Quickfur's renders

[Klitzing]

That's amazing Quickfur!

How much on the cutting edge (literally as well as figuratively I guess lol) are you with this rendering program of yours?

Are you breaking new ground that no other program has done before elsewhere in the world?

When you say that you "could've just downloaded precomputed coordinates off the 'Net somewhere" does that mean that this has already been done in great detail by other programs and supercomputers?

Hmmm, I do not know exactly, as I dont have R.Webbs "great stella" software myself, but as to what I remember of having read about it, it is able to export all its constructions as off-files, i.e. essentially providing coordinates and incidences. And it has plenty of uniform polychora already built in, that is you dont even have to construct them on your own. - Sure it is kind a competitor to your renderings. But I like your pix too. So there might be some crosswise input...

--- rk

[Klitzing]

Just to point out some errors in an older mail of yours:

[...]then I'll get to the 4_21 in 7D.[...]

4_21 is in 8d

[...]However, the snub 24-cell in 4D is not the alternation of an omnitruncate.[...]

Sure it is! Dont be misdirected from s3s4o3o, but it could be rewritten as s3s3s *b3s, i.e. it is the alternation of the omnitruncate of the bifurcated Dynkin graph.

BTW, even the tet is a true snub! Did you know? - Just consider s2s2s.
In fact just a special case of what you mentioned later down in that mail of yours: the antiprisms: s2sNs

--- rk

[quickfur]

[...] Well, "teddi" is the Bowers acronym for the tridiminished icosahedron. Even so "Teddy" has a slightly different spelling, its direct application onto its 4d counterpart seems missleading and confusing. I would prefer "Teddy" just being a petname for "teddi" itself. Yesterday I suggested the 4d fellow to be called a "4d Teddy", note the slight difference.

In that case, I should write "4-teddy" to refer to the 4D counterpart. And 3-teddy for the 3D counterpart, just to avoid ambiguity. We already have tigers in our forest, so why not have teddies as well?

As to "Teddies all the way up", yep, thats true. This is just Wendys sequence:
xfo&#xt > xfo3oox&#xt > xfo3oox3ooo&#xt > xfo3oox3ooo3ooo&#xt > ...
I never have verified whether those are possible without a kink at the middle layer, so.
But I dont see what could go wrong here.

What if the distance between the tips of pentagons (opposite the edges of the x3o3o) of two adjacent 3-teddies does not equal the edge length?

The same holds true for the xfo3oox4ooo&#xt you doubted to be valid:
there the bottom figure is an oct. Onto its faces teddis are attached and folded up to connect at their pentagons. This partial complex is fully valid. And even the tips of those pentagons would connect (as faces are planar). The top faces of those teddis are triangles again, but anti-aligned to the bottom ones. In that very case there are 4 teddies symmetrically around every lower lacing edge of those teddis. Accordingly there emanate 4 triangles from this upper lacing edge vertex onto those anti-aligned top-face triangles. And as those top-face triangels are connected tip-wise (teddis do connect at the pentagons), the remaining space can be filled by squippies, and all then be completed by that final base, a co.
(Note that this o3x4o has its triangles truely ant-aligned to the triangles of the opposite layer, the x3o4o!)
Therefore I suppose you got some calculation error somewhere...[...]

My arithmetic/algebra skills are definitely very error-prone, I'll admit that. I'll try to redo the calculations just to be sure. But you have only proven that the construction is topologically possible (which I don't doubt at all); the question, though, is whether all edges can be made equal length, particularly the edges between the tips of pentagons of two adjacent 3-teddies, since otherwise, the bottom cell o3x4o will have edge lengths unequal to the rest of the polychoron, so it will not be CRF.

EDIT: apparently my analytical skills are just as bad as my arithmetic skills. I inverted the sign of the height of the cuboctahedron, that's why the convex hull algo produced a non-CRF. You are absolutely right, the octahedral counterpart of 4-teddy exists, and is CRF.

[Keiji]

To get this back on topic, can we have a render of this newfound OctoTeddy™?

[quickfur]

So, to redeem myself from the embarrassment of flipping the sign of the height of the cuboctahedron layer in xfo3oox4ooo&#xt, resulting in the convex hull being non-CRF, here's solid proof that the thing exists and is CRF:



The cuboctahedral cell is outlined in red; the 3-teddy's slightly overhang it, so the square pyramids protrude from the cuboctahedron in this projection.

Here's a side-view:



I colored the cuboctahedron green so that it's easier to discern the cells. I purposely centered the projection on one of the edges surrounded by four 3-teddies, so that they're easier to pick out.

To get this back on topic, can we have a render of this newfound OctoTeddy™?

Haha, I was just writing this post when you replied. Great minds think alike? (Or in my case, fools seldom differ? )

Anyway, images are not proof of anything. The real proof is concrete Cartesian coordinates:

Code: Select all

# Top octahedron of edge length 2
< sqrt(2),  0,  0,  sqrt(2*phi)>
<-sqrt(2),  0,  0,  sqrt(2*phi)>
< 0,  sqrt(2),  0,  sqrt(2*phi)>
< 0, -sqrt(2),  0,  sqrt(2*phi)>
< 0,  0,  sqrt(2),  sqrt(2*phi)>
< 0,  0, -sqrt(2),  sqrt(2*phi)>

# Middle octahedron: scaled by golden ratio
< sqrt(2)*phi,  0,  0,  0>
<-sqrt(2)*phi,  0,  0,  0>
< 0,  sqrt(2)*phi,  0,  0>
< 0, -sqrt(2)*phi,  0,  0>
< 0,  0,  sqrt(2)*phi,  0>
< 0,  0, -sqrt(2)*phi,  0>

# Bottom cuboctahedron:
<  0,  sqrt(2),  sqrt(2), -sqrt(2/phi)>
<  0,  sqrt(2), -sqrt(2), -sqrt(2/phi)>
<  0, -sqrt(2),  sqrt(2), -sqrt(2/phi)>
<  0, -sqrt(2), -sqrt(2), -sqrt(2/phi)>
<  sqrt(2),  0,  sqrt(2), -sqrt(2/phi)>
<  sqrt(2),  0, -sqrt(2), -sqrt(2/phi)>
< -sqrt(2),  0,  sqrt(2), -sqrt(2/phi)>
< -sqrt(2),  0, -sqrt(2), -sqrt(2/phi)>
<  sqrt(2),  sqrt(2),  0, -sqrt(2/phi)>
<  sqrt(2), -sqrt(2),  0, -sqrt(2/phi)>
< -sqrt(2),  sqrt(2),  0, -sqrt(2/phi)>
< -sqrt(2), -sqrt(2),  0, -sqrt(2/phi)>


[Keiji]

Okay, I'm sold:

tridiminished icosahedron = triangular teddy or trigonal teddy
"4-teddy" = tetrahedral teddy or pyrohedral teddy
OctoTeddy™ = octahedral teddy or aerohedral teddy

And of course, there's the icosahedral teddy or hydrochoric teddy, which was mentioned on the wiki page.

I think the naming issue is resolved now

[quickfur]

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).

You not even are restricted to even faces. You just have to apply alternation on a local level, and doing so then running around the original polytope. If that one somewhere has a odd face, you accordingly will have to run twice around the original polytope in order to come back to the beginning place with the same parity. - This is what Norman has contributed to snubbing theory, and what he calls a holosnub.

That is cool. But I would think the result would (usually) be non-convex?

[...] Yes, there is a gap between the application of an alternated facetting (which is applicable always) and its afterward topological deformation towards an uniform variant (which depends mainly on the degrees of freedom, but also, esp. when applied to starry forms, where the "edges underneath" get rather small, become zero or get retrograde, the "uniformisation", even if possible, becomes umbiguous). Cf. the corresponding outline within my article on that topic: "Snubs, Alternated Facetings, and Stott-Coxeter-Dynkin Diagrams", by Dr. R. Klitzing, Symmetry: Culture and Science, vol. 21, no.4, 329-344, 2010
(pdf available at http://bendwavy.org/klitzing/pdf/Stott_v8.pdf.

Thanks for the link; I'll have to read that paper sometime.

Wow, so much to do all of a sudden! This forum was dead quiet over the summer and the past winter term, and now suddenly there's so much to catch up on. My free time is all soaked up!

[...]

This is interesting; recently on Wikipedia somebody came up with 4D cupolas which are generated by expanding (runcinating) 4D pyramids.

Hmmm. There are different possible extrapolations of 3d concepts into 4d. Not only with respect to Johnson solids, leading either to the Blind polychora or to the CRFs, but even in what to call a 4d antiprism, and, what is relevant here, a cupola. I had outlined this within my original paper on segmentochora, starting at the bottom of page 6 of http://bendwavy.org/klitzing/pdf/artConvSeg_7.pdf. There too are arguments provided why to use them in the way I did - which contradict the usage you cited!

Yeah, generalizing 3D terminology has a history of ambiguity. I didn't realize you were using "cupola" in a different sense.

I'm partial to Stott's construction -- I think expansion makes more sense in general, esp. for things like the "truncated cuboctahedron" where expansion automatically yields coordinates with the required uniform proportions, whereas truncating a cuboctahedron requires some ill-defined deformation afterwards. Expansion is also easier to generalize to higher dimensions, since the kernel of intersection of higher-dimensional polytope duals may not yield CRFs of the requisite proportions. But this is just my opinion, of course.

In any case, it's clear from your list of segmentochora that the division of segmentotopes into pyramids, cupola, etc., is really quite artificial, since in 4D and higher there are multiple constructions that generalizes the categories in different ways. In fact, shortly before I read your paper on segmentochora, I had realized that any pair of uniform polytopes with the same symmetry group whose circumradii have a difference less than the edge length will have a corresponding "cupola-like" CRF in the next higher dimension. This general concept subsumes pyramids (if we consider the CD diagram ooo...o to be a point), cupola in the expansion sense, cupola in the kernel of intersection sense, and perhaps includes a host of other less-obvious constructions as well.

Perhaps the best thing to do is to extend the scope of the term "cupola" to cover all of these constructions, rather than impose artificial limitations on what it applies to?

[...] 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).

Hehe, there are such special thingies. Those are called scaliform polychora (formerly negatively being attributed as weakly uniform ones). Scaliform polychora are allowed to use Johnson solids for faces, yet require to be vertex transitive in their overall symmetry.

Yes, thanks to wintersolstice, since the time I wrote that I've learned about the bi-icositetradiminished 600-cell (the polychoron with 48 transitive 3-teddies), and 2 or 3 others that I've yet to spend time to study.

To provide an example consider prissi (http://bendwavy.org/klitzing/incmats/prissi.htm), which uses ikes, trips, tuts, and tricues for facets. (That one occured within my alternated faceting research. It indeed can be deformed to have unit edges throughout.)

An other example is the segmentochoron tut || inv tut (http://bendwavy.org/klitzing/incmats/tut=invtut.htm). That one has tuts, tets, and tricues for facets.

--- rk

Yes I believe wintersolstice listed these as well. I just haven't gotten to study them yet. So many interesting CRFs, so little time!

[quickfur]

[...]
Hmmm, I do not know exactly, as I dont have R.Webbs "great stella" software myself, but as to what I remember of having read about it, it is able to export all its constructions as off-files, i.e. essentially providing coordinates and incidences. And it has plenty of uniform polychora already built in, that is you dont even have to construct them on your own. - Sure it is kind a competitor to your renderings. But I like your pix too. So there might be some crosswise input...

--- rk

I don't have R. Webb's "great stella" mainly because it requires Windows, but I primarily use Linux. My wife's laptop has windows, but most of the time she's using it and I don't really like using windows anyway.

As for competition, I don't think my renderer would tally up. From what I can tell, Webb's "great stella" (or stella4d) is much more advanced than my renderer, being able to handle non-convex polytopes, and analyzing them to identify cells with different symmetry roles, etc.. And having a user-friendly interface to boot. My renderer is just a command-line tool that requires some complicated scripting to use; the output is povray models which is good for models with lots of transparency, that I like to use, but it's not exactly the most interactive thing. I would imagine most people prefer more interactive software where they can play around with the model in real-time.

I did have plans for writing an interactive application for exploring 4D polytopes (and higher), but I found programming 3D hardware directly quite difficult and requires too much effort -- whereas using povray I can just set the colors, set the transparencies, generate the mesh, and off I go. No need to worry about sorting triangles back-to-front, how to compose overlapping polygons to simulate transparency, optimizing GPU performance, and that sorta thing.

[quickfur]

Just to point out some errors in an older mail of yours:

[...]then I'll get to the 4_21 in 7D.[...]

4_21 is in 8d

Yes, yet another symptom of me posting without thinking things through.

[...]However, the snub 24-cell in 4D is not the alternation of an omnitruncate.[...]

Sure it is! Dont be misdirected from s3s4o3o, but it could be rewritten as s3s3s *b3s, i.e. it is the alternation of the omnitruncate of the bifurcated Dynkin graph.

True! I didn't think of that!

BTW, even the tet is a true snub! Did you know? - Just consider s2s2s.
In fact just a special case of what you mentioned later down in that mail of yours: the antiprisms: s2sNs
[...]

This one is interesting. I've noticed that in some 4D CRFs, the tetrahedral cells function as a kind of "line segment antiprism", for example in segmentochora of the form polyhedron||dual_polyhedron, where they serve to connect edges of one orientation in one layer to edges rotated 90° in the other layer. The rest of the lacing cells are then just pyramids that connect the faces of one layer to vertices of the other. So in some sense, these segmentochora are kinda like 4D counterparts of antiprisms.

[quickfur]

Okay, I'm sold:

tridiminished icosahedron = triangular teddy or trigonal teddy
"4-teddy" = tetrahedral teddy or pyrohedral teddy
OctoTeddy™ = octahedral teddy or aerohedral teddy

And of course, there's the icosahedral teddy or hydrochoric teddy, which was mentioned on the wiki page.

I think the naming issue is resolved now

Heh. So now we have tigers and teddies in our forest. What other animals are out there?

I'm not fully convinced with this naming scheme, though. For one thing, we don't know if other similar shapes exist that don't have 3-teddies as cells. The teddies seem to be modified forms of the rotundae, where the base cell has been "contracted" or shrunken in some way. Are there any other kinds of contractions of rotundae that will yield CRFs?

And don't forget the "runcinated" tetrahedral teddy that I wrote about in the other thread -- I'm going to construct the coordinates for that one soon -- where you basically insert pentagonal prisms between the 3-teddies. The pyramids then turn into cupolae, and the top and bottom cells get expanded. In fact, this isn't limited to the tetrahedral teddy; I believe the construction also yields CRFs for the octahedral and icosahedral teddies. So what should we call these things? Fat-teddies?

[Marek14]

Is there any way to make a shape we could call "polar teddy"?

[quickfur]

Is there any way to make a shape we could call "polar teddy"?

Well let's see. A "pole" can be represented by a line segment, and so to use Wendy's construction of the teddies, we start with a line segment of length 1, which is our first layer, then a line segment of length phi (phi=golden ratio), which is our second layer, then a rectified line segment, which is a point. Adjusting the separations between the layers to get uniform edge lengths, we get ...

... a pentagon!

So it looks like a polar teddy is a pentagon.

[Keiji]

I'm not fully convinced with this naming scheme, though. For one thing, we don't know if other similar shapes exist that don't have 3-teddies as cells.

That didn't stop the tigers.

And don't forget the "runcinated" tetrahedral teddy that I wrote about in the other thread -- I'm going to construct the coordinates for that one soon -- where you basically insert pentagonal prisms between the 3-teddies. The pyramids then turn into cupolae, and the top and bottom cells get expanded. In fact, this isn't limited to the tetrahedral teddy; I believe the construction also yields CRFs for the octahedral and icosahedral teddies. So what should we call these things? Fat-teddies?

Well, you could just call them expanded teddies.

What happens if you perform this operation on the 3D teddy, or is it invalid for some reason?

[quickfur]

I'm not fully convinced with this naming scheme, though. For one thing, we don't know if other similar shapes exist that don't have 3-teddies as cells.

That didn't stop the tigers.

True. Though eventually there were tigroid shapes. Does that mean we will find teddroid shapes too?

On another note, we could adopt the Latin root for bears: ursa. So a tetrahedral teddy can be a tetrahedral ursachoron, an octahedral teddy can be an octahedron ursachoron, etc.. Or use "ursamorph" for analogy with xylomorph, hydromorph, etc. of Tamfang's nomenclature. At least we won't get laughs in the audience when we bring up a page titled "tetrahedral ursamorph" (as opposed to "tetrahedral teddy").

And don't forget the "runcinated" tetrahedral teddy that I wrote about in the other thread -- I'm going to construct the coordinates for that one soon -- where you basically insert pentagonal prisms between the 3-teddies. The pyramids then turn into cupolae, and the top and bottom cells get expanded. In fact, this isn't limited to the tetrahedral teddy; I believe the construction also yields CRFs for the octahedral and icosahedral teddies. So what should we call these things? Fat-teddies?

Well, you could just call them expanded teddies.

Actually, on second thought, maybe we don't need a separate name; specifying the top cell already determines what form it will be. A cuboctahedral teddy can only be an expanded tetrahedral teddy, since trying to apply Wendy's construction to a cuboctahedron will create 3D square teddies which are not CRF.

What happens if you perform this operation on the 3D teddy, or is it invalid for some reason?

The only teddy that's CRF in 3D is the triangular teddy. The square teddy has isosceles triangles at the bottom, and the pentagonal teddy is a dodecahedron with 5 vertices deleted. You get a pentagonal face with large edge length than the others.

I think the reason is because the only CRF pyramid in 3D is the line pyramid, whereas in 4D, you have three pyramids to go along with the 3 regular deltahedra (tetrahedron, octahedron, icosahedron): triangular pyramid, square pyramid, pentagonal pyramid.

I don't know if the 5D teddies will be CRF as well, but since we have CRF tetrahedral, octahedral, and icosahedral pyramids, I assume things should fit in properly. Can't be sure until actual coordinates are computed, though.

[Keiji]

That didn't stop the tigers.

True. Though eventually there were tigroid shapes. Does that mean we will find teddroid shapes too?

On another note, we could adopt the Latin root for bears: ursa. So a tetrahedral teddy can be a tetrahedral ursachoron, an octahedral teddy can be an octahedron ursachoron, etc.. Or use "ursamorph" for analogy with xylomorph, hydromorph, etc. of Tamfang's nomenclature. At least we won't get laughs in the audience when we bring up a page titled "tetrahedral ursamorph" (as opposed to "tetrahedral teddy").

Although this is now going to make me think of Pokémon every time.

Well, you could just call them expanded teddies.

Actually, on second thought, maybe we don't need a separate name; specifying the top cell already determines what form it will be. A cuboctahedral teddy can only be an expanded tetrahedral teddy, since trying to apply Wendy's construction to a cuboctahedron will create 3D square teddies which are not CRF.

I'd rather not; who says shapes must be CRF? A cuboctahedral teddy should be a cuboctahedral teddy regardless of the fact that it's non-CRF, just like a heptagonal pyramid is still a heptagonal pyramid. The unambiguous way to name it would be tetrahedral expanded teddy - specifically not expanded tetrahedral teddy because "expanded" operates on the word or phrase after it, so putting it first would create ambiguity.

What happens if you perform this operation on the 3D teddy, or is it invalid for some reason?

The only teddy that's CRF in 3D is the triangular teddy. The square teddy has isosceles triangles at the bottom, and the pentagonal teddy is a dodecahedron with 5 vertices deleted. You get a pentagonal face with large edge length than the others.

Yes, I recognise this, but it's not what I asked - is there such a thing as a trigonal expanded teddy (whether CRF or not)?

[quickfur]

That didn't stop the tigers.

True. Though eventually there were tigroid shapes. Does that mean we will find teddroid shapes too?

On another note, we could adopt the Latin root for bears: ursa. So a tetrahedral teddy can be a tetrahedral ursachoron, an octahedral teddy can be an octahedron ursachoron, etc.. Or use "ursamorph" for analogy with xylomorph, hydromorph, etc. of Tamfang's nomenclature. At least we won't get laughs in the audience when we bring up a page titled "tetrahedral ursamorph" (as opposed to "tetrahedral teddy").

Although this is now going to make me think of Pokémon every time.

lol... maybe we should stick with "ursachoron"? Or is that still too pokémon like?

Well, you could just call them expanded teddies.

Actually, on second thought, maybe we don't need a separate name; specifying the top cell already determines what form it will be. A cuboctahedral teddy can only be an expanded tetrahedral teddy, since trying to apply Wendy's construction to a cuboctahedron will create 3D square teddies which are not CRF.

I'd rather not; who says shapes must be CRF? A cuboctahedral teddy should be a cuboctahedral teddy regardless of the fact that it's non-CRF, just like a heptagonal pyramid is still a heptagonal pyramid. The unambiguous way to name it would be tetrahedral expanded teddy - specifically not expanded tetrahedral teddy because "expanded" operates on the word or phrase after it, so putting it first would create ambiguity.

OK. But I don't like the word "expand" because it's ambiguous. A 5D teddy can be runcinated and still be a teddy-like shape, for example.

What happens if you perform this operation on the 3D teddy, or is it invalid for some reason?

The only teddy that's CRF in 3D is the triangular teddy. The square teddy has isosceles triangles at the bottom, and the pentagonal teddy is a dodecahedron with 5 vertices deleted. You get a pentagonal face with large edge length than the others.

Yes, I recognise this, but it's not what I asked - is there such a thing as a trigonal expanded teddy (whether CRF or not)?

Certainly -- you'll get something with a hexagon on top, 3 pentagons around every other edge, with squares in between, and skirting the bottom you'll get trapeziums, and another hexagon at the bottom.

In fact, if you cantellate the tetrahedral teddy, you'll get a shape that has this trigonal expanded teddy as cells (along with a bunch of trigonal prisms and pentagonal prisms, etc). So "expanded" is ambiguous above 3D.

[Keiji]

lol... maybe we should stick with "ursachoron"? Or is that still too pokémon like?

Both were, but it doesn't matter. Mainly because of my next comment:

OK. But I don't like the word "expand" because it's ambiguous. A 5D teddy can be runcinated and still be a teddy-like shape, for example.

That's why I put the "expanded" immediately before the "teddy".

However, I accept that there's no way to distinguish between a teddy that has subsequently been expanded ("expanded (________ teddy)") and a teddy of an expanded polytope ("(expanded ________) teddy") with that nomenclature. Therefore, I now propose:

teddy = ursatope or ursamorph
expanded teddy = ursa____tope or ursa____morph where ____ is some word meaning "outside", which I also need to use for the xoo...oox uniforms.

My first thought, the obvious one, is "exo", but that doesn't go well (it creates "ursaexomorph", which gets shortened to "ursexomorph", which is rather unpleasant), although Google provides a number of alternatives. I considered Japanese, but that is "soto", and "ursasototope" is... just a little too alliterate for my liking, although "ursasotomorph" isn't so bad.

So, if we can find a decent word here, we're all good

Certainly -- you'll get something with a hexagon on top, 3 pentagons around every other edge, with squares in between, and skirting the bottom you'll get trapeziums, and another hexagon at the bottom.

Ah yes, of course. I pretty much knew all that, but for some reason I couldn't piece it together and visualise it until you put it in your own words. And this isn't even 4D we're talking about!

[quickfur]

[...]

OK. But I don't like the word "expand" because it's ambiguous. A 5D teddy can be runcinated and still be a teddy-like shape, for example.

That's why I put the "expanded" immediately before the "teddy".

However, I accept that there's no way to distinguish between a teddy that has subsequently been expanded ("expanded (________ teddy)") and a teddy of an expanded polytope ("(expanded ________) teddy") with that nomenclature. Therefore, I now propose:

teddy = ursatope or ursamorph
expanded teddy = ursa____tope or ursa____morph where ____ is some word meaning "outside", which I also need to use for the xoo...oox uniforms.

How does that deal with cantellated, runcinated, etc., teddies? Isn't it just still equivalent to "expanded ... ursatope"?

My first thought, the obvious one, is "exo", but that doesn't go well (it creates "ursaexomorph", which gets shortened to "ursexomorph", which is rather unpleasant), although Google provides a number of alternatives. I considered Japanese, but that is "soto", and "ursasototope" is... just a little too alliterate for my liking, although "ursasotomorph" isn't so bad.

So, if we can find a decent word here, we're all good

Well, let's run through the list of languages that I know or am somewhat familiar with:

English: ursa-out-tope > ursaouttope (sounds funny and looks weird)
Greek: ursa-ex-tope - you already discounted this one.
Latin: ditto (N.B. I don't actually know any Latin).
Russian: вы- (vy "out, out of"), внешный (vneshnyi "outside"), наружный, снаружи (naruzhnyi, snaruzhnyi): none of them looks (or sounds) particularly attractive in English.
Mandarin: wai (pronounced like "why") - not that great either: ursawaitope looks and sounds weird.
Hokkien: gua - ursaguatope. Surprisingly, this one actually doesn't look too bad.
Malay/Indonesian: luar, diluar: ursaluartope. Doesn't sound too bad, but looks a bit strange. Or "pinggir" (edge, boundary): Ursapinggirtope - looks totally bizarre.
Tagalog: labas - ursalabatope? Seems workable.
Latvian: ara: ursaaratope > ursaratope. Hmm. This might actually work! (Caveat: I don't know any Latvian. I got this from Google Translate. )
Croatian: vanjski: maybe vanatope? Ursavanatope. Sounds nice! (except for the unfortunate substring coincidence with "savana", which might give the wrong idea).
Estonian: välis-: hmm, valitope? Ursavalitope. Not too bad.
Finnish: ulko-: ulkotope. Ursaulkotope > ursalkotope. Not too bad, just looks a bit strange.
Swedish: ute-: utetope. Ursautetope > ursutetope. Not too bad, but a bit awkward.
Romanian: periferic: maybe peritope? Peri- is also nice for "outside": perimeter. Ursaperitope. Hey I like that! Plus, "peri" is also a Greek prefix, so it's not too incongruous with the rest of the derivations (which are generally drawn from Latin/Greek). Huh, why didn't I think of this earlier.

I vote for ursaperitope! Ursavalitope would be my 2nd choice. Then maybe ursalabatope.

P.S. Romanian also has marginas-: so maybe ursamarginotope might work too. But it's a bit verbose, and maybe confusing if you start thinking of marginalized polytopes...

(OK, I lied. I don't know most of the above languages. I was just using Google Translate. )

Certainly -- you'll get something with a hexagon on top, 3 pentagons around every other edge, with squares in between, and skirting the bottom you'll get trapeziums, and another hexagon at the bottom.

Ah yes, of course. I pretty much knew all that, but for some reason I couldn't piece it together and visualise it until you put it in your own words. And this isn't even 4D we're talking about!

I have the opposite problem. I can visualize things quite well, but trip over the simplest calculations that's anything harder than 1+1=2. (Even that I sometimes get wrong!) I know how to do the algebra, but I keep making silly mistakes everywhere.

[Keiji]

expanded teddy = ursa____tope or ursa____morph where ____ is some word meaning "outside", which I also need to use for the xoo...oox uniforms.

How does that deal with cantellated, runcinated, etc., teddies? Isn't it just still equivalent to "expanded ... ursatope"?

Uh... hm. It doesn't at all. I don't know why I thought that.

I vote for ursaperitope! Ursavalitope would be my 2nd choice. Then maybe ursalabatope.

I like ursaperitope. The other two make me think of saliva. I make associations too easily...

I have the opposite problem. I can visualize things quite well, but trip over the simplest calculations that's anything harder than 1+1=2. (Even that I sometimes get wrong!) I know how to do the algebra, but I keep making silly mistakes everywhere.

I mess up my calculations all the time to. But I probably double and triple check them a lot more than you (for instance, in that counting edges issue, I checked everything at least once before posting it on the wiki, again when you said there were 216 edges, and another two times before posting the new topic for it which still has no replies, cough cough).

Then again, I also spend far too much time checking that doors are locked and things I need are with me, so I'm just a generally paranoid person

[quickfur]

expanded teddy = ursa____tope or ursa____morph where ____ is some word meaning "outside", which I also need to use for the xoo...oox uniforms.

How does that deal with cantellated, runcinated, etc., teddies? Isn't it just still equivalent to "expanded ... ursatope"?

Uh... hm. It doesn't at all. I don't know why I thought that.

I vote for ursaperitope! Ursavalitope would be my 2nd choice. Then maybe ursalabatope.

I like ursaperitope. The other two make me think of saliva. I make associations too easily...

lol... OK, then ursaperitope it is. (It does make me think of bears peering through periscopes, though. Not that it's a bad thing.)

And I suppose now you'd need a whole system of infixes to deal with whether the teddies were expanded before, after, or somewhere in between, etc..

I have the opposite problem. I can visualize things quite well, but trip over the simplest calculations that's anything harder than 1+1=2. (Even that I sometimes get wrong!) I know how to do the algebra, but I keep making silly mistakes everywhere.

I mess up my calculations all the time to. But I probably double and triple check them a lot more than you (for instance, in that counting edges issue, I checked everything at least once before posting it on the wiki, again when you said there were 216 edges, and another two times before posting the new topic for it which still has no replies, cough cough).

Well, I wasn't ignoring that topic, if that's what you're hinting at. I was going to reply that perhaps you miscounted due to not taking into the wrapping around of the 3-sphere, but after thinking it through (surprising for me!) again, I realized that that wasn't the source of the discrepancy. So I still have no idea what went wrong with your calculation.

The only thing I can think of is that maybe there are more than two kinds of edges after all? I do note that the polychoron is chiral. Maybe that would cause some edges to be different, that would otherwise be considered the same?

Then again, I also spend far too much time checking that doors are locked and things I need are with me, so I'm just a generally paranoid person

Haha... I think I'm pretty paranoid too... in certain things but not others. I keep online passwords in files encrypted with GPG using 1024-bit keys with 25+ character passphrases, but then the passwords themselves are easy-to-guess ones. I'm weird.

[Keiji]

The BXD is even more interesting...

I believe this is the first polytope I've heard of which actually identifies element types that were distinct in its cells - namely, the way the top and bottom faces of the teddies become one and the same, and the way the edges redistribute themselves from four categories into two.

What this means is that if you try to generate the incidence matrix for the polytope, it's not so intuitive, because your reference to the cell has less cell types than the original definition of the cell! Compare on my polytope explorer: BXD, TDI (the important thing is that the C1a line should be the same in both, or at least longer in the BXD, but it's actually longer in the TDI).

Additionally, because I went to the effort of putting these shapes into the polytope explorer, I was required to investigate their duals (you cannot insert a polytope without its dual, and when inserting a polychoron you must figure out what its 3D cells are first). The dual of the BXD contains only one type of cell (because the BXD is vertex-transitive); this cell has 4 triangles and 2 quadrilaterals as faces, and when you draw a diagram of it, it looks like a triangular prism with one of the square faces cut diagonally into two:

I've highlighted the various components so that you can see that there are 3 types of vertices (in pairs), 6 types of edges (four pairs and two individually) and there are also 3 types of faces in pairs, although these aren't highlighted. The amazing thing about this polytope is that it's self dual, as you can see from its polytope explorer page. It is not CRF, but it should be embeddable in 3D space with plane faces. I do wonder what kind of symmetry a 3D embedding of this polytope would exhibit.

Perhaps the structure of this polytope can explain more about the structure of the BXD and maybe even lead to the discovery of other BXD-like figures?

[quickfur]

The BXD is even more interesting...

I believe this is the first polytope I've heard of which actually identifies element types that were distinct in its cells - namely, the way the top and bottom faces of the teddies become one and the same, and the way the edges redistribute themselves from four categories into two.

The n,n-duoprisms have a similar property too. The n-prism has two kinds of edges, those on the top/bottom faces, and those between them. In the n,n-duoprism, these two kinds of edges become the same: all edges of the n,n-duoprism are equivalent under its symmetry group (because the top/bottom edges of prisms in one ring become the in-between edges of prisms in the other ring, and vice versa).

What this means is that if you try to generate the incidence matrix for the polytope, it's not so intuitive, because your reference to the cell has less cell types than the original definition of the cell! Compare on my polytope explorer: BXD, TDI (the important thing is that the C1a line should be the same in both, or at least longer in the BXD, but it's actually longer in the TDI).

OK, one of these days I have to spend some time to understand your polytope explorer notation. I have to admit I don't fully understand what it is at the moment.

[...] The dual of the BXD contains only one type of cell (because the BXD is vertex-transitive); this cell has 4 triangles and 2 quadrilaterals as faces, and when you draw a diagram of it, it looks like a triangular prism with one of the square faces cut diagonally into two:

I've highlighted the various components so that you can see that there are 3 types of vertices (in pairs), 6 types of edges (four pairs and two individually) and there are also 3 types of faces in pairs, although these aren't highlighted. The amazing thing about this polytope is that it's self dual, as you can see from its polytope explorer page. It is not CRF, but it should be embeddable in 3D space with plane faces. I do wonder what kind of symmetry a 3D embedding of this polytope would exhibit.

Whoa. Now that's interesting. Would it have any relationship with the tridiminished icosahedron (in the 3D sense)?

I should do some renders of the dual BXD sometime. I've a weakness for cell-transitive polychora, and this one is both cell-transitive and vertex-transitive! (The other two that I'm aware of, besides the regular polychora, are the bitruncated 5-cell and bitruncated 24-cell, the latter of which must be one of my all-time favorite polychora.)

Perhaps the structure of this polytope can explain more about the structure of the BXD and maybe even lead to the discovery of other BXD-like figures?

Perhaps! I have to admit I'm rather weak in the non-CRF visualization department, though. I tried to render some 4D catalans a while ago, and besides a few with easily-recognizable cells (like the dual rectified 600-cell, with its pentagonal bipyramids), I have trouble "seeing" the structure in my mind.

Another related direction I was hoping to study one of these days is whether there are any other CRF non-uniform polychora that exhibit Hopf-fibration style symmetries. I doubt BXD is the only one out there. Given the vast diversity of ways in which you can put CRF polyhedra together, I expect that there has to be more CRFs out there that exhibit this kind of swirling symmetry.

[Keiji]

I believe this is the first polytope I've heard of which actually identifies element types that were distinct in its cells - namely, the way the top and bottom faces of the teddies become one and the same, and the way the edges redistribute themselves from four categories into two.

The n,n-duoprisms have a similar property too. The n-prism has two kinds of edges, those on the top/bottom faces, and those between them. In the n,n-duoprism, these two kinds of edges become the same: all edges of the n,n-duoprism are equivalent under its symmetry group (because the top/bottom edges of prisms in one ring become the in-between edges of prisms in the other ring, and vice versa).

Ooh - I'll have to work out the imats for the duotrianglinder then

OK, one of these days I have to spend some time to understand your polytope explorer notation. I have to admit I don't fully understand what it is at the moment.

It's not that amazing - it just displays the incidence matrix (imat for short). The top half and diagonal are left out, but the bottom row is kept (whereas Wikipedia keeps the diagonal but notes that you can omit the first and last row and column - the diagonal is equivalent to all of those). You can use the I_iiI_ij = I_jiI_jj invariant seen on that Wikipedia page to calculate the top half from the bottom half (or vice versa); then the imat for a dual is easily calculated by rotating the original imat 180 degrees. So you know that a shape is self dual iff rotating the imat 180 degrees gives you the imat you started with. Do note, however, that permuting the rows and columns of elements of the same rank (dimension) affects which elements are considered equivalent, and therefore a figure which is self dual may have representations which don't register as self dual. This was the case with the dual BXD cell I calculated, for example.

The polytope explorer does not calculate any of the imats itself, except for duals (and shapes in less than 3 dimensions, since they are trivial). I have to manually work them out and add them, however the polytope explorer does extremely stringent checking on whether the imat entered describes a valid polytope, to prevent typos and silly mistakes (like me thinking the black and purple edges in the dual BXD cell were equivalent, for instance - that's how I realised they weren't!).

One of my ambitions with the polytope explorer project was to find an algorithm that calculated the imat for a brick product, given the imats for its operator and operands. I've managed to find algorithms for the specific cases of prisms and pyramids, but haven't yet managed to find the fully generalized algorithm. I also believe there exists an algorithm that given an arbitrary imat, and the requirement for a CRF embedding, will either find the unique CRF embedding of that imat (up to scaling the polytope), or determine that there is no such CRF embedding. If we can accomplish these two goals, then we can do a computer search for all CRF polytopes in a given dimension which are also brick products. However, as you know, there are many CRF polytopes which are not brick products (except, trivially, of themselves), and we've probably already discovered most/all of the 4D CRF brick products anyway.

[...] The dual of the BXD contains only one type of cell (because the BXD is vertex-transitive); this cell has 4 triangles and 2 quadrilaterals as faces, and when you draw a diagram of it, it looks like a triangular prism with one of the square faces cut diagonally into two:
[...]
I've highlighted the various components so that you can see that there are 3 types of vertices (in pairs), 6 types of edges (four pairs and two individually) and there are also 3 types of faces in pairs, although these aren't highlighted. The amazing thing about this polytope is that it's self dual, as you can see from its polytope explorer page. It is not CRF, but it should be embeddable in 3D space with plane faces. I do wonder what kind of symmetry a 3D embedding of this polytope would exhibit.

Whoa. Now that's interesting. Would it have any relationship with the tridiminished icosahedron (in the 3D sense)?

I'm not sure, I haven't been able to find a decent symmetry for it yet, except by working with the edge graph in 2D and finding a nice symmetrical representation that probably makes the tetragonal faces nonplanar:



Perhaps it has a more optimal symmetry if some faces are folded into 4D?

I've a weakness for cell-transitive polychora

Me too

Another related direction I was hoping to study one of these days is whether there are any other CRF non-uniform polychora that exhibit Hopf-fibration style symmetries. I doubt BXD is the only one out there. Given the vast diversity of ways in which you can put CRF polyhedra together, I expect that there has to be more CRFs out there that exhibit this kind of swirling symmetry.

Yep - that's exactly what I mentioned I was looking for earlier in the thread.

[quickfur]

I just noticed something interesting about the BXD's degree-4 edges. Look at this render I did in the edge-counting topic again:



(Cross-eyed viewing recommended, otherwise you may not be able to see the pattern very clearly.)

Notice that the orange edges form skew polygons that are disjoint from each other? This means that these edges must trace out great circles around the BXD. (You may notice some breaks in the above image 'cos I have visibility clipping on, but you can tell from the cell transitivity that the pattern continues on the far side of the polytope too. I'll try to do a render with just these edges tomorrow, perhaps without visclip, so we can see the spiralling edges.) So this means they form a kind of spiralling that traces out more Hopf-fibration-like patterns.

So if we now swell up these edges into cells until they touch each other, we should get another polytope with the BXD's swirling symmetry! I don't know if it will be CRF, though -- I suspect not.

[Keiji]

So would this resulting polytope have 72 identical cells, each formed from a deg-4 edge of the BXD? Or would intermediate cells be necessary to join it all together? Would this count as some kind of truncation? If the edges are expanded into cells, what happens to the vertices?

So many questions...

[Klitzing]

[...]So "expanded" is ambiguous above 3D.

Surely not! While a regular polytope always has Dynkin symbol xPoQo...oRoSo, the dual regular polytope has symbol oPoQo...oRoSx. And the expanded polytope has xPoQo...oRoSx, i.e. always first and last node ringed. So this differs from one dimension to the next, as those nodes are to be counted from opposite ends. While runcination, cantellation etc. always does count from a single end on.

--- rk

[Keiji]

Ah, I think what he meant was the same as me there - it caused notational ambiguities between (expanded _____) ursatope and expanded (_____ ursatope). Nothing to do with the expansion operation itself

[quickfur]

So would this resulting polytope have 72 identical cells, each formed from a deg-4 edge of the BXD? Or would intermediate cells be necessary to join it all together? Would this count as some kind of truncation? If the edges are expanded into cells, what happens to the vertices?

So many questions...

The vertices become ridges (2D faces). Also, intermediate cells are not necessary, since you could just define the cells to lie in the hyperplane whose normal vector is the midpoint of the edge -- a convex hull algo will then give you whatever cell shape it is that arises from the intersections with the adjacent hyperplanes.

Of course, it might be interesting to add intermediate cells so that the result becomes CRF, if that is possible.

[quickfur]

[...]So "expanded" is ambiguous above 3D.

Surely not! While a regular polytope always has Dynkin symbol xPoQo...oRoSo, the dual regular polytope has symbol oPoQo...oRoSx. And the expanded polytope has xPoQo...oRoSx, i.e. always first and last node ringed. So this differs from one dimension to the next, as those nodes are to be counted from opposite ends. While runcination, cantellation etc. always does count from a single end on.
[...]

What I meant by ambiguous was that the naming convention doesn't allow you to distinguish between (expanded X) ursatope vs. expanded (X ursatope), since the two are distinct.

[wendy]

expand is Mrs stott for drawing out surtopes of the same kind, eg x3o3o can be edge-expanded to x3xo, and hedro-expanded to x3o3x.

Conway used it for the simplest example: face-expand (what i call 'runcinated'), ie x3o...o3o -> x3o...o3x.

The various kinds of other t_x,y,z used by Coxeter, and a_x,y,z by Conway (ambi-) are the actual notation that wythoff suggested. You can write, meaningfully, something like "t_v,e,h Cube", or t_0,1,2 120-cell. It was Coxeter who couppled Wythoff's t- notation with Schläfli's {3,3,5}.

The modern system is based on an entirely different thing, where one drops mirrors at the sites of the mirrors in a ascii-ised dynkin graph, vis x3x3x5o.

Norman johnson invented 'cantellated' and 'runciated', but i suggest these are around the wrong way. In any way, the shift of meaning is entirely my work.

cantellated, by johnson = second node, by krieger, the 'rectified n-rectate'. These are equal when alone or with truncate.
runciated, by johnson = third node, by krieger, the first and last node (the general expand by faces, or 'prism-circuit'.) These are equal in 4D.