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 Naming Polytopes With Sdn, was: Multiplicative SDN prefixes
Double sharp
Posted: Nov 12 2017, 03:39 AM


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{c}

Yesterday it occurred to me that the multiplicative prefixes in the names of the Catalan solids (the duals of the Archimedean solids) are a pretty rare occurrence of those in traditional terminology, and would provide a good introductory illustration of their use in SDN.

The tetrahedrals and octahedrals work wonderfully:

triakistetrahedron = trinatetrahedron (3*4=10) [sic; my mistake, should be trinaquadrahedron]
rhombic dodecahedron = rhombic unquahedron
triakisoctahedron = trinanoctahedron (3*8=20)
tetrakishexahedron = quadrahexahedron (4*6=20)
deltoidal icositetrahedron = deltoidal binunquahedron
disdyakisdodecahedron, hexakisoctahedron = binabinunquahedron (2*2*10), hexanoctahedron (6*8=40)
pentagonal icositetrahedron = pentagonal binunquahedron

But there is a little problem with the icosahedrals, being that we see both "pentakisdodeca-" and "hexeconta-" as prefixes: the first means 5*10 (five faces corresponding to one of the regular dodecahedron), and the second just means 50. In the obvious SDN transform these both become pentanunqua- and the distinction is lost. (So kind of like the traditional additive distinction between triacontakaidigon and triacontadigon.)

Perhaps we might then suggest that if there is any distinction between multiplication and a simple number, like in these names, or in base names (base 5-on-10 or pure base 50?), then we should instead use the nil forms. This means that we might have to edit a few of the octahedral Catalan names:

icositetrahedron = binilihedron

And then we can finish up the job:

rhombic triacontahedron = rhombic bihexahedron
triakisicosahedron = trinanunoctahedron (3*18=50)
pentakisdodecahedron = pentanunquahedron (5*10=50)
deltoidal hexecontahedron = deltoidal pentnilihedron
disdyakistriacontahedron, hexakisicosahedron = binabinabihexahedron (2*2*26=a0), hexanunoctahedron (6*18=a0)
pentagonal hexecontahedron = pentagonal pentnilihedron

I think I much prefer "hexanoctahedron" and "hexanunoctahedron" to the other names with the repeated "bina". (It comes from the fact that the faces being atopped with pyramids are not regular; indeed they are the rhombi of the rhombic unquahedron and rhombic bihexahedron.)

I guess this would work, except that I am not sure which form feels more natural in dozenal for a number like 50, which is a multiple of the dozen. I don't think I feel the multiplicativeness inherent in it much more strongly than in "hexeconta-" in decimal; it fades into the background as it is inherent in the base. So the "pentnili-" forms seem slightly odd to me, as though they are telling me something, and only then is the multiplicativeness of "pentanunqua-" revealed. But I can't very well reverse things because the multiplicativeness is at least more present in "pentanunqua-" than in "pentnili-".

So I'm not sure what best to do here, even though this system seems to work, to distinguish a "five-times-dozen-hedron" from a "five-dozen-hedron" in SDN.

OP's edit: I changed the original title of this thread to better match what its topic ended up moving to. The original was: "Multiplicative SDN prefixes: Five dozen or five times dozen?".
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Kodegadulo
Posted: Nov 13 2017, 01:49 AM


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The fact is, every prefix in SDN constitutes a multiplicative factor modifying whatever noun it is attached to. That holds regardless of whether we are talking about a simple multiplier prefix like bina·, or bihexa·, or penta·, or even pentnili·; or a power prefix like unqua· or uncia·; or even a composite of both, like penta·unqua· (with or without an audible -n- in between).

That is why, in my latest description of the system, I have elected to terminate each of these prefixes with a middle dot (·) character (Unicode U+00B7x), just as you can see in the preceding paragraph. Each of these middle dots can literally be interpreted as multiplication operators, even though you can leave them silent when pronouncing the final construct.

Now, the way I interpret it, pentnili· and penta·unqua· both mean exactly the same thing: "fivzy" (five dozen, 50z) ... times whatever. The simple multiplier pentnili· indicates "fivzy" by its digits: a five (pent) in the dozens' place and zero (nil) in the ones' place. The composite penta·unqua· indicates "fivzy" by multiplying five (penta·) times dozen-to-the-first-power (unqua·). But this just means exactly the same thing, really, and the choice between the two forms becomes nothing more than a matter of taste.

But for these "-akis" polyhedra, I think there is a bit more than just multiplication going on here. Note that in each case, the traditional name of such a beast is a two-word phrase: an "\(n\)-akis" modifier word followed by a "\(p\)-hedron" base-polytope name, e.g. "triakis tetrahedron". This indicates how each of the \(p\) faces of the base polyhedron get replaced with a pyramid of \(n\) new polygonal faces. Yes, it's true that the total number of faces in the result is \(n \times p\), but this means more than just the fact that the final number can be expressed as a multiple of some numeric base. It means that we still have an intimate relationship to that underlying \(p\)-hedron, but with an \(n\)-pyramidalization, so to speak, on each of its faces.

So, even if we decide to use SDN, rather than traditional decimal Latin or Greek, to regularize our nomenclature for polytopes, I see no reason why we wouldn't simply retain this "-akis" construct as part of the morphology of the terms for these special types of "pyramidalized" polyhedra:

Traditional (Decimal) NameSystematic Dozenal Name
triakis tetrahedrontrina·kis quadra·hedron
rhombic dodecahedronrhombic unqua·hedron
rhombic unnili·hedron
triakis octahedrontrina·kis octa·hedron
tetrakis hexahedronquadra·kis hexa·hedron
deltoidal icositetrahedrondeltoidal bina·unqua·hedron
deltoidal binili·hedron
disdyakis dodecahedron

hexakis octahedron
bina·bina·kis unqua·hedron
bina·bina·kis unnili·hedron
hexa·kis octa·hedron
rhombic triacontahedronrhombic bihexa·hedron
triakis icosahedrontrina·kis unocta·hedron
pentakis dodecahedronpenta·kis unqua·hedron
penta·kis unnili·hedron
deltoidal hexecontahedrondeltoidal penta·unqua·hedron
deltoidal pentnili·hedron
disdyakis triacontahedron
hexakis icosahedron
bina·bina·kis bihexa·hedron
hexa·kis unocta·hedron
pentagonal hexecontahedronpenta·gonal penta·unqua·hedron
penta·gonal pentnili·hedron


In short, SDN is a fine system that gets a lot of mileage out of treating concatenation as multiplication. But let's not take this multiplicative hammer and engage in "hammer syndrome", where we try to reduce everything we do into multiplicative nails...
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Double sharp
Posted: Nov 13 2017, 02:37 AM


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{c}

Well, concatenation as multiplication is certainly most people's gut impression, but just like the concatenation means something different in "2x" and "23", so SDN replicates this: in SDN, concatenation means place value when used with digits (e.g. bihexahedron), but multiplication with the special prefixes (e.g. hexanoctahedron). Your proposal to retain "kis" with Conway's meaning makes a good deal of sense and I shall be adopting it!

Now how would one use SDN for additive constructions, like the Archimedean solids? Consider for example the icosidodecahedron, t_1{5,3}. This has twenty triangular faces derived from the icosahedron and twelve pentagonal faces derived from the dodecahedron. What do we do about that? "Unocta-unquahedron" sounds like a 180-hedron, and "unqua-unoctahedron" sounds similarly as if "unqua-" was being applied as a power prefix; the "unnili-" forms have the same problem. Does that mean we need one more glue syllable to mean addition instead of multiplication or place value, or have I just missed an obvious solution?
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Oschkar
Posted: Nov 13 2017, 03:02 AM


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SDN already has a syllable to indicate addition: -et-, so this can be called an unqua-et-unoctahedron.
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Kodegadulo
Posted: Nov 13 2017, 03:05 AM


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Yes, concatenation of digit root syllables within an SDN prefix signifies positional-place-value, whereas concatenation of full SDN prefixes (each terminated with a particular glue syllable) signifies multiplication. One reason I've adopted this middle dot spelling is to help readers perceive the multiplication between prefixes, but yet perceive the digit root sequence within each prefix as a coherent blob.

I did at one point suggest the Latin et as an additive glue syllable (perhaps set off by hyphens rather than middle dots). I think it was in the context of attempting to emulate the style of numerals in natural language, but doing so doesn't seem to be as fruitful as simply sticking to concatenated digit-roots and place-value.

However, et might come in handy for cases like these Archimedean solids. For instance, the icosidodecahedron could be SDN-ified as the unocta-et-unnili·hedron, or the unocta-et-unqua·hedron. In other words, it's a combination of an eightzeen-sided (one-dozen-eight-sided) polyhedron and a dozen-sided polyhedron.

Of course, given that addition is commutative, this could just as easily be the unnili-et-unocta·hedron, or the unqua-et-unocta·hedron.

(EDIT: Oops, cross-posted with Oschkar!) smile.gif
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Double sharp
Posted: Nov 13 2017, 03:38 AM


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{c}

All right, that solves the problem, I think! happy.gif So I was right that another syllable was needed, but didn't realise that we already had it.

Now let's consider regular tilings. It seems to me that the concatenation in the traditional name "trihexagonal tiling" for r_1{6,3} means something different again: it means that there are trigonal faces (from the {3,6} triangular tiling) and hexagonal faces (from the {6,3} hexagonal tiling). We can't port this name directly to SDN because it would mean a tiling of 36-gons. So what to do? I'm not sure if "trina-et-hexagonal tiling" works, because "trina-et-hexagon" seems to mean a "three-plus-six-gon".

Also, how do we name the hyperbolic regulars? Wikipedia names {3,7} the "order-7 triangular tiling": I'm still thinking of a good SDN-ification. Then things get worse when the cells of the tiling are themselves tilings: is there something better for {6,3,3} than "order-3 hexagonal tiling tiling"? Most authors I've seen just throw up their hands and use the Schläfli symbols as names, but I wonder if we can do better.

How do we name infinite-sided polygons? (These make sense in hyperbolic geometry where they can be inscribed in horocycles or hypercycles. They also make sense in Euclidean geometry, but look rather degenerate, with their interiors being half-planes, kind of like the hemispherical polygons in the hemipolyhedra.) Do we just port into SDN the traditional Greek prefix apeiro?
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wendy.krieger
Posted: Nov 13 2017, 08:57 AM


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I prefer a different structure of names to the catalan figures.

Conway's names only work in 3 dimensions, but the notation works in every dimension. Thus i have revamped the names along the lines of Johnson, Kepler, et al.

The raising of peaks on a face is 'apiculation'. This is bi-, tri-, &c, according to the draught-dimension of the tip, viz the dimension of the simplex with those vertices. In this way, the x-apiculated figure is the dual of the x-truncated dual figure.

The n-apiculation ends in the n-surtegmate (the correct form of 'rhombic') and then procedes onto the (n+1)-apiculate. The zero-apiculate and zero-surtegmate are understood and have a clear meaning.

kis is cognate to, and means, the same as -ce in once, twice, thrice. The sense in kepler is that a trikis octahedron is a three-times octahedron. Conway converted this into an operator 'k'. But conway's names are really not suited to anything other than 3d, whereas mine are.

"apeiron" is a greek word, used for an expanse of desert or sea, the borderlessness of it. It comprises of -a- without, + peri aroundness. It is without aroundness. In my pg, it is used for tilings, since the periphery or perimeter of a figure, is taken as the limit of extent in the space where the figure is solid. It can be used of say, the square faces on a cube. A tiling has no boundary where the elements cease to be solid in that space. Thus it is without enclosure, or an 'aperion'.

Tilings in some spaces are apeiron (with a limit), but still finite. You can also have infinite polytopes that are polytope-like, such as a recent discussion on some {5,3,3} that is closed, with right-angles margin-angles. The symmetry of polytopes like {8,3} with right angles, or {8,4,A}, have one or more infinite cells, which are more polytope than tiling.



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wendy.krieger
Posted: Nov 13 2017, 09:03 AM


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If you mean to describe words for conjunction of numbers, there are a number of useful words to consider.

1. A sum of different things, such as twenty + twelve (icosa-dodecahedron),

2. A stem for 'in x ways', as 'the cube is a square prism in three ways'

3. A stem for 'for x times'. 'One repeats this process three times'.

4. apposition of number and property, 'twenty eight-pound guns.'

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Kodegadulo
Posted: Nov 13 2017, 12:48 PM


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Some truisms for ya, Wendy:

(1) If it ain't broke, don't fix it.

(2) If you think you got a better way of doing something, don't just assert that your way is better and all other ways are broken. Prove it. At the very least, describe it, in something more generally intelligible than your own pet idioms. Otherwise, all you're doing is tooting your own horn again.

But if you're going to do (2), try doing it in your own forum.
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Kodegadulo
Posted: Nov 13 2017, 01:04 PM


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QUOTE (Double sharp @ Nov 13 2017, 03:38 AM)
Now let's consider regular tilings. It seems to me that the concatenation in the traditional name "trihexagonal tiling" for r_1{6,3} means something different again: it means that there are trigonal faces (from the {3,6} triangular tiling) and hexagonal faces (from the {6,3} hexagonal tiling). We can't port this name directly to SDN because it would mean a tiling of 36-gons. So what to do? I'm not sure if "trina-et-hexagonal tiling" works, because "trina-et-hexagon" seems to mean a "three-plus-six-gon".

Okay but realize the nesting involved here. The form "trina-et-hexa·gonal = (3+6)·gonal = a three-and-six=nine cornered shape. On the other hand, a "trina·gonal / hexa·gonal tiling" = (3·gonal) + (6·gonal) tiling = a tiling of interleaved 3-cornered and 6-cornered shapes.

QUOTE
Also, how do we name the hyperbolic regulars? ...

Gotta run for now but will pick this up again this evening.
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Kodegadulo
Posted: Nov 13 2017, 03:48 PM


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QUOTE (Double sharp @ Nov 13 2017, 03:38 AM)
Also, how do we name the hyperbolic regulars? Wikipedia names {3,7} the "order-7 triangular tiling": I'm still thinking of a good SDN-ification. Then things get worse when the cells of the tiling are themselves tilings: is there something better for {6,3,3} than "order-3 hexagonal tiling tiling"? Most authors I've seen just throw up their hands and use the Schläfli symbols as names, but I wonder if we can do better.

To some extent, all of these polytope naming questions go beyond the scope of a "systematic numeric nomenclature", because an SNN really only deals with numbers. One needs to introduce other nomenclature to express ideas beyond pure number, be it quantitels and so forth for physical quantities, or in this case suffixes like -gon, -hedron, -choron, etc to signify polytopes of different dimensionality. Double sharp, it seems to me your difficulties are with how this latter sort of nomenclature is structured. There's nothing particularly difficult about inserting SDN into the slots where pure numbers can be fitted, the trouble seems to be with where these latter nomenclatures have placed such slots.

One easy avenue is to note that any SNN prefix can easily be formed into an ordinal, or more generally into an adjectival word, by terminating it with the suffix -al. So something like a "{6,3,3} order tiling" could simply be rendered as a "hexal-trinal-trinal-order tiling".

Now, if you want to read that backwards (or perhaps from top to bottom) as a "trinal-order tiling of trinal-order tilings of hexal-order polygons", that's perfectly feasible. You might even get away with truncating it to "trinal tiling of trinal tilings of hexagons". But if you're looking for something more compact than that, or perhaps more evocative and less sterile, that's more of a question for the wordsmiths of geometric/topological terminology than for SDN.
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Double sharp
Posted: Nov 13 2017, 03:56 PM


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"Hexal-trinal-trinal tiling" sounds quite fine, thank you! Certainly better than just resignedly saying "Schläfli 6, 3, 3". I guess I'll also say "trinagonal-hexagonal tiling" for r{6,3} in SDN.

I agree that the hyperbolic regular naming scheme is kind of a vaguely different question, but it came up in the same train of thought, so I decided to just throw it out there to. I guess we just need a canonical "infinite" marker for SDN to complete the whole set. "Apeiral"?
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Kodegadulo
Posted: Nov 14 2017, 12:19 AM


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QUOTE (Double sharp @ Nov 13 2017, 03:56 PM)
I guess we just need a canonical "infinite" marker for SDN to complete the whole set. "Apeiral"?

How is infinity a dozenal number? Or even a number in any base? Or even a number at all? This is again something that is not really in the scope of an SNN. It seems to me the prerogative of what to call a polytope with infinite facets belongs to those who have already thought on this. If the consensus of the geometry field is to call it an "apeirotope", then I see no reason to diverge from that.
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Double sharp
Posted: Nov 14 2017, 02:50 AM


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It kind of can be a number when you want it to, but mostly, if we're going to read {6, 3, 3} as a hexal-trinal-trinal tiling, I would want to a way to read {infinity, 3, 3} similarly. I suppose the best option is probably to call it an apeiral-trinal-trinal tiling, for consistency, going on the already consensus "apeirogon" and "apeirohedron", using "apeiro-" as a fake SDN prefix.

Anyway, the main purpose of this was to get rid of some of the collisions that seemed to result when directly porting the standard polytope names to SDN (and as a sideline to be able to say some of them clearly without just saying the Schläfli symbol number by number), and I think all the collisions have been solved by now in 3D.

More on topic, it seems to me that the ugly names like "hexa-5-tope" for higher than 4D, where there are no suffixes, don't work if you want to use SDN to read the number in "5-tope, 6-tope, ...", and so on. This is kind of the same problem as our old idea of "unciabimagnitudel" that got fixed by "unciamagnibinal", getting the numbers away from each other. So..."hexatopepental"? "Septatopehexal?" "Octatopeseptal?".

Seems to me that there's an inconsistency between what the prefixed numerals sometimes mean. An 8-gon has 8 facets, but an 8-cube has 8 dimensions. So a 8-polytope with 256 facets would seem to be a 256-8-tope, which sounds stupid. I think the reason this doesn't come up very much is that the usual names for the regular 8-polytopes are "8-simplex, 8-cube, 8-orthoplex".

Well, I'm not sure this can be helped. I guess it may seem a bit deflated after being able to use SDN to say things like "unennpentagon" quickly and obviously, rather than creating a long, incomprehensible Greek prefix for 257, but I guess we could just say (when we needed it) that we had an "undecoctafacetted octatope", which gets the point across.

P.S. I like the pun "diverge from that"... wink.gif
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Kodegadulo
Posted: Nov 14 2017, 03:12 AM


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Here's an idea:

Common Name SDN Name Schläfli Symbol Schläfli SDN Description
triangle trina·gon {3} trina·gonal bina·tope
square quadra·gon {4} quadra·gonal bina·tope
pentagon penta·gon {5} penta·gonal bina·tope
hexagon hexa·gon {5} hexa·gonal bina·tope
...
tetrahedron quadra·hedron {3,3} trina·gonal trina·hedronal trina·tope
hexahedron hexa·hedron {4,3} quadra·gonal trina·hedronal trina·tope
octahedron octa·hedron {3,4} trina·gonal quadra·hedronal trina·tope
dodecahedron unqua·hedron
unnili·hedron
{5,3} penta·gonal trina·hedronal trina·tope
icosahedron unocta·hedron {3,5} trina·gonal penta·hedronal trina·tope
triangular tiling trina·gonal tiling {3,6} trina·gonal hexa·hedronal trina·tiling
hexagonal tiling hexa·gonal tiling {6,3} hexa·gonal trina·hedronal trina·tiling
square tiling quadra·gonal tiling {4,4} quadra·gonal quadra·hedronal trina·tiling
...
pentachoron penta·choron {3,3,3} trina·gonal trina·hedronal trina·choronal quadra·tope
octachoron octa·choron {4,3,3} quadra·gonal trina·hedronal trina·choronal quadra·tope
hexadecachoron unquadra·choron {3,3,4} trina·gonal trina·hedronal quadra·choronal quadra·tope
icosatetrachoron binili·choron
bina·unqua·choron
{3,4,3} trina·gonal quadra·hedronal trina·choronal quadra·tope
hecatonicosachoron decnili·choron
deca·unqua·choron
{5,3,3} penta·gonal trina·hedronal trina·choronal quadra·tope
hexacosichoron quadbinili·choron
quadbina·unqua·choron
{3,3,5} trina·gonal trina·hedronal penta·choronal quadra·tope


Get it? The "-hedronal" adjectives don't indicate the number of polygons in the polytope, but rather the number of polygons that meet at each vertex. The "-choronal" adjectives don't indicate the number of polyhedra in the polytope, but rather the number of polyhedra that meet at an edge. The "-tope" and "-tiling" nouns don't indicate the number of facets in the shape but rather the dimensions of the space it is embedded in. The Schläfli symbol, and its SDN-ization, are descriptions of the characteristics of the shape, but neither is really its name.

Does that work?

Edit: Hmm, cross-posted with Double sharp this time. As to how to deal with higher than four dimensions, we would need to extend the sequence -gon, -hedron, -choron ... But I really despise the bastardized terms -teron, -peton, -exon, etc. I'd rather use something more transparent, let's say, -quadron, -penton, -hexon, -septon, -octon, etc. In fact, -unon, -binon, -trinon could be synonyms for -gon, -hedron, -choron. A nili·tope, or nilion, would be a zero-dimensional vertex.

So, for example, a "6-simplex" or "septa-6-tope" would simply be a "simplex hexa·tope" or "septa·penton", and its Schläfli symbol {3,3,3,3,3} would indicate that it was a "trina·gonal trina·hedronal trina·choronal trina·quadronal, trina·pentonal hexa·tope".

Common Name SDN Name Schläfli Symbol Schläfli SDN Description
6-simplex
septa-6-tope
simplex hexa·tope
septa·penton
{3,3,3,3,3} trina·gonal trina·hedronal trina·choronal trina·quadronal trina·pentonal hexa·tope
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Kodegadulo
Posted: Nov 14 2017, 05:02 AM


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Looking at this in more detail:

QUOTE (Double sharp @ Nov 14 2017, 02:50 AM)
More on topic, it seems to me that the ugly names like "hexa-5-tope" ...

Make that a hexa·faceted penta·tope, where the facets are quadrons -- a.k.a. a hexa·quadron.

QUOTE
... for higher than 4D, where there are no suffixes, don't work if you want to use SDN to read the number in "5-tope, 6-tope, ...", and so on.

Actually, I don't see a problem with calling those penta·tope, hexa·tope, septa·tope, etc. And I don't see a problem with rendering those into suffixes penton, hexon, septon, etc.

QUOTE
This is kind of the same problem as our old idea of "unciabimagnitudel" that got fixed by "unciamagnibinal", getting the numbers away from each other. So..."hexatopepental"? "Septatopehexal?" "Octatopeseptal?".

I don't think this issue actually applies here. First, I'd rather say the following:

"a trina·unon (or trina·gon) is a simplex bina·tope"
"a quadra·binon (or quadra·hedron) is a simplex trina·tope"
"a penta·trinon (or penta·choron) is a simplex quadra·tope"
"a hexa·quadron is a simplex penta·tope"
"a septa·penton is a simplex hexa·tope"
"an octa·hexon is a simplex septa·tope"
"an ennea·septon is a simplex octa·tope"
...
"an unbina·unnilion is a simplex ununi·tope"
... etc.

I don't see any of these as being a "collision" because in each case the prefix is clearly a multiplier indicating how many facets, while the suffix indicates the dimensionality of the facets, with the multiplier glue syllable clearly demarked.

QUOTE
Seems to me that there's an inconsistency between what the prefixed numerals sometimes mean. An 8-gon has 8 facets, but an 8-cube has 8 dimensions. So a 8-polytope with 256 facets would seem to be a 256-8-tope, which sounds stupid.

I would call this a "hypercube octa·tope", or an "unennquadra·septon".

QUOTE
I think the reason this doesn't come up very much is that the usual names for the regular 8-polytopes are "8-simplex, 8-cube, 8-orthoplex".

I think we can call these "regular octa·topes", and specifically the "simplex octa·tope", "hypercube octa·tope", and "orthoplex octa·tope".

QUOTE
Well, I'm not sure this can be helped. I guess it may seem a bit deflated after being able to use SDN to say things like "unennpentagon" quickly and obviously, rather than creating a long, incomprehensible Greek prefix for 257, but I guess we could just say (when we needed it) that we had an "undecoctafacetted octatope", which gets the point across.

That works, but it could also be the undecocta·septon. Or did you mean the unennpenta·septon?
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Double sharp
Posted: Nov 14 2017, 06:01 AM


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I don't remember what numbers I was thinking of (probably 14 + 194 = 1a8 for the midpoint of the truncation series involving the hypercube octatope and orthoplex octatope), but I think your approach (as usual) solves the problem with elegant simplicity and clarity. Where I was getting stuck is that "pentatope" and "hexatope" sound like multiplicative prefixes, except that (of course) the name "hexa-5-tope" has a dimensional problem; the 5-simplex is 5D, but its facets are 4D, so it should of course be a hexaquadron. So we could have "nilion, unon = gon, binon = hedron, trinon = choron, quadron, penton, hexon, septon, octon, enneon, decon, levon, unnilion (unquon?), ununon", and so forth.

BTW, the n-cube has 2n facets, two for each axis. The n-orthoplex has 2^n facets, one for each (whatever the higher-dimensional equivalent of "quadrant" and "octant" is; is there even a word?).

I guess "gon" really refers to the corners, because it comes from Greek γωνία with that meaning. Hence we have the term "isogonal", which confusingly means "vertex-transitive". "Edge-transitive" is more standardly "isotoxal" (from τοξον "arc"), and only from then on do we have "isohedral" and "isochoral". So perhaps using "gon" to mean "unon" is a little fraught once we get past polygons. "Toxon" might work - hence a cube is a "quadratoxal trinahedronal trinatope". It is more opaque, but more precise, and does at least harmonise with standard terminology.
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wendy.krieger
Posted: Nov 14 2017, 08:58 AM


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The regular measure and cross polytopes go by the name of prismaterid, and tegmaterid, in 4d. prismpetid and tegmapetid in 5d, and so forth. Calling a polytope a 216-zetton in 8d, is rather a hard way around. Instead, this is a "tegmayottid". The sphere is a glomoyottid (for the ball), or glomozettix (for the surface). The simplex is given as a "trachmayottid" (drawn-8d-solid).

The roots are tegma (tegum-product) yott (8d) id (solid).

Names for the remaining nine classes of root polytopes is still in debate.
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Kodegadulo
Posted: Nov 14 2017, 04:16 PM


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Deriving higher polytope names from SI prefixes is incredibly obtuse.

tera- > teron
peta- > peton
exa- > exon
zetta- > zetton
yotta- > yotton

Aside from these all being unnecessarily abominable and opaque mutations of classical numeric roots, the worst part is that they are simply not scalable. You really want to stop at just eight dimensions?And then wait how many decades for some turgid international committee in Brussels or Geneva or Copenhagen or wherever, just to come up with something equally laughable for the ninth power of thousand? Fecc, string theory and M-theory already talk about symmetries and dualities in eleven dimensions. Where are you going to get nomenclature for that?

Whereas it's simplicity itself to use SDN to talk about, say, a levadimensional simplex, or a simplex levon, being an unnili•decon; a levadimensional cube, or measure levon, being an undeca•decon; and a levadimensional orthotope, or cross levon, being an unbibiocta•decon.

I even gave the example of an enzeen- (one-dozen-one-) dimensional simplex, being the unbina•unnilon, or simplex ununon.

You want a fivzy (five dozen) dimensional cube, formed out of tenzy (ten dozen) cubes of forzy-leven (four dozen eleven) dimensions? Sure: it's the decnili•quadlevon, aka the measure pentnilon.

I can do this all day smile.gif
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icarus
Posted: Nov 14 2017, 04:55 PM


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Deriving higher polytope names from SI prefixes is incredibly obtuse.

ba-dum-dump!

This sort of humor is acutely appropriate and I laughed such that my coffee got on my shirt and I have to go home before going out now. I love it!

pentachoron / pentatope / simplex
octachoron / tesseract / "measure polytope"
hexadecachoron / orthoplex / "cross polytope" / hyperdiamond
icositetrachoron / octaplex
hecatonicosachoron / dodecaplex
hexacosichoron / tetraplex

I've always liked these names, but I would presume were we to name the shapes after its one-dimension-lesser facets, they might be briefer in dozenal. I do like the briefer "plex" names.
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Kodegadulo
Posted: Nov 14 2017, 07:58 PM


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QUOTE (icarus @ Nov 14 2017, 04:55 PM)
This sort of humor is acutely appropriate and I laughed such that my coffee got on my shirt and I have to go home before going out now. I love it!

Such humor does tend to have and edge to it, but the point is to make a brave face of it, take care not to peak too early, and understand your limits lest you wake up in a cell. It's a solid pastime with any number of intriguing facets. As long as you know some good lines, you'll do just fine.

-- Paulie Taupe

P.S. Simplex \(\Psi\)-mon met a \(\pi\)-man, circling the square ...
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Double sharp
Posted: Nov 14 2017, 11:25 PM


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I like the "plex" names too in 4D (and you can extend them to the star polytopes as well, with the addition of icosaplex for the icosahedral 120-cell, which shares the vertices and edges of the 600-cell; it is one of the 4D regular stars that could be considered analogous to the great dodecahedron).

In 5D and up, I would still normally just say n-simplex, n-cube, and n-orthoplex.
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icarus
Posted: Nov 15 2017, 04:31 AM


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I'm sorry I'm dyin' hahaha! Paulie Taupe!! Masterful! Gotta get my inhaler cuz out of breath from laffin

Yes once we are above the fourth dimension we have those three polytopes, and one honeycomb I think (going from foggy memory) above the fifth.
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Double sharp
Posted: Nov 15 2017, 05:46 AM


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In most dimensions (three and five and up) we do indeed have just the n-cubic honeycomb in R^n (which I guess is better called E^n in this context for Euclidean n-space). However, in E^4 we additionally have the 16-cell honeycomb {3, 3, 4, 3} and the dual 24-cell honeycomb {3, 4, 3, 3}.

Any other Schläfli symbol defines a valid hyperbolic honeycomb, although many have their vertices at "ideal" points at infinity or "ultra-ideal" points beyond it, using those terms with their standard meanings in hyperbolic projective geometry. (These are represented by points on and outside the boundary hypersphere respectively in the Poincaré hyperball model.)
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