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

Re: Bitruncated 5-cell

Postby quickfur » Tue Dec 09, 2008 4:40 pm

Hayate wrote:Can you say "comic relief"? :XD:

Ehh. Would you mind changing "pentatope" to "pentachoron"? A pentatope is a 5D shape with flat facets. Very different to a pentachoron :D

Hmm. According to my understanding, pentatope is quite a common name for the 4-simplex. A 5-polytope is a polyteron; I've never seen pentatope refer to a 5-polytope.
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Re: Bitruncated 5-cell

Postby Keiji » Wed Dec 10, 2008 12:33 am

Yes, "polyteron" is a more common name, but logically speaking, "pentatope" is to "pentashape" as "polytope" is to "shape", with a pentashape being a 5D shape (curved or not). I've never heard the word "pentatope" be used for a pentachoron anyway. :)
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Re: Bitruncated 5-cell

Postby quickfur » Wed Dec 10, 2008 1:51 am

Hayate wrote:Yes, "polyteron" is a more common name, but logically speaking, "pentatope" is to "pentashape" as "polytope" is to "shape", with a pentashape being a 5D shape (curved or not). I've never heard the word "pentatope" be used for a pentachoron anyway. :)

It is listed second on the list of alternative names for the 5-cell on George Olshevsky's page, y'know... :)

Speaking of which, has Olshevsky gotten around to reviving his site? It is sorely missed.
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Re: Bitruncated 5-cell

Postby wendy » Wed Dec 10, 2008 6:00 am

The name 'polytope' is supposed to apply to members of the family "polygon,polyhedron". Since there is no fixed name for the 4d member, it is usually taken that polytope = 4d, and that pentatope = 5-faced polytope,

Polycell is likewise a misuse of the word cell (which elsewhere means a solid element of a tiling, eg hexagon), taking into account that a dodecahedron of 5,3,3 is a cell of a spherical tiling. Coxeter uses this, but the use is depreciated in the PG.

Polychoron is the accepted word for 4d polytopes, this is by G.O. with modifications by Norman Johnson.

For higher dimensions (5-9), the series runs polyteron, polypeton, polyecton, polyzetton, and polyyotton, based on the metric prefixes, with some euphonics (good-soundings) applied.

There is an extention past 9 dimensions, but i prefer a different strategy here.
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Re: Bitruncated 5-cell

Postby quickfur » Wed Dec 10, 2008 5:46 pm

wendy wrote:The name 'polytope' is supposed to apply to members of the family "polygon,polyhedron". Since there is no fixed name for the 4d member, it is usually taken that polytope = 4d, and that pentatope = 5-faced polytope,

Yes, this is the usage I'm familiar with.

Polycell is likewise a misuse of the word cell (which elsewhere means a solid element of a tiling, eg hexagon), taking into account that a dodecahedron of 5,3,3 is a cell of a spherical tiling. Coxeter uses this, but the use is depreciated in the PG.

I've never heard of polycell except as Olshevsky's online nickname.

Polychoron is the accepted word for 4d polytopes, this is by G.O. with modifications by Norman Johnson.

What's "G.O."?

For higher dimensions (5-9), the series runs polyteron, polypeton, polyecton, polyzetton, and polyyotton, based on the metric prefixes, with some euphonics (good-soundings) applied.

I think this terminology is clearest. Using prefixes such as "penta" in "pentatope" is confusing, because it is unclear whether the "penta" applies to the dimension of the shape, or to the number of facets. I guess "pentachoron" is the better name here. I somewhat dislike names like "5-cell" because of visual confusion when speaking of a multitude of 5-cell's: one writes "5 5-cells" which can be visually confused with "55-cell". However, I find names like "icositetrachoron" a mouthful to pronounce and cumbersome to type, so I'd still prefer a shorter nickname if there is one. I could use Bowers' acronyms, I suppose, but they lack the euphony of a "sophisticated" English word (whatever that means), and give very little clue as to what they refer to.

There is an extention past 9 dimensions, but i prefer a different strategy here.

Are you referring to Bowers' extension to thousands and millions of dimensions? I saw that on his page once. Not that we'd ever get around to enumerating the elements of such high-dimensional shapes, although I suppose we could theoretically speak of the 1,000,000,000,000-cube and its dual, the 1,000,000,000,000-cross.
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Re: Bitruncated 5-cell

Postby Keiji » Wed Dec 10, 2008 9:13 pm

quickfur wrote:
wendy wrote:The name 'polytope' is supposed to apply to members of the family "polygon,polyhedron". Since there is no fixed name for the 4d member, it is usually taken that polytope = 4d, and that pentatope = 5-faced polytope,

Yes, this is the usage I'm familiar with.


But a polytope is surely of any dimension.

poly {tri, tetra, penta, hexa, hepta, octa, ...} = number of facets
tope {gon, hedron, choron, peton, ecton, zetton, ...} = dimensionality of facets

So a "polytope" is a shape with any number and dimension of facets, a "polychoron" is a shape with any number of cells (= 3D facets), a "pentatope" is a shape with 5 facets of any dimension, and a "pentachoron" is a shape with 5 cells.

quickfur wrote:What's "G.O."?


George Olshevsky. :P

quickfur wrote:However, I find names like "icositetrachoron" a mouthful to pronounce and cumbersome to type


I like the word "icositetrachoron", though...

quickfur wrote:I'd still prefer a shorter nickname if there is one. I could use Bowers' acronyms, I suppose, but they lack the euphony of a "sophisticated" English word (whatever that means), and give very little clue as to what they refer to.


Then go by their Kana mnemonics: Icositetrachoron = コシ = koshi :D
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Re: Bitruncated 5-cell

Postby quickfur » Wed Dec 10, 2008 10:35 pm

Hayate wrote:
quickfur wrote:
wendy wrote:The name 'polytope' is supposed to apply to members of the family "polygon,polyhedron". Since there is no fixed name for the 4d member, it is usually taken that polytope = 4d, and that pentatope = 5-faced polytope,

Yes, this is the usage I'm familiar with.


But a polytope is surely of any dimension.

Yes, but how many 5-polytopes do you know of that has only 5 facets?

poly {tri, tetra, penta, hexa, hepta, octa, ...} = number of facets
tope {gon, hedron, choron, peton, ecton, zetton, ...} = dimensionality of facets

This I agree with.

So a "polytope" is a shape with any number and dimension of facets, a "polychoron" is a shape with any number of cells (= 3D facets), a "pentatope" is a shape with 5 facets of any dimension, and a "pentachoron" is a shape with 5 cells.

quickfur wrote:What's "G.O."?


George Olshevsky. :P

Ah, of course. I thought she was referring to some theorem or mathematical construct. :oops:

quickfur wrote:However, I find names like "icositetrachoron" a mouthful to pronounce and cumbersome to type


I like the word "icositetrachoron", though...

It has 7 syllables and 16 letters to type, as opposed to "24-cell" which has only 4 syllables and 7 characters to type. Guess which one I prefer?

quickfur wrote:I'd still prefer a shorter nickname if there is one. I could use Bowers' acronyms, I suppose, but they lack the euphony of a "sophisticated" English word (whatever that means), and give very little clue as to what they refer to.


Then go by their Kana mnemonics: Icositetrachoron = コシ = koshi :D

Well, if you're going to use another language, maybe I should start using Russian letters. ;) Понял?

On a more serious note, just as chemical names have been standardized under the IUPAC, maybe we need an IUPAC for polytopes (International Unified Polytope Acronym Convention)... :lol:
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Re: Bitruncated 5-cell

Postby Keiji » Thu Dec 11, 2008 12:22 am

The romanizations of the Kana mnemonics are provided on HDDB on the table pages. This is quite regular, moreso than Bowers, and easy to pronounce and type. I can't see what you have against it =p

Also, Russian letters are ugly. D:
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Re: Bitruncated 5-cell

Postby wendy » Thu Dec 11, 2008 12:24 am

The PG seeks to set a way of constructing names for which the meaning is apparent from the stems of the word. For example, tetra+hedr+on gives 4 + 2d + patches, say a figure so bounded. Much research was undertaken in the linguistics of other fields of science, that made the transition to practical technology, a lot of the faults and triumphs of these endeavours form the basis of the polygloss.

That it is so dispart from the current termonology is more implication of the way that the current termonology is set. From three dimensions, 2d can appear as either 2d or N-1 d. A word like "face" or "hedral" can be freely applied to 2d or N-1 d, which match in three dimensions. In six dimensions, N-1 is 5, one speaks of a 'dihedral angle' meaning the angle that five-dimensional surtopes meet. Likewise, faces are not facing things, but facets are. So nice is it to divide at the outset 2d (hedron) vs N-1 d (face), so were one to encounter hedron, one is talking invariably of something of 2d, or made of 2d things.

The problem of apposition (placing two numbers side by side) is always present. One has several solutions to this: eg

twenty of four-pound guns (ie 20 in number, of guns that shoot 4 lb shots)
twenty-four-pound guns ( ie guns that shoot 24-lb shots)

One method is to use 'cerimonial numbers', such as latin or greek: eg 1992 XII 31 (christmas 1992). [month in latin numbers]

The problem with polytope count eg 24ch, is that one might end with three strung numbers, eg

eighteen 24tn is 18-24-4s, ie count of eighteen + sides at 24 + patch of 4d.

Some work is done on this.

In weights and measures, one might hear references to things like "metre-tenths" or metre-tens, being E-10 m and E10 m. The idea of postfix number of this kind has been looked on favourably. This would allow the use of a greater number of compound roots, subscripted by the dimension. Some terms, like 'margin' are descending, but the ordinals serve here, eg margin-sixth = M-6 = N-8 dimensions. margin-sixths dual into edge-sixes (E+6 = 6). The terminology is set that one and first refer to the unadorned word: a margin is a margin-first (or margin-prime), while an edge-one is an edge.

The matter awaits formal consideration though.
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Re: Bitruncated 5-cell

Postby quickfur » Thu Dec 11, 2008 1:52 am

Hayate wrote:The romanizations of the Kana mnemonics are provided on HDDB on the table pages. This is quite regular, moreso than Bowers, and easy to pronounce and type. I can't see what you have against it =p

Also, Russian letters are ugly. D:

Well, that's a subjective argument. :) I happen to think Russian letters are the prettiest things out there... esp. with such beautiful shapes as Ж, Я, or Ч. Or И, for that matter. One can suffix with ь or ъ to indicate convexity, and it fits beautifully into the spelling/pronunciation system. This is coming from someone who has learned the system, though. It would be unfair for me to say that kana is ugly, because I have not learnt how it works and therefore cannot appreciate what beauty it may have in its design.

The problem with using symbols from a different language alienates those who don't know the writing system, since it is being used in an English medium. Things would be a lot different if this forum were in, say, Japanese.
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Re: Bitruncated 5-cell

Postby quickfur » Thu Dec 11, 2008 2:17 am

wendy wrote:The PG seeks to set a way of constructing names for which the meaning is apparent from the stems of the word. For example, tetra+hedr+on gives 4 + 2d + patches, say a figure so bounded. Much research was undertaken in the linguistics of other fields of science, that made the transition to practical technology, a lot of the faults and triumphs of these endeavours form the basis of the polygloss.

That it is so dispart from the current termonology is more implication of the way that the current termonology is set.

Current terminology is one big self-contradictory tangle of inconsistencies. That is why I was only half-joking when I referred to the IUPAC--I think the study of polytopes would greatly benefit from a standardization of terminology.

From three dimensions, 2d can appear as either 2d or N-1 d. A word like "face" or "hedral" can be freely applied to 2d or N-1 d, which match in three dimensions.

Ah, yes, the classic example of the mess of self-contradicting terminology: the word "face". Traditionally, it refers to the 2D face of a polyhedron, of course, but people have extended it in completely inconsistent ways. "Facet" used to be a synonym for "face" (as in, facets of a diamond), but now it has come to mean (N-1)-face, whereas "face" itself could mean 2-face, (N-1)-face, or i-face for any i, as the abstract polytopists would have it (at least they're consistent). Coxeter, however, uses "cell" for (N-1)-face, whereas amongst 4D-centric folk, "cell" refers to a choron (a 3-face). I've seen "hypercell" being used for 4-face as well as (N-1)-face.

In six dimensions, N-1 is 5, one speaks of a 'dihedral angle' meaning the angle that five-dimensional surtopes meet.

Yes, "dihedral" is another one of those inconsistent terms: strictly speaking, a hedron should be a 2-face, so "dihedral" should be relating to two 2-faces. However, a dihedral angle thus defined has no meaning except in 3D, so it makes more sense to define it as "angle between two (N-1)-faces". Except that now "hedron" is taken to mean (N-1)-face rather than 2-face.

I have yet to see a widely-accepted term for this concept that does not contradict other uses of "hedron".

Likewise, faces are not facing things, but facets are.

Yes, so really, "face" shouldn't really apply except as an (N-1)-face (i.e., facet). Unfortunately you have the entire society of abstract polytopists contradicting such a proposition.

So nice is it to divide at the outset 2d (hedron) vs N-1 d (face), so were one to encounter hedron, one is talking invariably of something of 2d, or made of 2d things.

Would that we could apply this to "dihedral angle".

The problem of apposition (placing two numbers side by side) is always present. One has several solutions to this: eg

twenty of four-pound guns (ie 20 in number, of guns that shoot 4 lb shots)
twenty-four-pound guns ( ie guns that shoot 24-lb shots)

One method is to use 'cerimonial numbers', such as latin or greek: eg 1992 XII 31 (christmas 1992). [month in latin numbers]

The problem with polytope count eg 24ch, is that one might end with three strung numbers, eg

eighteen 24tn is 18-24-4s, ie count of eighteen + sides at 24 + patch of 4d.

The crux of the problem is that the three numbers come from three distinct domains: the first from the cardinality of the set of polytopes, the second from the cardinality of the set of facets, and the third from the ordinality of the dimension.

A straightforward solution suggests itself as using three distinct sets of number words for forming each of the three parts. A phrase like "24 icositetrachora" have this property, in that "24" comes from the Arabic numbers, "icositetra-" from the Greek prefix construction for 24, and "choron/-a" from a separate sequence -gon, -hedron, -choron, -teron, etc.. My only complaint here is the multiplicity of syllables in the second component; otherwise the system is impeccable.

Some work is done on this.

In weights and measures, one might hear references to things like "metre-tenths" or metre-tens, being E-10 m and E10 m. The idea of postfix number of this kind has been looked on favourably. This would allow the use of a greater number of compound roots, subscripted by the dimension. Some terms, like 'margin' are descending, but the ordinals serve here, eg margin-sixth = M-6 = N-8 dimensions. margin-sixths dual into edge-sixes (E+6 = 6). The terminology is set that one and first refer to the unadorned word: a margin is a margin-first (or margin-prime), while an edge-one is an edge.

The matter awaits formal consideration though.

Hmm. Is there a reason for using "margin" where others have used "ridge", for (N-2)-faces? The reciprocity of this system is nice, though. It makes comparisons of a polytope and its dual much less verbose.
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Re: Bitruncated 5-cell

Postby wendy » Thu Dec 11, 2008 9:09 am

quickfur wrote:Current terminology is one big self-contradictory tangle of inconsistencies. That is why I was only half-joking when I referred to the IUPAC--I think the study of polytopes would greatly benefit from a standardization of terminology.


I did something about it: the Polygloss.

quickfur wrote:Ah, yes, the classic example of the mess of self-contradicting terminology: the word "face".


Which is why i use 'face' roots for N-1 and 'hedron' for 2d. Still, the idiom for angle is related to the greek root (corner), is the fraction of angle around something. So a margin-angle is the angle a solid occupies at a margin (eg for {5,3,3}, 2/5 circle = 48.00, while a 'vertex angle' is the angle around a vertex, being 191/600, or 38.24 (very near 1/pi).

quickfur wrote:I've seen "hypercell" being used for 4-face as well as (N-1)-face.


hyper correctly is 'over', should only be used in the sense where an extra dimension is added to a calculation. One goes into 'hyperspace' when one invokes a third dimension to solve two dimensions. In six dimensions, it is rather silly to call the tesseract a 'hyper-cube', as it is to call a line segment a hyper-cube in three dimensions. Hyper is correctly N+1, but typically used where N=3.

quickfur wrote:Would that we could apply this to "dihedral angle".


The angle-idiom is the fraction 'around' an axis. So in the plane 'around' the margin, the faces appear as lines, and the margin as a point. This is a 2d angle, around the margin, and the angle is thus called 'margin-angle'.

quickfur wrote:Hmm. Is there a reason for using "margin" where others have used "ridge", for (N-2)-faces? The reciprocity of this system is nice, though. It makes comparisons of a polytope and its dual much less verbose.


Margin has a verb (mark), and applies to things where 'ridge' might be inappropriate. It is important to have a verb here, because delineate is related to N dimensions (ie to draw boundary marks on some N-1). So the 'margins' mark the face limits. Note that both /line/ and /edge/ have meanings of both 1 and N-2 dimensions: a deadline, to the edge and a line in the sand are actually marks (ie N-2), while a bus line, cube-edge, and railway line are edges (ie 1d).

There is also meant to be a decent-form for /frame/, that is, a latroframe is something that has 0,1 D, and a hedroframe has 0,1,2 D, but there ought be something that descends too:, ie N-1, or N-1, N-2, N-3 dimensions. This is the context that 'stellations' occur in. A stellation shares all surtopes except the vertices, while /great/ and /grand/ are for all dimensions down to 2, 3 dimensions.

The whole thing needs to be kept in duality.
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Re: Bitruncated 5-cell

Postby quickfur » Thu Dec 11, 2008 5:42 pm

wendy wrote:
quickfur wrote:Current terminology is one big self-contradictory tangle of inconsistencies. That is why I was only half-joking when I referred to the IUPAC--I think the study of polytopes would greatly benefit from a standardization of terminology.


I did something about it: the Polygloss.

Y'know, what would really help is a beginner's introduction to the polygloss. Something that will help the uninitiated into the system of terminology used in the polygloss, that maps some of the basic terms to the currently-accepted terms. While its current form is very good for reference, a reference isn't the best form of presentation to an audience that is unfamiliar with the basic concepts and basic terms. More examples of how each term is used will be very helpful. Say, use a 6-cube or something along those lines and describe concretely how each term applies to its elements. I think this will greatly help in the general adoption of the polygloss.

quickfur wrote:Ah, yes, the classic example of the mess of self-contradicting terminology: the word "face".


Which is why i use 'face' roots for N-1 and 'hedron' for 2d. Still, the idiom for angle is related to the greek root (corner), is the fraction of angle around something. So a margin-angle is the angle a solid occupies at a margin (eg for {5,3,3}, 2/5 circle = 48.00, while a 'vertex angle' is the angle around a vertex, being 191/600, or 38.24 (very near 1/pi).

Ah, I see.

quickfur wrote:I've seen "hypercell" being used for 4-face as well as (N-1)-face.


hyper correctly is 'over', should only be used in the sense where an extra dimension is added to a calculation. One goes into 'hyperspace' when one invokes a third dimension to solve two dimensions. In six dimensions, it is rather silly to call the tesseract a 'hyper-cube', as it is to call a line segment a hyper-cube in three dimensions. Hyper is correctly N+1, but typically used where N=3.

This makes sense. Still, I think "hyper" has been overused in general, and I think it's prudent to avoid it where possible.

[...]Margin has a verb (mark), and applies to things where 'ridge' might be inappropriate. It is important to have a verb here, because delineate is related to N dimensions (ie to draw boundary marks on some N-1). So the 'margins' mark the face limits. Note that both /line/ and /edge/ have meanings of both 1 and N-2 dimensions: a deadline, to the edge and a line in the sand are actually marks (ie N-2), while a bus line, cube-edge, and railway line are edges (ie 1d).

On a slightly unrelated note, it is interesting to surmise what the nature of writing in N dimensions would be, seeing that regardless of dimension, the analog of a hand holding a writing implement can only practically trace out a 1D curve (unless one goes back to fill in the hedrixes, chorixes, etc.). While it is convenient to assume the possibility of drawing in N-2 marks, I don't see how this could work in practice, since once you get past 1D marks, the great variety of possible shapes make it impossible to be simply drawn by the hypothetical (N-2) tip of the pen which must have a fixed shape. In the end, one can really only draw in 1D curves, albeit with much more possibility in the twistiness as a result of the (N-1)-surface on which it is drawn.

There is also meant to be a decent-form for /frame/, that is, a latroframe is something that has 0,1 D, and a hedroframe has 0,1,2 D, but there ought be something that descends too:, ie N-1, or N-1, N-2, N-3 dimensions. This is the context that 'stellations' occur in. A stellation shares all surtopes except the vertices, while /great/ and /grand/ are for all dimensions down to 2, 3 dimensions.

The whole thing needs to be kept in duality.

Yep.
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Re: Bitruncated 5-cell

Postby Keiji » Fri Dec 12, 2008 9:15 am

quickfur wrote:The problem with using symbols from a different language alienates those who don't know the writing system, since it is being used in an English medium. Things would be a lot different if this forum were in, say, Japanese.


As I keep saying, if you aren't familiar with Katakana, you can use the English equivalents, which are pronounced the same.
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Re: Bitruncated 5-cell

Postby wendy » Fri Dec 12, 2008 9:46 am

quickfur wrote:Y'know, what would really help is a beginner's introduction to the polygloss. Something that will help the uninitiated into the system of terminology used in the polygloss, that maps some of the basic terms to the currently-accepted terms.


I tried this on the various theme pages. There's one about time-dimensions, and about fabrics of space. More themes need to be weighed and loaded. Still a lot of work to do with glossing words. The version on the I:\local\gloss is somewhat more advanced, but some effort is being given to other pages (eg \local\os2fan2, the replacement for the rest of the pages. PG is intended to stand as a isolated work, so i am looking for technology to prepare this as a linked PDF file, for off-line use.

If you look on the front page of the index, you will see some of these introduction matter presented. It takes time, though. I've been looking at ever new matter that is arising (such as the unitary plane, which gives a useful guide to rotations of 4d via the complex euclidean plane). Much depends on insight, and understanding what goes on. But time is limited for other reasons, and one is heavily involved in DOS projects, (google os2fan2, for example), and even into things like the Wikipedia (user: Wendy.krieger, also mainly heavy into DOS) [one does not put original research onto the Wiki]. Metrology is also something one is keen into, since metrology leads to many dimensions.
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Omnitruncated 24-cell

Postby quickfur » Mon Dec 15, 2008 7:45 pm

After having computed "nice" coordinates for the omnitruncated 24-cell by means of extrapolation from the 2D orthogonal projection and suitable applications of a certain double rotation matrix derived from the symmetry of the 24-cell, I present the projection images for:

Omnitruncated 24-cell!

My favorite thing about this polychoron is the alternating structure of its hexagonal prisms, which form a beautiful network of octagonal cycles.
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Re: Omnitruncated 24-cell

Postby Keiji » Tue Dec 16, 2008 8:49 am

Wow, this one is truly beautiful. Good work! Is there any chance you'll do a side view sequence for this one?

I love those hexagonal prisms too :D
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Re: Omnitruncated 24-cell

Postby quickfur » Tue Dec 16, 2008 5:30 pm

Hayate wrote:Wow, this one is truly beautiful. Good work! Is there any chance you'll do a side view sequence for this one?

Thanks! I'm not sure about doing the side-view sequence, but it's a possibility. Right now I want to get all the uniform polychora posted first, and then I might come back and add more images for each one.

I love those hexagonal prisms too :D

I'm wondering if I should make an image with only the hexagonal prisms, with the great rhombicuboctahedra hidden. (That name is such a mouthful: great rhombicuboctahedra. There really should be a shorter name for it... I do have a pet name for it, but it involves such a horrible mangling of spelling conventions that I'd rather not inflict it upon the world. :P There's always truncated cuboctahedron, but it's not that much less verbose, and is rather inaccurate, since a literally truncated cuboctahedron is not uniform.)

(On that note, I learned recently that deriving the snub polyhedra by alternating the omnitruncate---e.g., deriving the snub cube by omnitruncating the cube and then alternating it---only gives a topological snub cube, not a uniform snub cube! I was pretty disappointed.)
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Re: Omnitruncated 24-cell

Postby wendy » Wed Dec 17, 2008 10:28 am

You can of course derive snubs by alternating the vertices of a omnitruncate.

The snub-faces form by the removal of alternate vertices = "diminishing", which typically form simplexes. The degrees of freedom correspond to the number of dimensions (ie the marked dots in the dynkin graph), while the variables to set correspond to the edges of the triangle thus formed. When N=4, E=6, so you are attempting to solve four variables in six equations.

None the same, you can make vertices "the same" in symmetry, and one has to reckon only those edges which link different vertices, eg consider the {3,3,3}, where nodes become a3b3B3A. You have vertices types a=A, b=B,and edge types aA, bB, and aB=Ba. This is three variables in two unknowns, which is usually not solved. This is a shame, because the faces of this creature consist of ten icosahedra, twenty octahedra, and sixty tetrahedra, but only topological.

You can alternate some of the vertices, as long as an even (or unmarked = 2), branch connects marked and unmarked vertices. This gives,

s3s3s4o = s3s4o3o = snub 24choron, s4o3o3o = x3o3o4o.

You can apply stott-style addition to these things too, which gives the rather interesting, although not uniform s3s4o3x. The faces of this figure consist of 24 truncated tetrahedra, 24 icosahedra, and 96 triangular cupola (cuboctahedra cut across a hexagon). Allowing for non-uniform faces, one can have the equalateral figure formed by 3xo*xx2%o, or, as Richard Klitzing writes, cube || icosahedorn. This is a lace-prism, formed by a parallel planes containing a cube and an icosahedron, with assorted triangular prisms, tetrahedra, and square pyramids forming the lacing faces.
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Re: Omnitruncated 24-cell

Postby quickfur » Wed Dec 17, 2008 5:36 pm

wendy wrote:You can of course derive snubs by alternating the vertices of a omnitruncate.

Yes, I understand that it is a general operation that works for any polytope (since an omnitruncate is always even).

The snub-faces form by the removal of alternate vertices = "diminishing", which typically form simplexes. The degrees of freedom correspond to the number of dimensions (ie the marked dots in the dynkin graph), while the variables to set correspond to the edges of the triangle thus formed. When N=4, E=6, so you are attempting to solve four variables in six equations.

None the same, you can make vertices "the same" in symmetry, and one has to reckon only those edges which link different vertices, eg consider the {3,3,3}, where nodes become a3b3B3A. You have vertices types a=A, b=B,and edge types aA, bB, and aB=Ba. This is three variables in two unknowns, which is usually not solved. This is a shame, because the faces of this creature consist of ten icosahedra, twenty octahedra, and sixty tetrahedra, but only topological.

I'm more interested in actual uniform snubs, though. What is the general approach for deriving the coordinates of, say, the uniform snub cube? And for polytopes that do not have uniform snubs, is there a way to derive the "most symmetric" form of the snub generated by the alternation of its omnitruncate?

You can alternate some of the vertices, as long as an even (or unmarked = 2), branch connects marked and unmarked vertices. This gives,

s3s3s4o = s3s4o3o = snub 24choron, s4o3o3o = x3o3o4o.

You can apply stott-style addition to these things too, which gives the rather interesting, although not uniform s3s4o3x. The faces of this figure consist of 24 truncated tetrahedra, 24 icosahedra, and 96 triangular cupola (cuboctahedra cut across a hexagon). Allowing for non-uniform faces, one can have the equalateral figure formed by 3xo*xx2%o, or, as Richard Klitzing writes, cube || icosahedorn. This is a lace-prism, formed by a parallel planes containing a cube and an icosahedron, with assorted triangular prisms, tetrahedra, and square pyramids forming the lacing faces.

This is interesting; recently on Wikipedia somebody came up with 4D cupolas which are generated by expanding (runcinating) 4D pyramids. I wonder how many Johnson polychora there are, whose facets, to retain consistency with the generalization of the Archimedean polyhedra, are allowed to be any Johnson solid.

In particular, I'm quite curious as to whether there are vertex-transitive polytopes whose facets are not uniform (this is a possibility, e.g., if you take a Johnson solid whose vertices aren't transitive, say they are of two types X and Y, you can, when forming the polychoron, join the type X vertices of one facet with the type Y vertices of another, and thereby make the result vertex transitive---at least, this is why I think it may be possible, but I don't know if there are actual examples of such polytopes).
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Re: Omnitruncated 24-cell

Postby Keiji » Wed Dec 17, 2008 6:02 pm

quickfur wrote:That name is such a mouthful: great rhombicuboctahedra. There really should be a shorter name for it...


koto is its English kana mnemonic. I really don't see why you don't use them =p
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Re: Omnitruncated 24-cell

Postby quickfur » Wed Dec 17, 2008 6:48 pm

Hayate wrote:
quickfur wrote:That name is such a mouthful: great rhombicuboctahedra. There really should be a shorter name for it...


koto is its English kana mnemonic. I really don't see why you don't use them =p

For the same reason I don't use my own pet name for it---I do have short names for these things, y'know! :)
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Re: Omnitruncated 24-cell

Postby Keiji » Wed Dec 17, 2008 9:44 pm

You can't exactly complain that there should be a shorter name when we have what, probably 3 short names for it now, but you just refuse to use them :\
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Re: Omnitruncated 24-cell

Postby wendy » Thu Dec 18, 2008 12:11 am

quickfur wrote:In particular, I'm quite curious as to whether there are vertex-transitive polytopes whose facets are not uniform (this is a possibility, e.g., if you take a Johnson solid whose vertices aren't transitive, say they are of two types X and Y, you can, when forming the polychoron, join the type X vertices of one facet with the type Y vertices of another, and thereby make the result vertex transitive---at least, this is why I think it may be possible, but I don't know if there are actual examples of such polytopes).


There are indeed examples, such as s3s4o3x. This has uniform vertices, and 24 truncated tetrahedra, 96 triangular cupola, and 24 icosahedra as faces. The vertices are all uniform. The icosahedra do not touch, so there are 12*24 vertices. At each of these vertices, we see one icosahedron, three cupola (one top and two bottoms), and 96 triangular prisms = 248 * 2 / 6. The vertex figure is formed by assembling these peices: one pentagon. one rectangle 1*r2, two triangles 1:r2:r2; two triangles 1:r2:r3, and one triangle 1:r3:r3.

Assemble so that the pentagon and 1:r3:r3 triangle are opposite short sides of the rectangle. These can be folded to touch. The length between the unconnected vertices of the pentagon and the triangle-rectangle vertex is r2, which leads to slots for the remaining four faces.
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Re: Omnitruncated 24-cell

Postby quickfur » Thu Dec 18, 2008 6:41 pm

wendy wrote:
quickfur wrote:In particular, I'm quite curious as to whether there are vertex-transitive polytopes whose facets are not uniform[...]

There are indeed examples, such as s3s4o3x. This has uniform vertices, and 24 truncated tetrahedra, 96 triangular cupola, and 24 icosahedra as faces. The vertices are all uniform. The icosahedra do not touch, so there are 12*24 vertices. At each of these vertices, we see one icosahedron, three cupola (one top and two bottoms), and 96 triangular prisms = 248 * 2 / 6. The vertex figure is formed by assembling these peices: one pentagon. one rectangle 1*r2, two triangles 1:r2:r2; two triangles 1:r2:r3, and one triangle 1:r3:r3.

Assemble so that the pentagon and 1:r3:r3 triangle are opposite short sides of the rectangle. These can be folded to touch. The length between the unconnected vertices of the pentagon and the triangle-rectangle vertex is r2, which leads to slots for the remaining four faces.

Oooh, shiny! I'm going to try computing the coordinates for this so that I can render it with my polytope viewer.
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Re: Omnitruncated 24-cell

Postby wendy » Fri Dec 19, 2008 9:03 am

The polytope s3s4o3x is formed by a Stott sum, being s3s4o3o + o3o4o3x. This leads to an easy way to find the coordinates.

The coordinate system for this figure is pyritochoral, [3,3+, 4], that is, all values are given with All changes of sign, and Even permutations.

The vertices of s3s4o3o are (0,v,1,f), where v = 0.618033&c, f = 1.618033&c. The required Stott sum corresponds to placing an icosahedral prism of some height, onto each of the icosahedral faces of the s3s4o3o. This means, for each vertex near a given length, we add either x,x,x,x or 2x,0,0,0, such that the distance between the tops of the prisms become that of the icosahedron, ie 2v. So we have eg

0,v,1,f + x,x,x,x => x, v+x, 1+x, f+x ; which gives 192 vertices (twe 172)
0,v,1,f + 0,0,0,2x => 0, v, 1, f+2x. ; which gives 96 vertices (twe 96)

All vertices are thus accounted for. The edge of the icosahedron is, eg 0,v,1,f to 0,-v,1,f; some 2v.

The vector between adjacent icosahedra is the difference between these, which is x, x, x, -x. The length of this vector is 2x, and therefore we see 2x=2v, makes x=v. We then replace x with v in the above, to give

(v, 2v, f, r5) and (0, v, 1, r5+v), where r5+v gives 3.854101966.
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Truncated 5-cell

Postby quickfur » Sat Dec 27, 2008 1:43 am

And now, for another little diversion, here are the projections of the truncated 5-cell

http://eusebeia.dyndns.org/4d/trunc5cell.html

Check out the cool animation at the end!

The 5-cell truncates are nice and simple polytopes, and quite pretty in their own way.
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Re: Quickfur's renders

Postby Keiji » Sat Dec 27, 2008 2:43 am

Since this forum was getting a bit cluttered, I decided to merge all the topics about your renders together and sticky it. Hope you don't mind!
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Re: Quickfur's renders

Postby quickfur » Tue Dec 30, 2008 1:44 am

Hayate wrote:Since this forum was getting a bit cluttered, I decided to merge all the topics about your renders together and sticky it. Hope you don't mind!

I get a sticky? Aww that's nice. :)

I've added a few more uniform polychora (bitruncated tesseract and runcinated tesseract); update news are on the main page.

My goal is to eventually cover all 64 convex uniform polychora (64 not counting the infinite families of duoprisms and antiprism prisms). After I get basic renders for all of them, I'm planning to revisit each one and add more renders from different viewpoints, etc..
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Omnitruncated 5-cell

Postby quickfur » Mon Jan 19, 2009 5:33 pm

I've done some renders of the omnitruncated 5-cell:

http://eusebeia.dyndns.org/4d/omni5cell.html

It has a similar network of alternating hexagonal prisms as the omnitruncated 24-cell, although not quite on the same scale. Still, it's a cute little uniform polychoron.
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