New nomenclature for regular polytopes

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

New nomenclature for regular polytopes

Postby quickfur » Mon May 24, 2010 4:17 am

Continuing this thread in a new topic, 'cos I think it deserves attention. From another thread:

Tamfang wrote:[...]
Suppose the elements, rather than numbers, were the basis of polytope nomenclature: pyromorph, geomorph, aeromorph, hydromorph, cosmomorph (or pick your own suffix so long as it's not hedron). These names carry over naturally to higher dimensions; that is, any measure polytope can be called a geomorph (or geon for short); if context does not specify the dimension, say geochoron for the cube and geoteron for the tesseract. These roots are a lot shorter than hecatonicosa– and no two have the same first letter (in Greek or in Latin).

But, you say, my beloved 24-cell is out in the cold! So we extend the scheme by arbitrarily borrowing one of the Chinese elements (fire, water, earth, metal, wood). Wood grows from earth and air, as the 24-cell has properties of the aeromorph and geomorph. Thus: XYLOTERON.


I fell in love with this new nomenclature at first sight. I'm a stickler for concise names, and something like hecatonicosichoron is just too big a mouthful for my tastes. But cosmoteron for the 120-cell? Absolutely awesome!

Using these names as the base names for deriving uniform polytope names is much better than the existing system, IMHO. (I mean, c'mon, as though hecatonicosichoron wasn't bad enough, now we have to deal with runcitruncated hecatonicosichoron? And once you start going to higher dimensions, the names might as well span an entire sentence. Or a paragraph.) The only thing is, we need another systematic base name for the alternated hypercubes, which form their own family of uniform polytopes.

I suggest using sidero- (iron), since it derives from earth (much of the Earth's composition is iron, but not all of it - so the alternated cubes derive from the "full" cubes). Hence, sideromorph for the alternated hypercubes.

What say yous?
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Re: New nomenclature for regular polytopes

Postby wendy » Mon May 24, 2010 7:09 am

I have given this also much thought.

The cube, the octahedron, and the sphere are the third moments of coherent products, that is, they represent in N dimensions, valid expressions of line^N, and are thus coherent units.

The simplex is a power of its vertex, so the simplex in three dimensions, is the fourth (drawing) over the vertices. It is a surtope product, but not coherent.

The polygons represent a unified iteration over the 'wrap' operator, with 2 as minimal, and like the products, represent an item for every power.

There are five remaining regular solids (icosa, dodeca, and in 4d, 24ch, 120ch and 600ch). I use 'twelftychoron' for 120c.

Some thought have been given to the regular name forms for:
  • icosahedral, dodecahedral, and 24ch
  • The gosset figures, by k_21, 2_k1 and 1_k2 (see eg gossetoicosa, gossetaocta and gosetadodeca in the PG

There is room for naming these figures in my short-form: /n/ [SPC], \n\, 1/nQ (nd CO), 1\nQ "double-cube", /En "halfcube", /1/1/../1/ [horton cell], and the varieties like \nB, (gosset icosa-duals), \4B, 2/2B etc.

So far, only "octagonny" 1/Q/1 and "octagrammy 1/Qi/1 have been accepted in the list.
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Re: New nomenclature for regular polytopes

Postby Keiji » Mon May 24, 2010 10:28 pm

quickfur wrote:
Tamfang wrote:[...]

I like this nomenclature! I find the current facet-count-based system very cumbersome. (I mean, c'mon, hecatonicosachoron? That's quite a mouthful.) Hydroteron is a way better name, and cosmoteron is just awesome. Xyloteron is a cool exotic name that befits the 24-cell's special place in the ranks of regular polytopes. I think I shall adopt this nomenclature!


I agree with Nick quickfur! I definitely like these names, they are short, sweet and simple :D

There's just a couple of odd things about Tamfang's scheme. First, the suffixes are offset by one (a 4D object is a -choron, not a -teron; a 5D object is a -teron). Secondly, icosahedron and dodecahedron are apparently swapped around. So correcting those, we have:

pyron (pyrotope) = simplex
pyrogon = equilateral triangle = regular triangle = regular trigon
pyrohedron = regular tetrahedron
pyrochoron = regular pentachoron = regular 5-cell
pyroteron = regular hexateron

geon (geotope) = measure polytope
geogon = square = regular quadrilateral = regular tetragon
geohedron = cube = regular hexahedron
geochoron = tesseract = regular octachoron = regular 8-cell
geoteron = penteract = regular decateron

aeron (aerotope) = cross polytope
(aerogon = diamond = dual square)
aerohedron = regular octahedron
aerochoron = regular hexadecachoron = regular 16-cell
aeroteron = regular icosidodecateron

xylochoron = regular icositetrachoron = regular 24-cell

hydrohedron = regular dodecahedron
hydrochoron = regular hecatonicosachoron = regular 120-cell

cosmohedron = regular icosahedron
cosmochoron = regular hexadecachoron = regular 600-cell

Do go ahead and correct me if the "errors" I "corrected" were actually intentional, but I think this is a really good naming scheme as is! :)

What about the uniform polytopes? The four rectates would analogously be geoaerohedron, geoaerochoron, hydrocosmohedron and hydrocosmochoron, but can we have some prefixes for truncate, rectate, snub and whatever the 4D ones are?
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Re: New nomenclature for regular polytopes

Postby quickfur » Tue May 25, 2010 3:10 pm

Keiji wrote:[...]There's just a couple of odd things about Tamfang's scheme. First, the suffixes are offset by one (a 4D object is a -choron, not a -teron; a 5D object is a -teron).

True, although do keep in mind that a 4D object being -choron is based on the dimensionality of its surtopes (surface elements). In this new naming scheme, we're referring to the entire polytope as opposed to the count of its surface elements, so arguably Tamfang's suffixes are correct.

However, I agree that this could become a source of confusion, so perhaps it's best if we stick with the current convention and call 4D polytopes -chora(on), not -tera(on).

Secondly, icosahedron and dodecahedron are apparently swapped around.

No, they are not. The ancient Greeks believed that the pentagon was quintessential, and compose the heavens; water was believed to be icosahedral because that was the closest to a spherical shape (in their view). Therefore, astro- refers to the dodecahedral polytopes, and hydro- refers to the icosahedral ones.

[...]What about the uniform polytopes? The four rectates would analogously be geoaerohedron, geoaerochoron, hydrocosmohedron and hydrocosmochoron, but can we have some prefixes for truncate, rectate, snub and whatever the 4D ones are?

Be careful there. From 4D onwards, the geotope (geomorph?) rectates are distinct from the aerotope (aeromorph?) rectates. In 3D, the rectates are identical, but in 4D, it's the bitruncates that are identical. In 5D, the truncation needs to go yet deeper before they are equivalent.

What are the Greek names for the Archimedean polyhedra? They may be a good source of prefixes for naming the uniform polytopes.

Also, I've already suggested sidero- for the alternated geomorphs. (Tamfang suggested using -morph; perhaps that might be better than -tope in this naming scheme? Because, as I've mentioned, the names refer to the polytope themselves rather than their surface elements which consist of multiple (poly) facets (topes). In this sense, aeromorph - air-shaped - seems to make more sense than aerotope - air-place?.)
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Re: New nomenclature for regular polytopes

Postby wendy » Wed May 26, 2010 7:13 am

The term 'pyritohedral' is applied to the group orbifold = 3*2, represented by 'even perm+all change of sign', which is one of the subgroups of order 2 of the octahedral group. The other two are * 2 3 3 = all permutations, even change of sign (halfcubic), and 2 3 4 even (permutations + change of sign) = rotational. There is a subgroup of order 4, given by even perm, even chs.

In four dimensions, there is a second set of five over this, represented by [Coxeter] 3+,4,3+ (288), 3+,4,3 (576), (3,4,3)+ = 576, 3,4,3 of 1152, and 3,4,3= of 2304. The group 3,4,3+ = 3+,4,3 is the symmetry of the snub 24ch, the common group of [3,3,5] and [3,4,3].
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Re: New nomenclature for regular polytopes

Postby Keiji » Thu May 27, 2010 1:12 pm

quickfur wrote:
Secondly, icosahedron and dodecahedron are apparently swapped around.

No, they are not. The ancient Greeks believed that the pentagon was quintessential, and compose the heavens; water was believed to be icosahedral because that was the closest to a spherical shape (in their view). Therefore, astro- refers to the dodecahedral polytopes, and hydro- refers to the icosahedral ones.


Right, fair enough. I would have been fine with either order {3,3} {3,4} {4,3} {3,5} {5,3} (vertex count) or {3,3} {4,3} {3,4} {5,3} {3,5} (facet count), though the order suggested is {3,3} {4,3} {3,4} {3,5} {5,3}. Well, why not use the vertex count order, but keep the names the same? So then it goes pyro - aero - geo - (xylo) - hydro - cosmo.

[...]What about the uniform polytopes? The four rectates would analogously be geoaerohedron, geoaerochoron, hydrocosmohedron and hydrocosmochoron, but can we have some prefixes for truncate, rectate, snub and whatever the 4D ones are?

Be careful there. From 4D onwards, the geotope (geomorph?) rectates are distinct from the aerotope (aeromorph?) rectates. In 3D, the rectates are identical, but in 4D, it's the bitruncates that are identical. In 5D, the truncation needs to go yet deeper before they are equivalent.


I know. I refer to ox+o (where x+ = one or more 'x', as in regexp) as rectate for any dimension because of that property, and you can see my names for them in the "conventional" column here. That said, I'm not particularly convinced that my scheme gives good names for the others, but this is because I can't properly visualize most of the non-parent polytopes (yet) so I have no way of knowing what the most natural scheme would be.

Also, I've already suggested sidero- for the alternated geomorphs. (Tamfang suggested using -morph; perhaps that might be better than -tope in this naming scheme? Because, as I've mentioned, the names refer to the polytope themselves rather than their surface elements which consist of multiple (poly) facets (topes). In this sense, aeromorph - air-shaped - seems to make more sense than aerotope - air-place?.)


Well, we currently use -tope for a general shape of any dimension and -*on for a specific dimension: regardless of the family (e.g. "tapertope"). I think, to change that convention, would be a bit confusing, as with changing them up by one dimension (choron to teron, etc).
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Re: New nomenclature for regular polytopes

Postby quickfur » Thu May 27, 2010 3:03 pm

Keiji wrote:[...]Right, fair enough. I would have been fine with either order {3,3} {3,4} {4,3} {3,5} {5,3} (vertex count) or {3,3} {4,3} {3,4} {5,3} {3,5} (facet count), though the order suggested is {3,3} {4,3} {3,4} {3,5} {5,3}. Well, why not use the vertex count order, but keep the names the same? So then it goes pyro - aero - geo - (xylo) - hydro - cosmo.

I'm more used to face count order, but either way is fine. :-)

[...]Be careful there. From 4D onwards, the geotope (geomorph?) rectates are distinct from the aerotope (aeromorph?) rectates. In 3D, the rectates are identical, but in 4D, it's the bitruncates that are identical. In 5D, the truncation needs to go yet deeper before they are equivalent.


I know. I refer to ox+o (where x+ = one or more 'x', as in regexp) as rectate for any dimension because of that property, and you can see my names for them in the "conventional" column here. That said, I'm not particularly convinced that my scheme gives good names for the others, but this is because I can't properly visualize most of the non-parent polytopes (yet) so I have no way of knowing what the most natural scheme would be.

Well, all the uniform polytopes arise from the possible combinations of circled and non-circled nodes in the Coxeter-Dynkin diagram for the parent polytope. So that gives us a way of checking whether our naming schemes cover all the possibilities. As for visualizing them... it's really not that hard in the case of 4D; take a look at http://en.wikipedia.org/wiki/Uniform_polychoron - many of the uniform polytopes, especially the pentatopic/tesseractic ones, have nice projection images on their respective pages.

Also, I've already suggested sidero- for the alternated geomorphs. (Tamfang suggested using -morph; perhaps that might be better than -tope in this naming scheme? Because, as I've mentioned, the names refer to the polytope themselves rather than their surface elements which consist of multiple (poly) facets (topes). In this sense, aeromorph - air-shaped - seems to make more sense than aerotope - air-place?.)


Well, we currently use -tope for a general shape of any dimension and -*on for a specific dimension: regardless of the family (e.g. "tapertope"). I think, to change that convention, would be a bit confusing, as with changing them up by one dimension (choron to teron, etc).

Fair enough. Using -tope is fine by me.
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Re: New nomenclature for regular polytopes

Postby Keiji » Sun May 30, 2010 12:12 pm

I've split out all the posts about constructing uniform polytopes.

Can we have a prefix for "snub", as opposed to "snub geohedron", "snub cosmohedron" and "snub xylochoron"?
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Re: New nomenclature for regular polytopes

Postby Tamfang » Wed Nov 17, 2010 3:18 am

I am gratified that Quickfur expressed my position on morphemes, above (before dismissing it; one can't have everything). It's a bit puzzling that no one consulted me, or brought this thread to my attention, before putting my name on http://teamikaria.com/hddb/wiki/Table_of_regular_polytopes_by_Tamfang_name.
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Re: New nomenclature for regular polytopes

Postby Tamfang » Wed Nov 17, 2010 3:31 am

snub is itself a translation of Latin simus, so how about simo–? Unfortunately the Wikipedia articles on snubs have no links to Greek Wikipedia. For truncation I like pen–.

Also. For figures with hydro/cosmo symmetry that aren't more clearly akin to one than to the other, how about rhodo– (after a family of plants whose flowers and fruits have fivefold symmetry)? And for the aero/geo family, how about stauro– (cross)?
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Re: New nomenclature for regular polytopes

Postby Tamfang » Wed Nov 17, 2010 3:59 am

Keiji wrote: I would have been fine with either order {3,3} {3,4} {4,3} {3,5} {5,3} (vertex count) or {3,3} {4,3} {3,4} {5,3} {3,5} (facet count), though the order suggested is {3,3} {4,3} {3,4} {3,5} {5,3}. Well, why not use the vertex count order, but keep the names the same? So then it goes pyro - aero - geo - (xylo) - hydro - cosmo.

Huh? What has the sequence to do with anything? Does this refer to something that was removed from the thread?
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Re: New nomenclature for regular polytopes

Postby wendy » Wed Nov 17, 2010 8:06 am

Arranging polytioes by increasing vertices also arranges them by increasing volume and size. This seems a pretty standard order, so something like the 'fifth' regular polychoron means 335.

Three of the regular polytopes arise by regular product, along with the sphere. Of these, the simplex in N dimensions, is the N+1 power of a point, while the cross, measure and sphere is the corresponding Nth power of a line. The other thing is that the -hedron or -hedrix stem can not be used, because this is a feature of the surface, rather than the solid. The form for the 3D solid is chorid.

This is relatively important, because there are measures that carry this name. A cubic foot becomes a prismachorid foot or pc ft. One also has tegmatochorid feet (tc ft: 1 pc ft = 6 tc ft), and a crindachorid foot (cc ft). Of the radian and its derivitives, the measure is of space, but the scale is of surface, so one might talk of a prismahedral radian or a tegmahedral radian. These refer to a surface (hedron) in space (chorid).

Since we have already names for the product, this leads immediately to, eg 'tegmachorid (octahedron), prismachorid (cube). The current name for the sphere-product is /crind/, which would lead to crindachorid (3-sphere). We are then left wuth a name pyrachorid (tetrahedron).

This leaves something like 5+8 = 13, and prehaps more, for names. The general groups correspond to 343. 3..5, 5..3 (for the regular ones), and the six families of gossets ( /3..3B = k_21), 3..3/B = 2_k1 and 3..3B/ = 1_k2, and their duals. These do indeed form a common third trigonal group with reflexes down as far as 2d. One sees attempt to name these in the PG under entries at gosseto-

The five outstanding regular ones then form something like 5, 35, 335 vs 5, 53, 533 and by itself 343. Of 35, 335 we already have icosahedron, fifhundchoron, while 53, 533 gives dodecahedron, twelftychoron.

Names for the regular stars (eg 3+3 stella octangula), in four dimensions, gives stella tegmata (3+34 = 3 tegmatoterids in a 24ch), stella prismato (43+3 = 3 prismaterids in a 24ch), and the stella metrica (120 pyraterids in a twelftychoron).
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Re: New nomenclature for regular polytopes

Postby quickfur » Wed Nov 17, 2010 4:08 pm

Tamfang wrote:snub is itself a translation of Latin simus, so how about simo–? Unfortunately the Wikipedia articles on snubs have no links to Greek Wikipedia. For truncation I like pen–.

Also. For figures with hydro/cosmo symmetry that aren't more clearly akin to one than to the other, how about rhodo– (after a family of plants whose flowers and fruits have fivefold symmetry)? And for the aero/geo family, how about stauro– (cross)?

Maybe rhodo- and stauro- can be collective? So the rhodotopes include the hydrotopes and cosmotopes, and the staurotopes comprise the aerotopes and geotopes. Just a thought. I particularly like rhodo-, it's a very pretty name for such pretty polytopes.
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Re: New nomenclature for regular polytopes

Postby Tamfang » Wed Nov 17, 2010 5:20 pm

quickfur wrote:... So the rhodotopes include the hydrotopes and cosmotopes, and the staurotopes comprise the aerotopes and geotopes. Just a thought. ...

Rhodomorphs, if you please ....
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Re: New nomenclature for regular polytopes

Postby quickfur » Wed Nov 17, 2010 5:41 pm

Oh right, sorry. Forgot that we're replacing -tope with -morph. :-)
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Re: New nomenclature for regular polytopes

Postby Keiji » Thu Nov 18, 2010 10:15 pm

Tamfang wrote:It's a bit puzzling that no one consulted me, or brought this thread to my attention, before putting my name on http://teamikaria.com/hddb/wiki/Table_of_regular_polytopes_by_Tamfang_name.


I wanted to get the nomenclature on the wiki ASAP... what else would I have called it?

Is using your name a problem?

Tamfang wrote:snub is itself a translation of Latin simus, so how about simo–?


Simogeohedron, simocosmohedron and simoxylochoron all sound good to me. :)

Unfortunately the Wikipedia articles on snubs have no links to Greek Wikipedia. For truncation I like pen–.


Penaerohedron, pengeohedron? I dunno, it sounds a bit weird. Also penta starts with pen, so that might be a little confusing.

I initially thought there might be confusion with the word truncatriate (where the tri originates from 3, as in tri, tetra, penta), but then I realized this isn't a term on its own, it's part of the term cubic truncatriate (or similar), but under this scheme the 2-word term would be renamed to pengeohedrate anyway, so it wouldn't matter.

I guess I could get used to pen-, though - I can certainly think of a lot worse :D

Tamfang wrote:
Keiji wrote: I would have been fine with either order {3,3} {3,4} {4,3} {3,5} {5,3} (vertex count) or {3,3} {4,3} {3,4} {5,3} {3,5} (facet count), though the order suggested is {3,3} {4,3} {3,4} {3,5} {5,3}. Well, why not use the vertex count order, but keep the names the same? So then it goes pyro - aero - geo - (xylo) - hydro - cosmo.

Huh? What has the sequence to do with anything? Does this refer to something that was removed from the thread?


I think that was only to do with what order they appeared in the table, which is nothing particularly important.

Tamfang wrote:Also. For figures with hydro/cosmo symmetry that aren't more clearly akin to one than to the other, how about rhodo– (after a family of plants whose flowers and fruits have fivefold symmetry)? And for the aero/geo family, how about stauro– (cross)?


Sounds good, as does quickfur's idea about including the hydro/cosmo/aero/geotopes in them.

quickfur wrote:
Tamfang wrote:Rhodomorphs, if you please ....
Oh right, sorry. Forgot that we're replacing -tope with -morph. :-)


See several posts up for my argument against that:

I wrote:Well, we currently use -tope for a general shape of any dimension and -*on for a specific dimension: regardless of the family (e.g. "tapertope"). I think, to change that convention, would be a bit confusing, as with changing them up by one dimension (choron to teron, etc).
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Re: New nomenclature for regular polytopes

Postby Tamfang » Thu Nov 18, 2010 11:05 pm

Keiji wrote:... what else would I have called it?

How about "elemental names"?
Keiji wrote:Is using your name a problem?

It implies that I embrace a practice which I expressly opposed, in the first sentence quoted at the top of this page! I can't stop others from using such language but I can and do refrain from it myself.

Precedent may constrain us by creating opportunities for ambiguity, but it ought not to bind us more than that. Words are not so scarce as to oblige us to equate a body with its elements. I don't see how it's any more confusing to deviate from that sloppy convention than to abandon hecatonicosa–.
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Re: New nomenclature for regular polytopes

Postby quickfur » Fri Nov 19, 2010 12:58 am

Keiji wrote:[...]
Tamfang wrote:snub is itself a translation of Latin simus, so how about simo–?


Simogeohedron, simocosmohedron and simoxylochoron all sound good to me. :)

Eeek! Please, no. It sounds too close to being a derivative of simian. Add to that the fact that these are to refer to hemihypercubes, one would think that we were talking about half-brained monkeys!

Unfortunately the Wikipedia articles on snubs have no links to Greek Wikipedia. For truncation I like pen–.


Penaerohedron, pengeohedron? I dunno, it sounds a bit weird. Also penta starts with pen, so that might be a little confusing.

Yeah, I don't like pen- either. I would suggest a better scheme, but none comes to mind.

Well, we currently use -tope for a general shape of any dimension and -*on for a specific dimension: regardless of the family (e.g. "tapertope"). I think, to change that convention, would be a bit confusing, as with changing them up by one dimension (choron to teron, etc).

However, I have to say that I like -morph better than -tope; the latter derives from Greek topos, which means place; I don't find that a very useful way to think about these objects. Rather, -morph, which means shape, seems eminently more appropriate. The only advantage -tope has over -morph is its familiarity, which is really no more than a historical accident.
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Re: New nomenclature for regular polytopes

Postby Keiji » Fri Nov 19, 2010 10:28 am

quickfur wrote:However, I have to say that I like -morph better than -tope; the latter derives from Greek topos, which means place; I don't find that a very useful way to think about these objects. Rather, -morph, which means shape, seems eminently more appropriate. The only advantage -tope has over -morph is its familiarity, which is really no more than a historical accident.


Okay, fair enough. I guess it won't be that much of a problem, given we're renaming everything else anyway.
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Re: New nomenclature for uniform polytopes

Postby Tamfang » Fri Nov 19, 2010 11:31 pm

By the way, what law says that truncated (for example) must be an adjective or a prefix? Chemistry gets plenty of use out of suffixes, as does Esperanto. (Is there a layman-friendly book about how the chemical suffixes were adopted?)

You could have (arbitrary example) –ane for the 'parent', –ene for the rectate, –ine for the birectate; –ade for the truncate, –ede for the bitruncate ... –o– and –u– for alternations ....
On another hand, in English the vowels of unstressed syllables are rarely pronounced clearly, so this scheme has obvious room for improvement. Perhaps an extra consonant for bi–.

Ideally I'd want the system to be applicable to tilings, both Euclidean and hyperbolic.
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Re: New nomenclature for regular polytopes

Postby quickfur » Sat Nov 20, 2010 12:35 am

Tamfang wrote:By the way, what law says that truncated (for example) must be an adjective or a prefix? Chemistry gets plenty of use out of suffixes (as does Esperanto).

Oooh, I smell a fellow conlanger! ;)

You could have (arbitrary example) –ane for the 'parent', –ene for the rectate, –ine for the birectate; –ade for the truncate, –ede for the bitruncate ... –o– and –u– for alternations ....
On another hand, in English the vowels of unstressed syllables are rarely pronounced clearly, so this scheme has obvious room for improvement.

Who says we must adopt subtle vowel variations, and who says we must restrict it to only monosyllabic suffixes? Many natural languages do just fine with polysyllabic endings (e.g. -yami/-ami in Russian for plural instrumental--and almost all adjectives have disyllabic endings), and consonants provide a much wider range of distinct sounds.

Also, to maximize distinctiveness, it's often useful to restrict any vowels to apical vowels (/a/, /i/, /u/, or in English spelling, "ah", "ee", "oo"), which have maximal contrast with each other. It's the in-between stuff, like /e/, /æ/, etc., that starts bringing in potential ambiguity.

Having said that, though, you've to realize that the number of uniform truncates grow exponentially with dimension, arising after all from the number of ways to ring the nodes in their Coxeter-Dynkin diagrams. Unfortunately, human beings (and consequently human language or naturalistic language, which is apparently what we're trying to achieve here) are rather poor at dealing with combinatorial objects. Witness, for example, the absolutely atrocious tentative naming scheme for newly synthesized chemical elements: a 1-to-1 correspondence between the numbering system (atomic number) and syllables gives us such horrors as unununium for element 111, which grates on the ears when pronounced with 3 identical vowels (oon-oon-oon-ium), and suffers heavily from distortion when one tries to pronounce it more "naturally" (e.g. yoo-nuhn-yoo-nium). Any naming scheme that does such direct translations from a numbering scheme (e.g., using n-digit binary numbers to enumerate all possible uniform truncates of dimension n) is prone to exactly the same encumbrance as names like "hecatonicosachoron".

For such cases, I've always liked the IUPAC's method of prefixing with numbers, e.g., 1,3-dimethylbenzene. I adopted the same scheme in naming duoprisms: 3,5-duoprism instead of the wordy trigonal-pentagonal-duoprism. A similar scheme for naming the uniform truncates might therefore be more appropriate here: take the Coxeter-Dynkin diagram, orient it such that the edge of highest degree lies on the left, then read it from left to right with ringed node=1 and unringed node=0, using each number as a prefix. So for example, the cuboctahedron would be a 0,1,0-stauromorph truncate (or whatever suffix you may want to invent for it), and the great rhombicuboctahedron would be a 1,1,1-stauromorph truncate. This lets you name very high dimensional objects without running out of breath, e.g., 1,0,1,1,0-stauromorph truncate instead of cantibirectifiedblahblahified hexateron.

We can still use a suffix instead of "truncate", of course; but by relegating the combinatorial stuff to arabic numerals, which is best suited for the task, we have more syllables at our disposal for the special cases, such as snubs, alternations, etc..

And if we ever feel that binary numbers waste too many syllables to pronounce, we can always convert it into its decimal value, so the cuboctahedron becomes the 2-stauromorph truncate (or 2-staurotome, if we adopt -tom, from Greek tomo, to cut, as a general suffix for all uniform truncates), and the great rhombicuboctahedron becomes the 7-staurotome. The omnitruncated 600-cell would then become the 15-rhodotome, which you must admit is a very sweet and simple name indeed. And we don't even have to worry about the possibility of ugly names like 0-rhodotome, because at least one node in the Coxeter-Dynkin diagram must be ringed, otherwise there's only a single point.

We may even eliminate dimensional ambiguity (e.g., a 3-polytope 0,1,0-staurotome vs. a 4-polytope 0,0,1,0-staurotome) by inserting the dimension number right after the CD number: so cuboctahedron = 2,3-staurotome, rectified 16-cell = 2,4-staurotome. So the omnitruncated 600-cell becomes the 15,4-rhodotome, but since there aren't any rhodomorphs above 4D, and the value 15 can only occur in 4D because it requires 4 binary digits, we can elide the dimensional number and just write 15-rhodotome, as before. Similarly, since the xylomorphs only occur in 4D, we can simply write 15-xylotome for the omnitruncated 24-cell.

Using this scheme, even the regular polytopes themselves are represented: the 8-xylotome is just the 24-cell, and the 1-xylotome is its dual; the 8,4-staurotome is the tesseract, and the 1,4-staurotome is the 16-cell itself. By omitting the dimension, we can make dimension-independent statements like "the 8-staurotomes tile their respective spaces".

(Is there a layman-friendly book about how the chemical suffixes were adopted?)

From my (admittedly limited) reading, they were generalizations of existing ad hoc names.
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Re: New nomenclature for regular polytopes

Postby Tamfang » Sat Nov 20, 2010 1:03 am

quickfur wrote:Oooh, I smell a fellow conlanger! ;)

Rather manqué. My dad is a prominent Esperantist; at age ~13 I made some notes (now lost without a trace of regret) on a half-assed imitation of Sindarin; I kibitzed on the Conlang list when it was young — that's about all.

Having said that, though, you've to realize that the number of uniform truncates grow exponentially with dimension ...

And the number of possible suffixes grows exponentially with number of syllables. You needn't invent an infinite batch of them all at once.

For such cases, I've always liked the IUPAC's method of prefixing with numbers, e.g., 1,3-dimethylbenzene. I adopted the same scheme in naming duoprisms: 3,5-duoprism instead of the wordy trigonal-pentagonal-duoprism.

Jinx! or Snap! depending on your time-zone.

... And if we ever feel that binary numbers waste too many syllables to pronounce, we can always convert it into its decimal value ...

I'd be more inclined to use octal, to make related numbers more recognizable: it's a bit of a chore to divide by 64 mentally.

We may even eliminate dimensional ambiguity (e.g., a 3-polytope 0,1,0-staurotome vs. a 4-polytope 0,0,1,0-staurotome) by inserting the dimension number right after the CD number: so cuboctahedron = 2,3-staurotome, rectified 16-cell = 2,4-staurotome.

I'm not in love with this concept. In both the precedents above, the numbers are commutative. 0,1-truncation differs unavoidably from 1,0-truncation, but at least the bits are of the same kind.
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Re: New nomenclature for regular polytopes

Postby Keiji » Sat Nov 20, 2010 10:19 am

Do we really need to care about all the possible truncates?

The special cases, i.e. the interesting shapes, all occur in dimensions < 5.

Coming up with names for those that don't involve tedious numbers is a must.

However, for truncations only available in 5D and up, we can use numbers to represent them, since we usually won't care about an individual such case - if we ever talk about truncations in dimensions >= 5, we're likely talking in the general case.
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Re: New nomenclature for regular polytopes

Postby Tamfang » Sat Nov 20, 2010 5:01 pm

Keiji wrote:The special cases, i.e. the interesting shapes, all occur in dimensions < 5.

The E-series (up to 8 dim.) are not without interest to some.
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Re: New nomenclature for regular polytopes

Postby quickfur » Sat Nov 20, 2010 5:11 pm

I agree with using octal instead of decimal; it does make the mapping into binary easier. But how would that help with distinguishing between dimensions? I also agree that inserting the dimension number into the name is a bit of a "hack", to borrow from computerese. But I tend to dislike using number-based suffixes for dimensions, because once you get past a certain point, it starts becoming like "hecatonicosa-" all over again. I mean, "600-cell" is much more preferable to "hecatonicosachoron", in spite of whatever flaws it may have on its own. Sometimes a number is just a number, and best represented as a number. Unless, of course, we adopt both schemes: an ad hoc scheme for naming the lower dimensions based on suffixes, and a numerical scheme (written with actual numerals, not the equivalent of "unununium") for dimensions higher than the ad hoc names go.

As for the "special" truncations, we can invent a separate set of terminology for them. A little bit of redundancy won't kill us. From what I can tell, the "special" truncations are (1) the series of uniform truncations from a regular polytope to its dual -- this Keiji & I have found to be of the pattern 1,0,0,0,.. (the regular polytope itself); 1,1,0,0,...; 0,1,0,0,...; 0,1,1,0,...; 0,0,1,0,...; 0,0,1,1,...; and so on until ...0,0,1,1; ...0,0,0,1 (its dual). (2) The mesotruncate (the middle member of the previous series). (3) The "expanded" uniform polytopes obtained by expanding the facets of a regular polytope outwards - this is unique for each regular/dual pair (e.g. small rhombicuboctahedron is obtained by expanding either the cube or the octahedron; and runcinated tesseract == runcinated 16-cell). (4) The omnitruncated polytope (the 1,1,1,...,1-tome), also unique for each regular/dual pair.
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Re: New nomenclature for regular polytopes

Postby quickfur » Sat Nov 20, 2010 6:19 pm

Tamfang wrote:
quickfur wrote:Oooh, I smell a fellow conlanger! ;)

Rather manqué. My dad is a prominent Esperantist; at age ~13 I made some notes (now lost without a trace of regret) on a half-assed imitation of Sindarin; I kibitzed on the Conlang list when it was young — that's about all.
[...]

I discovered the Conlang list while searching for grammatical terms online... while trying to invent a non-English language for a fictional universe I've invented (I can never stand the mimesis-shattering trope of a completely alien universe whose inhabitants somehow manage to acquire English, as spoken on Earth in the 20th century, as their native tongue). I was there a number of years, during which I invented two languages, the first of which was a failure due to a lack on understanding of linguistics on my part, and the second of which was very promising; but both have been abandoned for a number of years now due to lack of time and other competing interests, and I've not been subscribed to the list since then.

So I'm more of an ex-conlanger by now.

Interestingly enough, Alkaline, the owner of the original version of this forum, was also a Conlang list subscriber (but now either inactive or unsubscribed). I wonder what it is about conlanging tendencies that leads one to interest in higher-dimensional geometry.
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Re: New nomenclature for regular polytopes

Postby quickfur » Sat Nov 20, 2010 7:03 pm

Tamfang wrote:
Keiji wrote:The special cases, i.e. the interesting shapes, all occur in dimensions < 5.

The E-series (up to 8 dim.) are not without interest to some.

They also diverge from the hypercube/simplex series of truncates beyond 5D, and thus require a separate naming scheme. They are sporadic, in the sense that they only occur for a limited number of dimensions, so they deserve their own names. Furthermore, there are a couple o' other sporadics in very high dimensions, corresponding with mathematical objects of high symmetry, but I forget the specifics.
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Re: New nomenclature for uniform polytopes

Postby Tamfang » Sat Nov 20, 2010 7:22 pm

quickfur wrote:From what I can tell, the "special" truncations are ....

I read this as implying that someone has set down a formal definition of 'special' but you don't know exactly what it is. Am I warm?
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Re: New nomenclature for uniform polytopes

Postby quickfur » Sat Nov 20, 2010 7:37 pm

Tamfang wrote:
quickfur wrote:From what I can tell, the "special" truncations are ....

I read this as implying that someone has set down a formal definition of 'special' but you don't know exactly what it is. Am I warm?

Funny you should mention that, 'cos I just read exceptional objects on Wikipedia, and exceptional polytopes are among those listed. As far as I know, no one on this forum has set down a formal definition of what constitutes special, but based on previous discussions here, I gathered that most people were interested in one of the objects I listed.
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Re: New nomenclature for regular polytopes

Postby Keiji » Mon Nov 22, 2010 8:28 pm

Tamfang wrote:
Keiji wrote:The special cases, i.e. the interesting shapes, all occur in dimensions < 5.

The E-series (up to 8 dim.) are not without interest to some.


Yes, the E-series deserve their own names, but this doesn't mean we should enumerate names for all uniform shapes up to 8 dimensions, because there are a vast number of (5..8)-dimensional shapes which are uninteresting.
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