e. all layers are pairwise to be laced accordingly) nor would form a circuit or ring (r). This is because Wendy aimed that staggered symbol without any extension for layers which are NOT shifted apart, i.e. for according compounds. [...]

I see. Well, at least for towers, I think the barred notation is easier to read. For circuits or rings, probably a different notation would be desirable, but I haven't worked enough with those to be able to judge what is most suitable.

[quickfur]

[...]Yep, the orientation matters. This I had to do wordingly: hex = tet || dual tet, but you simply would write xo3oo3ox&#x. OTOH that then clearly could be rewritten as x3o3o || o3o3x.

Yes, we could just adopt the convention of symmetry elements in each layer lining up with each other. I think that covers all the cases except for cube||ike.

But there is a further issue. My || notation allows only for multi-layered symbols, i.e. for linear sequences of stacks, that is: your lace prisms or lace towers. Never for lace rings (at least without any further looping symbol to be added at the end), and esp. not for lace simplexes. Neither for compounds.

Or we'd opt for |l=q| instead for just |q| (i.e. "l", the lacing edge length is to be set accordingly), and add further |h=0| (the height is to be set accordingly). Then the compounds could be included now: xo3oo4ox = x3o4o |h=0| o3o4x.

--- rk

I think the main advantage of the || notation is that it's very intuitive and readable for lace prisms and lace towers. I'm not so certain about stretching it to fit lace rings... but maybe ( ... || ... || ...) for rings?

[wendy]

One notes that Richard's "cube || ike" is "x4o3o || o4s3s" or xo4os3os&xt. Some calculations become correct with vo5o3ox&xt, (ie v*doe || ike), because resolving 's' is rather hard as a coordinate.

I like quickfur's abbreviation of x or o, if all positions have the same value. It somehow makes sense.

There is no inferred definition of 'height' in a lace-tower. It's more an accident. It is just as useful to lace up things like _3ox4_&xt, which is always flat, regardless of the space-fillers. Most of the -3-3-5- can be laced together by a chain of 'x', and there are two chains with -3-3-5x&ft, and -3-3-5o&ft, right down to the point in the latter case.

Like the rest of the CD system, the lacing is meant to be the edge, rather than the height it creates. That is, we are not looking for unit-height slabs in segmentotopes, but unit-lacings. The actual thing could end up flat, as oox6oxx&xt does. Still, this can be found by way of the spreadsheet, which basically implements a perfectly mechanical way of finding these things.

In essence, a lace prism like oxo8o&kt, might be thought as point, octagon, point, lace together by k-lacing. When you pick the thing up, you get the effect of two octahedra, rotated at a point. Of course, if the figures are o6o, o6x and x6x, and the lacing is x, then picking this lot up will make all the bits remain flat. This is picked up in the mechanics of the spreadsheet.

It is interesting that Richard mentions that || is 'parallel'. I always thought of it in terms of the REXX function, which joins two strings together without a space. So to me, it could be read, eg |x|. None the same, if the lacing is the same between subsequent layers, the meaning of || could be 'lacing as the previous level', with a default of 'x'. So, the octagon ball could come at o8o |q| x8o || q8o || x8o || o8o, where the q carries through as a new default. I use the same thing in added fractions.

[wendy]

The polygloss is currently under review, especially with possible rewrites to the notation page, and to several entries. 'Lace *' needs heavy rework. Of course, it has to be kept to line with its original aim: to survive six and eight dimensions without looking silly.

Other entries being reworked are the 4d rotation page (including a thing on how seasons work!). Some of Klitzing's notations will make it into the variations page. (the whole CD thing permits variations: that's it's abiding strength).

The actual source tape for the PG is around 648 kb (45,0000 b), but i have been using a folding editor, and inline marks to make editing easier. The program that builds it checks all of the internal links. :s Once it comes to a sufficient beta stage, the 'as nature intended' version will be updated. When i get it down pat, both bits will get updates.

[quickfur]

One notes that Richard's "cube || ike" is "x4o3o || o4s3s" or xo4os3os&xt. Some calculations become correct with vo5o3ox&xt, (ie v*doe || ike), because resolving 's' is rather hard as a coordinate.

Nice! I had forgotten that o5ox is equivalent to a "snubbed" o4ox.

I like quickfur's abbreviation of x or o, if all positions have the same value. It somehow makes sense.
[...]

Not sure what you're referring to; when I write these CD symbols I usually just adopt the same convention as the graphical version of the CD diagram of omitting the edge label when it's 3. Certainly, x4oxx is easier to type than x4o3x3x, and no ambiguity is introduced; and 3 is probably the most common edge label in the realm of spherical polytopes. Sometimes for simplicial polytopes I include the 3 in x3xoo, just to contrast it with x4xoo, though one could certainly just omit the 3 and write xxoo for x3x3o3o, and x4xoo for x4x3o3o.

On second thoughts, though, perhaps at least one 3 should be included for the simplicial polytopes, then that allows us another direction of abbreviation: a polystratic stack like x4oo||x4ox||x4ox||x4oo could then be simplified by dropping the 4's in the second and subsequent vertex layers, and be written as x4oo||xox||xox||xoo, with the understanding that the subsequent layers inherit the same symmetry group as the first. We may also use the digitless CD string xoxo to refer collectively to that particular ringing of the CD diagram in any symmetry group, so one could make general statements about xoxo polytopes (which includes x3oxo, x4oxo, xo4xo, and x5oxo).

[Klitzing]

One notes that Richard's "cube || ike" is "x4o3o || o4s3s" or xo4os3os&xt. Some calculations become correct with vo5o3ox&xt, (ie v*doe || ike), because resolving 's' is rather hard as a coordinate.

Nice! I had forgotten that o5ox is equivalent to a "snubbed" o4ox.

Not at all!
Ike = s3s3s = o4s3s, i.e. ike is the alternated faceting of the toe (made again uniform).
When Wendy uses a v-scaled doe (i.e. one of edge size 0.618) instead of the (unit sized) cube, then this is done because that cube is vertex inscribable into that doe. Not that the overall shape then would be identical, just things like the calculation of the circumradius etc. might also be done by means of that substitute.

Wendy did mention, that such calculations are rather easily done when dealing with Wythoffian bases, but become difficult for snubbed bases.

I like quickfur's abbreviation of x or o, if all positions have the same value. It somehow makes sense.
[...]

Not sure what you're referring to; when I write these CD symbols I usually just adopt the same convention as the graphical version of the CD diagram of omitting the edge label when it's 3. Certainly, x4oxx is easier to type than x4o3x3x, and no ambiguity is introduced; and 3 is probably the most common edge label in the realm of spherical polytopes. Sometimes for simplicial polytopes I include the 3 in x3xoo, just to contrast it with x4xoo, though one could certainly just omit the 3 and write xxoo for x3x3o3o, and x4xoo for x4x3o3o.

Yep. Your abbreviation thus is rather similar to Jonathan Bowers' one. He too does such a thing, replacing additionally the link marks by symbols too:
none for a link marked 3
, for a link marked 3/2
' for a link marked 4
" for a link marked 4/3
^ for a link marked 5
* for a link marked 5/2
^' for a link marked 5/4
*' for a link marked 5/3

In fact, Wendy was just mis-reading your x3xo (= x3x3o) as an abbreviation for xx3xo(&#x), i.e. reducing stacked node symbols, which are all alike, into one single one. - As you just outlined, this was not intended here at all. It moreover shows that this shouldn't be done either, because of the very issue of misinterpretation!

On second thoughts, though, perhaps at least one 3 should be included for the simplicial polytopes, then that allows us another direction of abbreviation: a polystratic stack like x4oo||x4ox||x4ox||x4oo could then be simplified by dropping the 4's in the second and subsequent vertex layers, and be written as x4oo||xox||xox||xoo, with the understanding that the subsequent layers inherit the same symmetry group as the first. We may also use the digitless CD string xoxo to refer collectively to that particular ringing of the CD diagram in any symmetry group, so one could make general statements about xoxo polytopes (which includes x3oxo, x4oxo, xo4xo, and x5oxo).

This remembers my rather to what George Olshevsky once was after (e.g. in his paper on panoploid tetracombs) just providing the decoration string separately from the mere symmetry string. So we might write "o3o4o:001||101||101||001".

He there uses 0=o and 1=x. - Wendy then might come up again with other shortchord values. Thus it perhaps might serve better then to replace o=2, x=3, q=4, f=5, h=6, k=8. (OTOH v=5/2 would become useless here for obvious reasons.) - Or we simply stick with her characters.

--- rk

[Keiji]

On second thoughts, though, perhaps at least one 3 should be included for the simplicial polytopes, then that allows us another direction of abbreviation: a polystratic stack like x4oo||x4ox||x4ox||x4oo could then be simplified by dropping the 4's in the second and subsequent vertex layers, and be written as x4oo||xox||xox||xoo, with the understanding that the subsequent layers inherit the same symmetry group as the first. We may also use the digitless CD string xoxo to refer collectively to that particular ringing of the CD diagram in any symmetry group, so one could make general statements about xoxo polytopes (which includes x3oxo, x4oxo, xo4xo, and x5oxo).

Of all the alternatives presented, I like this one the best.

[Klitzing]

... I think the main advantage of the || notation is that it's very intuitive and readable for lace prisms and lace towers. I'm not so certain about stretching it to fit lace rings... but maybe ( ... || ... || ...) for rings?

Nice idea, but I would not opt for that. It is just because of the need of parantheses for grouping reasons as well. I.e. you might want to give lace cities too in inline form. Then it might be desirable to give each tower within parantheses, and then aligning those towers (paranthesed things) by some secondary outer stacking.

E.g. the cube-pyramid could be then given as o4o || (x4o || x4o). - Sure in this special example parantheses are needless, this is just an perpendicular simplex (triangle), but even then it might serve for highlighting purposes...

I.e. we might keep that issue of looking for reconnections open, but I would ask not to abuse parantheses for that purpose.

Btw., we might consider my corresponding notion for virtual node symbols (used for revisited real ones) here too:
"*a" equates for revisiting the first real node (e.g. x3o3o3*a represents in inline form a closed loop)
"*b" equates for revisiting the second real node (e.g. x3o3o3/2o3*b represents a loop'n'tail Dynkin diagram), etc.
(Esp. the general 4 node Dynkin symbol cthus can be cut open into xPoQoRoS*aT*c *bU*d.)
So we might introduce here similarily:
"*A" equates for revisiting the first real layer,
"*B" equates for revisiting the second real layer, etc.

OTOH lace rings are not too often encountered. So I rather doubt that those would require some special notation extension at all.

--- rk

[Klitzing]

On second thoughts, though, perhaps at least one 3 should be included for the simplicial polytopes, then that allows us another direction of abbreviation: a polystratic stack like x4oo||x4ox||x4ox||x4oo could then be simplified by dropping the 4's in the second and subsequent vertex layers, and be written as x4oo||xox||xox||xoo, with the understanding that the subsequent layers inherit the same symmetry group as the first. We may also use the digitless CD string xoxo to refer collectively to that particular ringing of the CD diagram in any symmetry group, so one could make general statements about xoxo polytopes (which includes x3oxo, x4oxo, xo4xo, and x5oxo).

Of all the alternatives presented, I like this one the best.

Me not! As this mixes mere decoration strings with true Dynkin symbols. That is, x4oo||xox might well equate to x4o3o||x4o3x (cube atop sirco) or alternatively to x4o3o||x3o3x (cube atop co).

Even x4oo itself might mean x4o3o (neglecting 3's) or x4o4o (neglecting further equal numbers)...

--- rk

[Keiji]

As quickfur said the omission represents a 3, and using the previous symmetry group is only when there are no numbers at all (hence one 3 if they are all 3s).

[quickfur]

On second thoughts, though, perhaps at least one 3 should be included for the simplicial polytopes, then that allows us another direction of abbreviation: a polystratic stack like x4oo||x4ox||x4ox||x4oo could then be simplified by dropping the 4's in the second and subsequent vertex layers, and be written as x4oo||xox||xox||xoo, with the understanding that the subsequent layers inherit the same symmetry group as the first. We may also use the digitless CD string xoxo to refer collectively to that particular ringing of the CD diagram in any symmetry group, so one could make general statements about xoxo polytopes (which includes x3oxo, x4oxo, xo4xo, and x5oxo).

Of all the alternatives presented, I like this one the best.

Me not! As this mixes mere decoration strings with true Dynkin symbols. That is, x4oo||xox might well equate to x4o3o||x4o3x (cube atop sirco) or alternatively to x4o3o||x3o3x (cube atop co).

Even x4oo itself might mean x4o3o (neglecting 3's) or x4o4o (neglecting further equal numbers)...

--- rk

I think you misunderstood my abbreviation; the omitted edge labels are always 3. So x4oo always means x4o3o, never x4o4o. It's only when all numbers are omitted that it inherits the previous symmetry from context, so xoo may refer to x3o3o, x4o3o, x5o3o, x4o4o, etc., but x3oo is always x3o3o, and x4oo is always x4o3o, and x4o4o cannot be abbreviated. The reason for choosing 3 as the candidate for omission is because it occurs most frequently with spherical polytopes, which seems to be the main area of interest here.

Of course, 4 might be a better candidate for omission in the case of Euclidean tilings, where it is a frequent occurrence, but I rather not add confusion and just settle with explicit indication of anything except 3's.

[Klitzing]

As quickfur said the omission represents a 3, and using the previous symmetry group is only when there are no numbers at all (hence one 3 if they are all 3s).

I am fully aware of. Most of us here too. (E.g. except of Wendy, as I pointed out already. )
But that is due to the closeness of the definition right in this context.
I was hinting rather for trapstones in future applications of such a reduction.

My main point here was: too much of reduction does not always bring in clearness, it often brings in confusion! (If not at the writers, so it might occure at the readers!) This we always should be aware of. - I think, we won't loose too much, typing some few more numbers, and getting thereby a much wider spread readability.

(As far we would not switch to a completely different system, e.g. like Jonathan's, which I was mentioning already. That one kind of is closer related to Dynkin's previous lattice notation (using line types) rather than Coxeters take over to polytopes (introducing numerical line marks).)

--- rk

[quickfur]

One notes that Richard's "cube || ike" is "x4o3o || o4s3s" or xo4os3os&xt. Some calculations become correct with vo5o3ox&xt, (ie v*doe || ike), because resolving 's' is rather hard as a coordinate.

Nice! I had forgotten that o5ox is equivalent to a "snubbed" o4ox.

Not at all!
Ike = s3s3s = o4s3s, i.e. ike is the alternated faceting of the toe (made again uniform).

Sorry, I was unclear, by "snubbed" I meant the process of dividing edges in the golden ratio, which is the canonical construction of o5o3x's coordinates. In terms of alternation, you're right that it's the (topological) alternation of o4s3s, which, starting from a uniform o4x3x, would be non-regular. However, an appropriately rescaled truncated octahedron, say o4i3j for suitable values of i and j, would allow the alternation to be the regular o5o3x.

Anyway, here's an area where the notation is a bit lacking: how does one indicate unequal shortchords of an alternated CD symbol? AFAIK, 's' implicitly assumes a length of 'x', whereas here we need to combine 's' with different lengths. Perhaps o4si3sj? Looks rather ugly, though.

[...]
Not sure what you're referring to; when I write these CD symbols I usually just adopt the same convention as the graphical version of the CD diagram of omitting the edge label when it's 3. Certainly, x4oxx is easier to type than x4o3x3x, and no ambiguity is introduced; and 3 is probably the most common edge label in the realm of spherical polytopes. Sometimes for simplicial polytopes I include the 3 in x3xoo, just to contrast it with x4xoo, though one could certainly just omit the 3 and write xxoo for x3x3o3o, and x4xoo for x4x3o3o.

Yep. Your abbreviation thus is rather similar to Jonathan Bowers' one. He too does such a thing, replacing additionally the link marks by symbols too:
none for a link marked 3
, for a link marked 3/2
' for a link marked 4
" for a link marked 4/3
^ for a link marked 5
* for a link marked 5/2
^' for a link marked 5/4
*' for a link marked 5/3

In fact, Wendy was just mis-reading your x3xo (= x3x3o) as an abbreviation for xx3xo(&#x), i.e. reducing stacked node symbols, which are all alike, into one single one. - As you just outlined, this was not intended here at all. It moreover shows that this shouldn't be done either, because of the very issue of misinterpretation!

I think rewriting every possible edge label with a punctuation symbol is a bit excessive; why not just indicate the label directly? It adds nothing but obscurity.

What would be truly useful, though, would be a way to indicate a wildcard label that may be substituted with suitable values. If we use '.' as a wildcard, for example, then x.x3x can refer to x3x3x, x4x3x, or x5x3x, etc.. This affords us another way of referring to CD decoration independently of the underlying symmetry group; 4D omnitruncates, e.g., can be referred to categorically by writing x.x.x.x.

A wildcard node decoration would be useful too; suppose we use '*' to represent o, x, f, ... or whatever other lengths we may use; then we could write *3*4*3* to categorically refer to any 24-cell truncate, or x3*4*3* for any of the foregoing that has a marked first node. The two kinds of wildcards can be combined, so one may write *.*3*3o to refer to any polytope whose symmetry group matches the pattern {x,3,3} (where x=3, 4, 5, ...) and whose last node is unmarked.

I think this is a more useful way of making use of punctuation symbols, rather than just rewriting numerical labels.

On second thoughts, though, perhaps at least one 3 should be included for the simplicial polytopes, then that allows us another direction of abbreviation: a polystratic stack like x4oo||x4ox||x4ox||x4oo could then be simplified by dropping the 4's in the second and subsequent vertex layers, and be written as x4oo||xox||xox||xoo, with the understanding that the subsequent layers inherit the same symmetry group as the first. We may also use the digitless CD string xoxo to refer collectively to that particular ringing of the CD diagram in any symmetry group, so one could make general statements about xoxo polytopes (which includes x3oxo, x4oxo, xo4xo, and x5oxo).

This remembers my rather to what George Olshevsky once was after (e.g. in his paper on panoploid tetracombs) just providing the decoration string separately from the mere symmetry string. So we might write "o3o4o:001||101||101||001".

He there uses 0=o and 1=x. - Wendy then might come up again with other shortchord values. Thus it perhaps might serve better then to replace o=2, x=3, q=4, f=5, h=6, k=8. (OTOH v=5/2 would become useless here for obvious reasons.) - Or we simply stick with her characters.

--- rk

Olshevsky's notation above is encompassed by the two-wildcard scheme suggested above: *3*4*:o.o.x||x.o.x||x.o.x||o.o.x.

The first part gives the symmetry group without specifying node decoration (*3*4*); and the second part gives node decorations without specifying symmetry group (o.o.x, etc.). Putting the two together expresses the same thing as Olshevsky's notation. Of course, it can express much more; it basically lets one do pattern-matching on CD diagrams.

[quickfur]

As quickfur said the omission represents a 3, and using the previous symmetry group is only when there are no numbers at all (hence one 3 if they are all 3s).

I am fully aware of. Most of us here too. (E.g. except of Wendy, as I pointed out already. )
But that is due to the closeness of the definition right in this context.
I was hinting rather for trapstones in future applications of such a reduction.

My main point here was: too much of reduction does not always bring in clearness, it often brings in confusion! (If not at the writers, so it might occure at the readers!) This we always should be aware of. - I think, we won't loose too much, typing some few more numbers, and getting thereby a much wider spread readability.

(As far we would not switch to a completely different system, e.g. like Jonathan's, which I was mentioning already. That one kind of is closer related to Dynkin's previous lattice notation (using line types) rather than Coxeters take over to polytopes (introducing numerical line marks).)

--- rk

You're right, maybe it's worth the extra effort to type a few numbers to increase comprehensibility in the long run.

As for line types vs. numerical labels, I think numerical labels are more flexible. There are only so many line types to choose from, but numerical labels lets you write something that the authors of the notation may not have conceived before, e.g., x9o for a nonagonal hyperbolic tiling. I wouldn't want to use up 9 symbols just to be able to have enough line types to represent this, esp. since it's not likely to come up repeatedly!

By not using up punctuation symbols unnecessarily, we can save them for more useful purposes like wildcards that I mentioned, which also alleviates the need for abbreviation: one could just write x.o.o to refer to x4o3o if the symmetry group *4*3* is clear from context. It's easier to type and easier to read, too (at least to me ).

[Klitzing]

But getting back from meta-mathematics to more interesting polytopes:

Just reconsidered spid = x3o3o3x. Its lace city is

Code: Select all

   x3o   o3o   
              
x3o   x3x   o3x
              
   o3o   o3x   


This one shows, that we not only have its rotunda being tet || co, but also a fat and a narrow luna. Both are known segmentochora: tet || tricu (with lace city:

Code: Select all

x3o   o3o   
            
   x3x   o3x


), resp. {6} || trip (with lace city:

Code: Select all

   x3o   
        
x3o   x3x


).

So far nothing new. Then I calculated the dihedral angle of those wedge facets (tricues), being arccos(-1/4) resp. arccos(1/4). Thus it turns out, that this wedge angle is not generally a rational fraction (with respect to the full circle) but rather can be a real number (within the limits 0 resp. 1/2 for CRFs). - That idea of rational fraction just was implied by all the so far considered lunes. - But here we'd get rather an 0.290215-lune and an 0.209785-lune instead!

--- rk

[quickfur]

But getting back from meta-mathematics to more interesting polytopes:

Just reconsidered spid = x3o3o3x. Its lace city is

Code: Select all

   x3o   o3o   
              
x3o   x3x   o3x
              
   o3o   o3x   


This one shows, that we not only have its rotunda being tet || co, but also a fat and a narrow luna. Both are known segmentochora: tet || tricu (with lace city:

Code: Select all

x3o   o3o   
            
   x3x   o3x


), resp. {6} || trip (with lace city:

Code: Select all

   x3o   
        
x3o   x3x


).

So far nothing new. Then I calculated the dihedral angle of those wedge facets (tricues), being arccos(-1/4) resp. arccos(1/4). Thus it turns out, that this wedge angle is not generally a rational fraction (with respect to the full circle) but rather can be a real number (within the limits 0 resp. 1/2 for CRFs). - That idea of rational fraction just was implied by all the so far considered lunes. - But here we'd get rather an 0.290215-lune and an 0.209785-lune instead!

--- rk

Interesting!! Can these luna be reassembled in the "wrong" order to produce new CRFs?

[quickfur]

[...]
Interesting!! Can these luna be reassembled in the "wrong" order to produce new CRFs?

After thinking about this at the back of my mind through the whole evening, I'm almost certain that reassembling these luna in the "wrong" order produces the gyrated x3o3o3x (i.e. cut x3o3o3x in half, rotate one piece to orientation of dual simplex, and glue them back). That is, if we call the two types of lunae A and B, then A-B-A-B reconstitutes the x3o3o3x, and A-B-B-A will produce a gyrated x3o3o3x. In the latter construction, there is no actual gyration; but due to the underlying symmetry group, reversing A-B to B-A coincides with rotating A-B into dual orientation. This is analogous to how cutting the icosidodecahedron o5x3o into 4 lunae and gluing them back in the "wrong" order produces a pentagonal orthobirotunda (i.e. gyrated icosidodecahedron), because reversing the order of the two fragments of the pentagonal rotunda coincides with rotating it by 72°, due to the underlying symmetry.

I'm still curious though: are there any uniform polychora that can be cut into lunae (not necessarily CRF) and glued back in the "wrong" order to produce CRFs that aren't trivially derived by another manner (e.g. simple rotation)?

//

On a related note, I've been thinking again about finding a 4D analogue for the snub disphenoid. One particularly suggestive construction for the 3D snub disphenoid is to take two octahedra, cut off 1 quadrant from each (i.e. two triangular faces), then squeeze them along the axis perpendicular to the quadrant so that the remaining triangular net "fans out" a little bit, such that the skew polygon of the cut is equal to itself rotated by 90°. Then the two pieces can be glued together along this skew polygon to form the snub disphenoid. Unfortunately, the direct analogue of this procedure does not work on the 16-cell o4o3o3x; the skew polyhedron that results from cutting off a quadrant cannot be made equal to itself via any rotation/squeezing, so two 16-cells cut this way can never join up to form a closed shape.

However, this impasse may be surmounted if we insert additional cells between the two 3/4 16-cells; maybe some tetrahedra or square pyramids or maybe triangular prisms to bridge the gaps, so that a closed CRF shape would be formed. Based on the Blind couple's results (as pointed out by Klitzing), tetrahedra alone will not work, because all CRFs with only tetrahedral cells have been enumerated, and none of them correspond with a 4D snub disphenoid. But since our definition of CRF is more permissive, perhaps some combination of square pyramids or triangular prisms may work to close up the gaps between the 3/4 16-cells to make a CRF.