Johnsonian Polytopes

[wendy]

We see that Richard, by using 3,3.3,4 and 4,3..3,4, is confusing direction into the mix. This, and that the spheric/euclidean laminatopes only have 1 or 2 faces. We need to dismantle the whole thing by considering simething that has more.

The example here shows that x4o4o has four different steps.

Code: Select all



      *====eY====*====eY====*    y=6        The group x4o4r.
      |          :          |
     eX  --&    oX    & -- eX               The cells are squares with vertex *
      |    |     :    |     |               This corresponds to integer x and integer y
      *----oY----*----oY----*    y=5
      |    |     :    |     |               The object in the cell is &
     eX -- &    oX    & -- eX
      |          :          |                There are partial expansions to eX and oY
      *====eY====*====eY====*    y=4         shown by lines.

    x=2       x=3          x=4               The wall-mirrors are shown | : -- ===




Note here the object in the cell (here &), is pushed away from two of the walls (eY and oX), by strutts, but are held against the other two walls. This is exactly how the marking of nodes in the Wythoff mirror-edge figures works. We are in fact, dealing with the four degenerate groups of the mirror represented by 'r' in x4o4r.

The group x5o3o4r has in fact, two mirrors, the internal mirrors x5o3o, and the recto-dodecahedral group represented by the r, the twelve pentagons of the dodecahedron. We demonstrate first that the 12 walls are separate, that in fact, 'r' consists of 12 separate groups or nodes, in the way that r in x4o4r above is four groups, or r in x3o4r is three groups.

The ruse here is to number the faces of the dodecahedron from 1 to 12. On reflection in the pentagons, we get planes entirely of faces of type '1' and planes of face '2'.

You can then for a dodecahedron, in the face of x5o3o4r, push against any wall, to create a prismatic layer. This is not a spread like in the cubic lattice, but the bits are lifted away from the plane in the same way that we saw the x5o4o gives f5o4o, and hence the x5o fits in there with lots of room. It has to actually go further up, but you get the idea.

The vertices no longer fall in the plane, and like the conversion from x3o4o into a elongated square bypiramid, we get only one set of edges perpendicular to the r mirror-planes. And one can repeat as needed, giving 2, 3, ..., 12 sets of edges perpendicular to the r-mirror planes, in the way that continued partial expansions of the x3o4r gives one, two, three sets of edges perpendicular to the r2r2r mirror set.

I only did this with the x5o3o4r.

It is interesting to note that although 'r' stands for a mirror edge of zero, the difference between r and o, is that the o is a vertex on the mirror, while the r represents a pair of vertices on opposite sides of the mirror (ie it has a direction), rather like o = ø while r is o|o (with no edge). The partial stott expansions do in fact expand a subset of the edges appaar at r. (The actual intent was simply to mark the nodes that the recto-mirrors appear at, but good call, rk).

[Klitzing]

Partial Stott transitions going on!

So far I've investigated linear diagrams containing some link marked by 4.
At most its ends might be bifurcated.

But then, would it be possible to apply it to loops in the diagram too?


So I considered the easiest of those, the symmetry o3o4o3*a. Esp. I started with the classical Stott expansion from x3o4o3*a up to x3o4x3*a and looked whether there are partial ones, which would provide intermediate steps.

Here is what I've found.

x3o4o3*a does allow for a 4-coloring of either class of the triangles. Here it will be enough to 4-color just one. Let "0" be neutral (the other class), and the colors being 1,2,3, and 4. Then the vertex types e.g. would be 01020102, 01030103, 01040104, 02030203, 02040204, and 03040304.

It follows that we could expand either such color-class independently into hexagons. 2 opposite triangles then become a pair of edge-connected hexagons, again of the same color. - If we'd expand 2 such color-sets, the corresponding vertices using both these colored triangles would then expand into a square with the corresponding hexagons being adjacent.

Thus we finally would result in x3o4x3*a, if all 4 color classes will be expanded.


Then, I questioned myself, what would be the corresponding lower Wythoffian relative, i.e. the corresponding partial Stott transition series between o3o4o3*a (point) and o3o4x3*a?

Here is what I've found.

Here I started with the final tiling, i.e. o3o4x3*a. This one too allows for a 4-coloring of the triangles. Let "0" be the "color" of a square, and we'd use the colors 1,2,3, and 4 for the triangles, then the vertex types e.g. would be: 010203, 010204, 010304, and 020304.

Here we then would be allowed to reduce any set of colored triangles independently piecwise into points: Applying it once we get to a tiling with 2 different vertices (both, colored or un-colored) of the same frequency: vertex type [(3,4)^3] (as before) and vertex type [(3^2,4)^3]. - Applying that reduction to the next color would result in a uniform tiling again, having vertex type [(3^3,4)^4]. That one is an interesting one, as it neither is an hyperbolic tiling with a simplicial domain, nor it belongs to those Coxeter tilings (or have I overseen something?). - And then, a application of that reduction to the 3rd color would result in a well-known uniform tiling again, in x3o-infin-o with vertex type [3^infin] and all tiles being just the triangles of the remaining color. - Thus, the last reduction of those triangles to points, clearly would collapse the whole tiling into a single point. Thus we are indeed back to where we aimed to, i.e. at o3o4o3*a.


Right by construction, this symmetry did not lend to the preconditions of the recently provided proposition. But even the idea of proof would not apply here! So far we just have taken resort to some subsymmetry of some easy base series (fewest ringed nodes), which allows for some rewriting in that one, so that the desired partial Stott transitions just all become classical ones. This thus proves the possible corresponding transitions for the remainder of that set of partial Stott transition series, where the the starting, resp. final symbol, written in the higher symmetry, has more nodes being ringed. - But this would not apply here: because a necessary condition for this sort of proof to apply, would be that this easiest series would have uniform structures only. But we've seen, that in this case the partially contracted o3o4x3*a already uses 2 vertex types.

--- rk

[wendy]

The reason it works in groups like x5o3o4r, &c, is that the mirror marked 'r' is distinct from the x and o mirrors, and that the r itself is presented as a group of N mirrors, where x5o3o has N faces.

In the case of a group like o3o4o3o3z &c, which are loop-groups, it won't work, because one can see that there is only one kind of mirror. It needs two kinds of mirror ro work. It might work on something like o3o3x3o3z4r, which is a subgroup of something like {E,4,3,4} because the tail 4's are freely extendable.

[Klitzing]

The reason it works in groups like x5o3o4r, &c, is that the mirror marked 'r' is distinct from the x and o mirrors, and that the r itself is presented as a group of N mirrors, where x5o3o has N faces.

In the case of a group like o3o4o3o3z &c, which are loop-groups, it won't work, because one can see that there is only one kind of mirror. It needs two kinds of mirror ro work. It might work on something like o3o3x3o3z4r, which is a subgroup of something like {E,4,3,4} because the tail 4's are freely extendable.

Even so it most probably is diffusely related to what I've mentioned when speaking about full- and subgroup representations, I fear I have not fully sugested what you are saying by "the mirror marked 'r' is distinct from the x and o mirrors". (Supposedly in action, not as a mere letter, hehe.) - Further help?

Your second paragraph either has a typo (as my o3o4o3*a would be your o3o4o3z, not o3o4o3o3z), or you are already stating things about further investigations, which I've not looked in so far. - ?

--- rk

[wendy]

A group represented by a dynkin-graph is a reflective group, each node is a mirror. Some of these mirrors are identical by way of being images of a common thing. One can identify what mirrors are identical, because they are connected by odd branches. If there is no path by odd branches, then the mirrors are functionally different, and may occur with different sets of elements.

In a group like o5o3o4r, the mirror planes that contaon 'o', contain all three walls so marked, whlie the remaining wall is a different kind (part of the surface of a dodecahedral cell). A group like o3o4o3z or o3o4o3o3z, contains only one kind of mirror, since on removing the even branches, the thing remains connected. In short, it's not suffice to have a o4o branch in a loop. It needs to be an open chain.

The particular kind of 'partial stott expansion' is working, because the 'r' groups are themselves a composite group. In the simplest case, one has r dividing into X, Y, and Z, and the overall polytope of symmetry hr (ie o3o3o + r2r2r = o3o4r) gives mediate figures hx, hxy, hxyz = hr. In the case of the group o5o3o4r, the r is comprised of twelve subsets, but these are as the vertices of a icosahedron, so there is more than one way to make a group of 12 steps.

[Klitzing]

Nice attempt, Wendy. I now can follow what you were saying.
But your conclusion seems wrong, so.

Consider the tiling o3o4x3*a. It obviously allows for a consistent 4-coloring of the triangles, e.g. the one given in the attached pic with colors yellow, brown, black, and green.

pac-o3o4x3a-4colored.png

(124.05 KiB) Not downloaded yet

As shown in this pic, it would be possible (in contrast to what you wrote) to apply a partial Stott contraction here: reduce the yellow triangles each to points. Thereby too those squares, which are depicted in white, would reduce to edges. Thus that partial contracted thingy would use vertices of the form [3,3,4,3,3,4,3,3,4] (the former yellow triangles) plus the remaining vertices with incident brown, black, and green triangles, i.e. [3,4,3,4,3,4].

You then could apply that partial contraction again, say to the brown triangles. This would result in a uniform tiling again, the vertices all being [3,3,3,4,3,3,3,4,3,3,3,4,3,3,3,4]. - And applying that again to the black triangles would result in a tiling with green triangles only, vertices then being of type [3^infin]. - And sure, applying it once more to the green ones would result in a complete contraction of everything into a single point only.

That last one then could be re-described as o3o4o3*a, i.e. it surely does correspond to the so far mentioned setup.

That is, even so this tiling has a loop Dynkin symbol with just a single even numbered link, it none the less allows for a series of partial Stott transitions!

--- rk

[wendy]

I suppose here that the real symmetry involved is s4s4r4z, rather than x3o3o4z. This puts the rows of squares etc on the lines, and thus resolves in the same manner as the 'r' groups before.

If you eliminated first the white and yellow faces, you would get 3,3,4*3 as ye suggest, But then you have only the choice of the red squares and one of their adjacent triangles, which would end in a (3,12). But i see no connection between 3,8 and 3,12?

[Klitzing]

I suppose here that the real symmetry involved is s4s4r4z, rather than x3o3o4z. This puts the rows of squares etc on the lines, and thus resolves in the same manner as the 'r' groups before.

This would be an interesting solution, indeed!

If you eliminated first the white and yellow faces, you would get 3,3,4*3 as ye suggest, But then you have only the choice of the red squares and one of their adjacent triangles,...

wrong. I just colored those squares white, which correspond to the elimination of the yellow triangles. For the next elimination, say of the brown triangles, you would have to color some of the (remaining) red squares as according white ones first. I.e. not all would be eliminated in that step!

In fact this step results in a uniform tiling again: [3(green), 3(green), 3(green), 4, 3(black), 3(black), 3(black), 3(black), 4, 3(green), 3(green), 3(green), 4, 3(black), 3(black), 3(black), 4].

... which would end in a (3,12). But i see no connection between 3,8 and 3,12?

And in a 3rd reduction step, we would like to reduce the black triangles to points. This then reduces all remaining squares to edges then too. The (single) vertex type then would be [3^infinity].

And the 4th reduction would reduce any green triangle to a point each. But as there are just green triangles before, the whole tiling collapses into a single point (thus correctly being described by o3o4o3*a).

--- rk

[Klitzing]

I suppose here that the real symmetry involved is s4s4r4z, rather than x3o3o4z. This puts the rows of squares etc on the lines, and thus resolves in the same manner as the 'r' groups before.

This would be an interesting solution, indeed!

But then, this would suggest a 3-coloring of the squares, instead of the intended 4-coloring of the triangles. - Any further idea?

--- rk

[wendy]

No, the colouring of the squares and triangles would match s4s, which has four sides.

[Klitzing]

Sure, s4s has 4 sides.
But s4s4s4*a has 3 types of squares, and just one type of snub-triangles (connecting to either type of squares).

Whereas my coloring of o3o4x3*a distinguishes the triangles into 4 colores. (This then induces a similar (not shown) coloring of the squares into 6 types: those adjacent to yellow and green triangles, those adjacent to yellow and brown ones, those to yellow and black ones, those to green and brown, those to green and black, and those to brown and black.)

--- rk

[wendy]

I had another look at the picture. The triangles and squares correspond to the vertices and edges of a tetrahedron. One notes that {8,3} contains the same group that makes {4,3}, by identifying opposites of {8,3}.

What's being removed corresponds to a vertex and half-edge, and i suppose that after four such instances are removed, one is left with a {4,8}.

Ok, i was wrong. It's actually a vertex and three edges.

[Klitzing]

I had another look at the picture. The triangles and squares correspond to the vertices and edges of a tetrahedron. ...

In the sense that there are 4 colors of triangles used in my coloring, and then each squares uses equal opposite adjacencies, thus providing "2 differents out of 4" = 6 combinations. Thus there is an isomorphism, mapping the colored triangles per class onto the vertices of a tetrahedron, and the squares (likewise per class) onto its edges. - but where does that help?

... One notes that {8,3} ...

Hmmm, my tiling was o3o4x3*a = s8o3o. So I suppose you are with your mind still in group theory, not at the tilings.

... contains the same group that makes {4,3}, by identifying opposites of {8,3}.

What's being removed corresponds to a vertex and half-edge, ...

Suppose, you still stick here to that tetrahedron isomorphy of yours. The yellow triangles are reduced to points. Thus will be omitted. So, yes, that tetrahedron vertex will be omitted. And likewise all squares, which are incident to the yellow triangles (highlighted white in my pic), will be reduced to edges, i.e. are omitted as well. That is, in the language of your tetrahedron, those tetrahedron edges, which are incident to the yellow tetrahedron vertex, are to be omitted too - but I'd say in full, not just half-edges. I.e. you would reduce that tetrahedron thereby into its base triangle.)

(Btw. your tetrahedron idea mirrors then the other transitions too: the next partial Stott contraction would reduce the brown tiling triangles to points and the remaining adjacent squares into edges. This would mapped to the next (brown) tetrahedron vertex and the incident edges, which are to be omitted as well. Thus the former tetrahedron now would become a single edge. - The third partial Stott contraction then would reduce the black tiling triangles to points. Thereby all remaining squares would be reduced to edges as well. Mapping this information onto your tetrahedron, then would read: a further (black) vertex has to be removed together with all incident edges. Thus your tetrahedron would be reduced to a single vertex. - And finally, the 4th partial Stott reduction would reduce the green triangles to points as well. In view of your tetrahedron this would read as further reduction of the single vertex which was left over. That is, my tiling will be reduced to one single point, and your tetrahedron will be reduced to nothing.)

... and i suppose that after four such instances are removed, one is left with a {4,8}.

???

--- rk

[Klitzing]

Wendy, if you got problems to understand that final step when the whole tiling collapses, you might do just an ovarall (classical) Stott expansion (to the series as a whole): i.e. now consider the similar 4 step transition series from x3o4x3*a = x3o8s down to x3o4o3*a = x3o8o. Here too we have a 4-coloring of the hexagons (which then implies a similar 4-coloring of the triangles and a 6-coloring of the squares).

• x3o4x3*a = x3o8s is uniform and has the vertex type [3,6,4,6]. Here the adjacent hexagons would then be of 2 out of 4 colors.
• "pac-x3o4x3*a" would contract the red hexagons of the former tiling at their edges which connect to squares. It then still has some vertices of the former type, but new ones of the type [3,3,3,6,6].
• "pabex-x3o4o3*a" then would be derived by applying the same hexagon to triangle contraction to the brown ones, say. Again the squares between 2 brown triangles would be reduced to edges. In the special case of the previously reduced square (then just an edge) this reduction would transform it further down to a point. In that specific case we would get the vertex type [3^8], but the former 2 types would exist here as well.
• Next we would apply the same to the black hexagons. This results in "pex-x3o4o3*a", which then would not have squares any longer. Even so, the green hexagons would still be in place. Thus the vertex types here would be [3^8] and [3,3,3,6,6].
• The final contraction step would reduce those green hexagons into triangles as well. We then are back to uniformity. The vertex type is [3^8] only, the tiling would be x3o4o3*a = x3o8o.

--- rk

(PS: "pex" = partially expanded / "pabex" = partially bi-expanded / "pac" = partially contracted)

[wendy]

need to look at this somewhat closer. removed the offending comment off my last post, though. simply wrong. You actually remove a vertex and three edges off the tetrahedra.

[wintersolstice]

I wondered if this shape would qualify as a rotunda?

1 icosahedron (top) 12 Gyroelongated pentagonal pyramids 40 octahedra 30 square pyramids and 1 rhombicosadodecahedron (base)

it's made by taking the icosahedron anticupola (icosahedron semicupola) and the partially base rectified icosahedron/dodecahedron cupola (icosahedron/docecahedron mesocupola) and joining them together. it is found as a fragment of the rectified 600 cell

btw(1)

I noticed a few errors with the rotundae

icosadodecahedral rotunda:

missing a cell 1 truncated dodecahedron (its base)

icosahedral rotunda:

missing a cell a truncated icosahedron (it's base)

it's from the truncated 600 cell not the rectified 600 cell

other than that I think it's right (I'm not 100% sure if those are the bases) and as far as I now theres no more errors

btw(2)

I was wondering if I could put both sets of names for the cupola forms (since I had names aswell) besides there's loads of shapes with more than one name

also include in the table the 3 vertex transitive cases (just for completeness)

tetrahedron antiprism (16 cell)
truncated tetrahedral cupola prism (semi anti prism)
tetrahedron anticupola (semicupola) (rectified 5 cell)

btw(3)

I currently working on defining "rotunda" and "cupola forms" and see if there are any more rotunda (including outside the ones which come from uniform polychora )

and in general to classify all these shapes in way where it's easier to search

[Klitzing]

I wondered if this shape would qualify as a rotunda?

1 icosahedron (top) 12 Gyroelongated pentagonal pyramids 40 octahedra 30 square pyramids and 1 rhombicosadodecahedron (base)

it's made by taking the icosahedron anticupola (icosahedron semicupola) and the partially base rectified icosahedron/dodecahedron cupola (icosahedron/docecahedron mesocupola) and joining them together. it is found as a fragment of the rectified 600 cell

Well, i use "rotunda" in a specific sense, meaning apolytope being orbiform on the one hand and uses the equatorial (hyper)plane for bottom facet realm, i.e. being kind of hemispherical. - Using this for definition, then your described figure surely is not a rotunda.

I'd rather say its just the "bistratic icosahedron-first cap of the rectified hexadecachoron", or "x3o5o || (pseudo) o3x5o || x3o5x", i.e. xox3oxo5oox&#xt (in Wendy's lace tower notation).

As you can see from my rox side, in icosahedron-first orientation the rox (= rectified hexacosachoron) would have 17 vertex layers, i.e. the equatorial one would be the 9th (not the 3rd).

btw(1)

I noticed a few errors with the rotundae

where do you refer to?

icosadodecahedral rotunda:

missing a cell 1 truncated dodecahedron (its base)

Supposedly this won't be no "rotunda" as well. So I looked around and found that this might describe what would be best called the "bistratic icosidodecahedron-first cap of rectified hecatonicosachoron", or "o3x5o || (pseudo) o3o5f || o3x5x", i.e. ooo3xox5ofx&#xt. (As usual in Wendy's notations, an "f" edge would be one, which is tau=1.618.. times as large.)

icosahedral rotunda:

missing a cell a truncated icosahedron (it's base)

it's from the truncated 600 cell not the rectified 600 cell

Once again no "rotunda". But you're right with the last remark: its the "bistratic icosahedron-first cap of the truncated hexacosachoron", or "x3o5o || (pseudo) u3o5o || x3x5o", i.e. xux3oox5ooo&#xt. ("u" denotes an edge, which is 2 times as large.)

other than that I think it's right (I'm not 100% sure if those are the bases) and as far as I now theres no more errors

Your mentioned bases are correct.

btw(2)

I was wondering if I could put both sets of names for the cupola forms (since I had names aswell) besides there's loads of shapes with more than one name

also include in the table the 3 vertex transitive cases (just for completeness)

tetrahedron antiprism (16 cell)
truncated tetrahedral cupola prism (semi anti prism)
tetrahedron anticupola (semicupola) (rectified 5 cell)

"Pyramid" and "prism" do have a clear meaning in any dimension. The similar usage of "antiprism" and "cupola" for 4D where defined in my paper on convex segmentochora in 2000. The usage of "cupola" since then was asked to be allowed to apply in a looser sense to any segmentochoron (i.e. monostratic orbiform polychoron), provided it would not be covered by any of those other terms already. This might be acceptable, I think, as long as it becomes clear from the context whether one refers to the stricter (as there defined) or looser usage.

But meanwhile a whole bunch of terms have come up to refer to several special subclasses of those looser-sense cupolae. To me all those terms are neither memorable nor useful. They are just a mouthful of to be reminded wordings of what would be much more precise described by the actual segmetochoron description (a atop b) or by the corresponding lace prism notation (if applicable).

A total nonsense e.g. is the term for your 2nd example: "truncated tetrahedral cupola prism". First at all, there is no clearity how to apply parantheses here, i.e. to what applies the adjective "truncated"? Next: what does mean "cupola prism"? Is that a prism of a cupola? Or a cupola of a prism? (As I fiddled out from context: neither one would be meant here!)

Here are the polychora you probably meant:

• hexadecachoron
• truncated pentachoron
• rectified pentachoron

btw(3)

I currently working on defining "rotunda" and "cupola forms" and see if there are any more rotunda (including outside the ones which come from uniform polychora )

and in general to classify all these shapes in way where it's easier to search

I usually use the following wordings:

• segment : s.t. cut out from some larger entity, esp. between 2 parallel vertex layers (of arbitrary distance)
• monostratic / bistratic / ... : qualifier of segments, describing those which have no inner vertex layer / just one further one / etc.
• cap : segment with one vertex layer being tangential to the larger entity
• rotunda : cap with the other cut being equatorial

But note, that in the term "segmentotope" of my 2000 paper not only the "segment" definition, but also "monostratic" is subsumed. Thus, if using segment in the more general sense, it would be best to add the adjective "multistratic".

Btw. "stratic" just derives from "stratos": a (thick) layer of clouds.

--- rk

[wintersolstice]

Well, i use "rotunda" in a specific sense, meaning apolytope being orbiform on the one hand and uses the equatorial (hyper)plane for bottom facet realm, i.e. being kind of hemispherical. - Using this for definition, then your described figure surely is not a rotunda.

I'd rather say its just the "bistratic icosahedron-first cap of the rectified hexadecachoron", or "x3o5o || (pseudo) o3x5o || x3o5x", i.e. xox3oxo5oox&#xt (in Wendy's lace tower notation).

As you can see from my rox side, in icosahedron-first orientation the rox (= rectified hexacosachoron) would have 17 vertex layers, i.e. the equatorial one would be the 9th (not the 3rd).

Supposedly this won't be no "rotunda" as well. So I looked around and found that this might describe what would be best called the "bistratic icosidodecahedron-first cap of rectified hecatonicosachoron", or "o3x5o || (pseudo) o3o5f || o3x5x", i.e. ooo3xox5ofx&#xt. (As usual in Wendy's notations, an "f" edge would be one, which is tau=1.618.. times as large.)

Once again no "rotunda". But you're right with the last remark: its the "bistratic icosahedron-first cap of the truncated hexacosachoron", or "x3o5o || (pseudo) u3o5o || x3x5o", i.e. xux3oox5ooo&#xt. ("u" denotes an edge, which is 2 times as large.)

and in general to classify all these shapes in way where it's easier to search
I usually use the following wordings:

• segment : s.t. cut out from some larger entity, esp. between 2 parallel vertex layers (of arbitrary distance)
• monostratic / bistratic / ... : qualifier of segments, describing those which have no inner vertex layer / just one further one / etc.
• cap : segment with one vertex layer being tangential to the larger entity
• rotunda : cap with the other cut being equatorial

but that's how you define "rotunda" though, the people on this forum use a different definition: what you would call a "cap" it seems (just like they use different definition of "cupola" (here its the platonic||cantelate on your paper its platonic||rectate)

I was under the impression that there was no universal definintion of rotunda or cupola.

maybe we need to get together and make some definitions (for cupola rotunda etc) that are universal and that we all agree on and maybe come up with some new terms so there's no more argueing over what names to use

sorry I'm just trying to settle a dispute over names

A total nonsense e.g. is the term for your 2nd example: "truncated tetrahedral cupola prism". First at all, there is no clearity how to apply parantheses here, i.e. to what applies the adjective "truncated"? Next: what does mean "cupola prism"? Is that a prism of a cupola? Or a cupola of a prism? (As I fiddled out from context: neither one would be meant here!)

I meant "truncated tetrahedral cupoliprism"
non uniform scaliform polychoron and that wasn't a name I gave to it

Here are the polychora you probably meant:
•hexadecachoron
•truncated pentachoron
•rectified pentachoron

with the correction of the second these shapes qualify as cupola forms (on the table shown on the wiki) that's what I meant
I currently working on defining "rotunda" and "cupola forms" and see if there are any more rotunda (including outside the ones which come from uniform polychora )

[Klitzing]

Okay, you are right, there was no real definition for cupolae or rotundae (nor antiprisms) for higher dimensional spaces. This is why I mentioned pyramids and prisms in contrast. Those terms bear a clear meaning in 3D so. Thus when trying to use those other ones in 4D (and beyond) this should be a generalization of most of their properties at least.

In my paper of 2000 I went through this already, arguing why its best to take the generalizations I've given there (e.g. the platonic||cantelate would not exist in any case, using unit lacings). So my choice was done by purpose, well evaluating other possibilities of extrapolation too. And I hate to argue about this any 2 or 3 years again. - Up to then there was no fixed terminology in use, as far as I could say.

If some members on this forum decide to abuse those given definitions, even while being aware of previous ones using the same terms in a different sense, then this is quite sad. Such an behaviour just contributes to the babylonian confusion. I did already have some discussion on that here before a while. I even was willing to accept the term cupolae being used in a less strict sense for any non-pyramidal and non-prismatic segmentotope. But even that would imply monostraticness!

In fact quickfur then wanted to "correct" the forum-wiki. But he got rather quiet, lately.

I don't think we have to introduce names for every specific decoration of lace prism symbols so. (Here I'm speaking on family names, i.e. thereby generalizing apon the to be used link marks.) They all form a not too large set and the lace prism symbol even is usually much shorter (and also inline printable) than any aimed for name. - So, what is their use? Again this justs adds names into some dictionary, which have to be remembered... - This might even work fine within this forum. But going beyond, you'll would have this very discussion again and again. Esp. when others would like to do so as well on their own.

In fact we have to consider 2 points in here:

• Are you free to introduce some definition? Or were some others doing so before?
• Was that a well-considered choice of name, which adds more clairity, rather than introduces additional fog?

Here again some of my reasons for using those terms as they are in use already:

• A 3D antiprism uses gyrated polygons for bases. But this does not generalyse freely into higher dimensions. - But one could consider those polygons likewise as being vice versas duals. Therefore I had propagated the higher dimensional usage in that latter sense. (Speaking of duals this surely restricts the bases to be regular polytopes.)
• The cupolae of 3D all are monostratic. Thus this lends for generalization. The laterals of cupolae are triangles and squares. Those triangles well could be generalized into pyramids. But what for the squares? Do they generalize into prisms or into antiprisms? Both choices then have be considered, leading to xPoQo || xPoQx resp. xPoQo || oPxQo (in 4D). But as the former not in all cases does exist, it was decided to lend that term (cupola) to the greater class. Esp. as the other one well could be described by the term "cap".
• The single 3D rotunda is hemispherical. As hemispherical polytopes are a very specific type of segments, worth using a special name, that name was taken over in this sense. Other segments well can be termed as multistratic caps.

Btw. please don't mistake me to be personal, so.

--- rk

[Klitzing]

Back on Partial Stott Transitions again.

Recently considered the hyperbolic symmetry group o4o3o3o3*b = o3o3o3*a4o.

• The transitions from o4o3x3o3*b (cells being: oct (o4o3x .) and trat (. o3x3o3*b, considered an horotope here)) towards x4o3x3o3*b (cells being: sirco (x4o3x .), trip (x . x3o), cube (x4o . o3*b), and trat (. o3x3o3*b)) clearly is classical (one-step) Stott expansion. But you could color the trips of the latter consistently by 3 colors. Accordingly you could flatten those trips into triangles independantly. This leads to a 3-step partial Stott contraction! It further turns out, that the 2 medial forms even are scaliform here.
• The transition from o4o3x3x3*b (cells being: oct (o4o3x . & o4o . x3*b) and that (. o3x3x3*b, considered an horotope here)) towards x4o3x3x3*b (cells being: sirco (x4o3x . & x4o . x3*b), hip (x . x3x), and that (. o3x3x3*b)) again is a one-step classical. Here too the hips might be 3-colored. Thus either class of colored hips can be contracted independantly again, thus providing a 3-step partial Stott contraction series. Here the medial forms would ask for different vertex classes, so.
• The further transition from o4o3o3o3*b (zero-dimensional, i.e. a single point only) towards x4o3o3o3*b (cells being: cube (x4o3o . & x4o . o3*b), vertex figure being trat (. q3o3oo*b)) again is a classical Stott transition. - Despite the induced coloring of the former family members, this does not result in a corresponding partial Stott transition here: the partial contraction of the latter one would consist of some (locally) 2D structure of squares, but asking for infinitely many such at any edge. And this contradicts to dyadicity!

--- rk

[Klitzing]

Back to the main topic of this thread!

The rectified tesseract (o3o3x4o, rit) can be given as lace city either as

Code: Select all

x4o o4q x4o
           
o4q     o4q
           
x4o o4q x4o


(showing that the cuboctahedra (co) would have a dihedral angle of 90°) or, using orthogonal triangular symmetry, as

Code: Select all

  x3o x3x o3x 
               
o3o u3o o3u o3o
               
  x3o x3x o3x 


i.e. being (horizontally) a bistratic stack of co || q-cube || co, where q = sqrt(2) edge, or (diametrally) a tristratic stack of tet || tut || inv. tut || dual tet.

This latter display shows that the known scaliform segmentochoron "tut || inv tut" (a.k.a. tutcup) could be derived therefrom as a parabidiminishing:

Code: Select all

    x3x o3x
           
  u3o o3u 
           
x3o x3x   


Thus tutcup alternatively might be called "pabdirit" too.

Today it occured to me that rit likewise has a metabidiminishing as well:

Code: Select all

    x3x   
           
  u3o o3u 
           
x3o x3x o3x


That genuinely bistratic wedge, which might be abbreviated "mibdirit", then would have for cell count: 1 co + 2 tuts + 6 tets + 6 tricues.

--- rk

[quickfur]

Okay, you are right, there was no real definition for cupolae or rotundae (nor antiprisms) for higher dimensional spaces. This is why I mentioned pyramids and prisms in contrast. Those terms bear a clear meaning in 3D so. Thus when trying to use those other ones in 4D (and beyond) this should be a generalization of most of their properties at least.

In my paper of 2000 I went through this already, arguing why its best to take the generalizations I've given there (e.g. the platonic||cantelate would not exist in any case, using unit lacings). So my choice was done by purpose, well evaluating other possibilities of extrapolation too. And I hate to argue about this any 2 or 3 years again. - Up to then there was no fixed terminology in use, as far as I could say.

If some members on this forum decide to abuse those given definitions, even while being aware of previous ones using the same terms in a different sense, then this is quite sad. Such an behaviour just contributes to the babylonian confusion. I did already have some discussion on that here before a while. I even was willing to accept the term cupolae being used in a less strict sense for any non-pyramidal and non-prismatic segmentotope. But even that would imply monostraticness!

In fact quickfur then wanted to "correct" the forum-wiki. But he got rather quiet, lately.

I think you should understand that many (most?) of us here are hobbyists, not professional mathematicians, so one should not assume that we are completely up-to-date w.r.t. terminology in recently published papers. For one thing, my time is quite limited between my day job, family, and all the other fascinating hobbies I have that all demand more time, moar, MOAR time, so while I do try to catch up on stuff every now and then, I'm far more likely to read only what interests me and skip over "boring" details like terminology and nomenclature.

Of course, that should not be an excuse for conflicting or contradictory terminology, but you shouldn't feel so offended by the use of other terminologies on this forum. Many of us made a lot of independent rediscoveries before we realized that others have already found the same things, so we already have some amount of home-brew terminology that we feel quite attached to before we are exposed to more consistent terms. I'm not saying we should promote this situation, of course, but that's where we begin.

For that very reason, I fully agree that a single, consistent terminology accepted by everyone involved will be most beneficial. Which is why I wanted to update the wiki to rename some of the terms used there. But I have been too busy with other things for the past while, so I haven't gotten to anything on that front yet (I haven't even found enough time to update my own website, which is already a few months behind schedule on the Polytope of the Month series!). Plus, I don't run the wiki -- Keiji does -- and I don't want to step on his toes either by introducing or renaming terms that contradict the ones he used in other places on the wiki.

One peculiarity of higher-dimensional terminology is that most terms are extrapolated by different people for different reasons from 3D terms, which are really no good at capturing what really happens out there. So different people will prefer different ways of naming something, which leads to the present situation of confusing / contradictory terms. I've tried to promote wendy's Polygloss at various times, as it seems to be the most consistent system of higher-dimensional terms, but it does require a rather steep learning curve, and it seems that not many people have adopted it.

I don't think we have to introduce names for every specific decoration of lace prism symbols so. (Here I'm speaking on family names, i.e. thereby generalizing apon the to be used link marks.) They all form a not too large set and the lace prism symbol even is usually much shorter (and also inline printable) than any aimed for name. - So, what is their use? Again this justs adds names into some dictionary, which have to be remembered... - This might even work fine within this forum. But going beyond, you'll would have this very discussion again and again. Esp. when others would like to do so as well on their own.

I think given our demographics there will always be personal terms for things. I don't know if it's possible, or even beneficial, to try to purge that out. After all, different ways of approaching the subject often leads to new insights and new discoveries. But I do agree that we need to agree on a single, consistent set of terms that we use for communicating with each other, so that we avoid misunderstandings and lots of wasted time trying to explain what we really mean! It will require some effort from all parties involved, though, so we have to be willing to take the time to work out such terms and learn them.

For this purpose, I think the wiki is the best tool to use -- a lot of new discoveries and naming conventions, etc., tend to get lost in these very long topics, and very few people have the time/patience to read through everything. While the forum is good for discussing / debating over what terms should be adopted, the results really need to be posted to the wiki under a single page or a single set of pages, that should be treated as normative for the currently-accepted common terminology. That way, we don't have to hunt in 100 different posts to find the right term to use, and there is a single reference for deciding how to name something.

In fact we have to consider 2 points in here:

• Are you free to introduce some definition? Or were some others doing so before?
• Was that a well-considered choice of name, which adds more clairity, rather than introduces additional fog?

[...]

With the first point, having a common set of definitive pages on the wiki to describe the currently-accepted terminology will be very helpful to resolve these issues.

For the second point, I agree with the spirit of the statement -- names should help clarify rather than to obscure. Having said that, though, there are often different ways of looking at something: different patterns of generalizations or analogy, different ways of looking at something, which result in different systems of naming which may lead to new discoveries by making it obvious where gaps in the generalizations are. For the purposes of communication, though, we do need to stick to a single set of terms, no matter how cumbersome we think it is, or how much we think it sucks. We need a higher-dimensional equivalent of the IUPAC's nomenclature for chemicals -- it's very long, very verbose, and has rigid (and boring) rules for naming compounds, and one might argue that it's rather ugly. But it's also very precise and avoids miscommunications and misunderstandings. Nobody who hasn't learned IUPAC naming rules could pass as a chemist in this day and age -- they'd be laughed out of their jobs. I think we need a similar "official" nomenclature for higher-dimensional objects and concepts, that everyone can communicate in, even if they don't believe it's the best terminology.

Nevertheless, there still remain historical chemical names and "pet names" that are still in widespread use today -- sometimes it's useful to refer to a very common chemical by a common nickname. But that doesn't mean we don't need an "official", consistent terminology. I think it should be OK to allow "pet names" for higher-dimensional objects, as long as we also make it clear what it means in terms of the "official" terminology.

Well, the first step is to make an official terminology. Klitzing, since you seem most experienced in this area, maybe you could start a terminology page on the wiki to document a single, consistent set of terms to use? We can discuss / debate in the forum over terms that aren't obviously the best ones to use, but I think it will help to use the "official" wiki page as a fixed reference. Maybe wendy can chime in too, since her Polygloss, IMO, is one of the most consistent systems for referring to higher-dimensional concepts that doesn't suffer from the unnecessary bias in today's 3D-centric terms (not to mention the inconsistent usage of the same terms by different authors - even among peer-reviewed publications).

[wendy]

Most of us are like me, hobbyists. The polygloss is as much an experiment in 'latin and greek roots' and 'language ideas' as a dimensional dictionary. One can see that etymologies and rationales are dotted through the text. But it also has a number of notions that many mathematicians appear to miss, and generally make the field harder to understand. One plays "Heaviside" to Coxeter's "Maxwell" here. I tend to think of my work here akin to the works of Oliver Heaviside, who for the rationalisation of the theory, makes discoveries.

The key secret here is that the notion that for every ascent, there is a matching descent. While one sees lines might be made of points, and planes of lines, there is also the notion that a plane is an equal sign, and a line is two equal signs, and a point is three. So when one looks at three dimensions, one has to guess whether the root meaning is based on 'dimensions', or 'number of equal signs'. Most of the common words are set to 'number of equal signs', and new words are invented for dimensions.

Some words are made to fall back to their etymologies. An angle is 'around' something, rather than 'at something'. This is why i opted for (marginal angle), meaning around the margin (two equal signs), rather than 'dihedral angle' (angle at the meeting of two hedra). In part, the angle is a fraction of space against an orthogonal circle.

Still. We can comment on the lace-prisms.

One can build a pretty simple term for lace-prisms, without using cupolae etc. A figure like xo3oo5ox&x can be disected into word-like forms "xoo || oox 3,5 &x", where the first base is listed, and then the second base, the ruling symmetry, and a marker. This could, with the numbering of 1248... for the nodes, become something like "one on four icosahedral lace-prism". But someone else might decide on some better implementation of this sort of notation. It's sort of like the IUPAC names, but it translates pretty simply into what is needed to construct the dynkin-symbol.

Antiprisms and antitegums have meanings in every dimension, as the lace-prism and lace-tegum of a figure and its dual. There are expressions like the 'antiprismic sequence' and the 'antitegmic sequence' and the 'Hass antitegum'. It's not probably wise to change from this.

Sometimes, you have to watch names. John Conway's set of names as used in the conway-hart notation, gives the hint that it is restricted to three dimensions. In fact, it's a general dynkin-symbol, on the notion that the named figure is regular, and that its flags can be used like a symmetry group to generate wythoff-like constructions on it. It does indeed have a reflex in four dimensions. In essence, an operator like 'ambi' is oxo, and expand = aa gives xox. These exist perfectly in four dimensions, one can construct, eg an xoxo!oxxo!24ch. This is read like the standard Conway notation. The first one corresponds to a regular wythoff construction, ie 343 becomes o3x4x3o. The second one xoxo! then divides faces of the object into flags, and places the new vertices somewhere on the line connecting the old vertex and the centres of the old hedra. New faces etc are constructed. As in 3d, one can construct xoxo! from oxoo!oxoo! - the rectified rectate.

[Keiji]

For that very reason, I fully agree that a single, consistent terminology accepted by everyone involved will be most beneficial. Which is why I wanted to update the wiki to rename some of the terms used there. But I have been too busy with other things for the past while, so I haven't gotten to anything on that front yet (I haven't even found enough time to update my own website, which is already a few months behind schedule on the Polytope of the Month series!). Plus, I don't run the wiki -- Keiji does -- and I don't want to step on his toes either by introducing or renaming terms that contradict the ones he used in other places on the wiki.

Let's look at the (small) rhombicuboctahedron. Firstly it doesn't help that there's already confusion over whether you include the word "small" or not, and as for the great rhombicuboctahedron, even Wikipedia prefers to call it the less ambiguous name of truncated cuboctahedron, even though "the name truncated cuboctahedron, given originally by Johannes Kepler, is a little misleading. If you truncate a cuboctahedron by cutting the corners off, you do not get this uniform figure: some of the faces will be rectangles." Now for the main bit, rhombi-cub-octa- is one big portmanteau, cube and octa referring to the cuboctahedron which is half way between the two duals (a mesotruncate) while rhombi was chosen simply because "twelve of the square faces lie in the same planes as the twelve faces of the rhombic dodecahedron which is dual to the cuboctahedron", and rhombi- takes merely from the rhombus which really has nothing to do with the rhombicuboctahedron itself.

My name for this is the stauroperihedron. Now, there was a big fuss back when with Tamfang decided I'd misused his notation idea when I decided to extend it to everything under the sun (and used it a dimension higher or lower than intended), but there is no need to dwell on that. The concepts I latched onto were simple: pyro- refers to 3, stauro- refers to 4, rhodo- refers to 5, and geo,aero- and cosmo,hydro- refer to the two "ends" of the stauro- and rhodo- spectra. Mesotruncates are in the middle, pantomotruncates are all (=pan), and peritruncates are outside (=peri, like perimeter). And even though -hedron really means "faces" which are 2 dimensional, they are what bound a 3 dimensional shape and is already widely accepted as being a suffix for one. So what do we call a 3 dimensional shape which is the outside truncate (xx, xox, xoox, xooox etc.) in the 4 series? A stauro(4)peri(xox)hedron(3D).

I must admit I've been very pushy with my names for things, however I am sure if you looked at my system against the widely accepted names for convex uniform polyhedra, if you hypothetically ignored the fact that the current system has been around for pretty much all relevant history you'd agree mine makes more sense, even if you might use different words or parts of words for it. It is just a question of sticking to what has evolved, versus designing something new from scratch. Sure, it doesn't properly cover all of the 2^n possible truncations as you extend to higher dimensions, but it's a lot easier to say "cosmochoron" than "hecatonicosachoron", and I'd argue even if not complete, it is an improvement over the existing system.

[Keiji]

Re: an IUPAC-style nomenclature, I agree this would be very useful, and I wouldn't even begin to suggest that my "Tamfang-inspired" naming system be a part of this, mainly because it's restricted to uniform polytopes, and a generalised naming convention needs to work for anything that one might want to describe. Taking that to the extreme, it would have to include figures with less symmetry, maybe non-convex or self intersecting, having curved facets, being non-orientable, or even fractals.

Some of you may remember FLD, which was my attempt to come up with something similar to SMILES, but that only does convex figures with flat facets, and it does have a few undesirable quirks. Then there are incidence matrices, where my two main points of interest lie (finding a way to represent curved facets in incidence matrices, and finding an algorithm for the generalised brick product operating on incidence matrices). I do also very much like Wendy's lace city idea, and I would use it, if only I could properly understand it. But this probably too has its own limits.

At the end of the day I think coming up with one be all and end all notation or naming convention is an impossible task. These things are only there if they're useful, so it's often more appropriate to use a different convention depending on what your goal is.

[quickfur]

For that very reason, I fully agree that a single, consistent terminology accepted by everyone involved will be most beneficial. Which is why I wanted to update the wiki to rename some of the terms used there. But I have been too busy with other things for the past while, so I haven't gotten to anything on that front yet (I haven't even found enough time to update my own website, which is already a few months behind schedule on the Polytope of the Month series!). Plus, I don't run the wiki -- Keiji does -- and I don't want to step on his toes either by introducing or renaming terms that contradict the ones he used in other places on the wiki.

Let's look at the (small) rhombicuboctahedron. Firstly it doesn't help that there's already confusion over whether you include the word "small" or not, and as for the great rhombicuboctahedron, even Wikipedia prefers to call it the less ambiguous name of truncated cuboctahedron, even though "the name truncated cuboctahedron, given originally by Johannes Kepler, is a little misleading. If you truncate a cuboctahedron by cutting the corners off, you do not get this uniform figure: some of the faces will be rectangles." Now for the main bit, rhombi-cub-octa- is one big portmanteau, cube and octa referring to the cuboctahedron which is half way between the two duals (a mesotruncate) while rhombi was chosen simply because "twelve of the square faces lie in the same planes as the twelve faces of the rhombic dodecahedron which is dual to the cuboctahedron", and rhombi- takes merely from the rhombus which really has nothing to do with the rhombicuboctahedron itself.

I think we can all agree that the "standard" terminology sucks. There's nothing rhombic about the "rhombicuboctahedron", and "truncated cuboctahedron" is outright wrong (I've always argued that x4x3x is better understood as the Stott expansion of the hexagonal faces of the truncated octahedron, or the Stott expansion of the octagonal faces of the truncated cube, or the Stott expansion of the non-axial squares of the "rhombicuboctahedron").

I wasn't suggesting that the wiki should use the "standard" terminology.

My name for this is the stauroperihedron. Now, there was a big fuss back when with Tamfang decided I'd misused his notation idea when I decided to extend it to everything under the sun (and used it a dimension higher or lower than intended), but there is no need to dwell on that. The concepts I latched onto were simple: pyro- refers to 3, stauro- refers to 4, rhodo- refers to 5, and geo,aero- and cosmo,hydro- refer to the two "ends" of the stauro- and rhodo- spectra. Mesotruncates are in the middle, pantomotruncates are all (=pan), and peritruncates are outside (=peri, like perimeter). And even though -hedron really means "faces" which are 2 dimensional, they are what bound a 3 dimensional shape and is already widely accepted as being a suffix for one. So what do we call a 3 dimensional shape which is the outside truncate (xx, xox, xoox, xooox etc.) in the 4 series? A stauro(4)peri(xox)hedron(3D).

I think the core idea (having different stems for different symmetry groups) is sound. As far as the regular polytopes go, such a system is perfect. I'm on the fence about whether we should adopt "hedron" for 3D shapes, or Tamfang's preferred "morph". But that's a question of implementation; the core idea is sound.

I must admit I've been very pushy with my names for things, however I am sure if you looked at my system against the widely accepted names for convex uniform polyhedra, if you hypothetically ignored the fact that the current system has been around for pretty much all relevant history you'd agree mine makes more sense, even if you might use different words or parts of words for it. It is just a question of sticking to what has evolved, versus designing something new from scratch. Sure, it doesn't properly cover all of the 2^n possible truncations as you extend to higher dimensions, but it's a lot easier to say "cosmochoron" than "hecatonicosachoron", and I'd argue even if not complete, it is an improvement over the existing system.

Well, just about anything beats the existing system, I'm sure we all can agree.

What keeps me on the fence about the whole issue, though, is the issue of communication vs. representation. For the sake of communication, it is best if one uses pre-existing, well-known terminology so that newcomers can understand what we're talking about. To give a contrived example, imagine if we all agreed on this forum that what we called "cube" before should now be called, for whatever reason, "lumafyqonomus" (the actual term is irrelevant, you can substitute any arbitrary name here). As long as we all agree on this, there's no contention at all. But a newcomer coming to this forum, or an anonymous web surfer encountering the wiki, say, will have no idea what "lumafyqonomus" means, even though he may be very familiar with exactly what a "cube" is. We have only re-lexified our terminology, but the essence of the thing has not changed. Yet in doing so, we've alienated potential newcomers (not to mention made any materials we publish online unreachable via typical search engine terms). We may also potentially alienate other researchers, who may regard our neologisms as unnecessary.

On the side of representation, though, one has to concede that a consistently-derived name like "geohedron" is far better than an arbitrary, unanalysable term like "cube". In this case the benefit may not be obvious, but once we start talking about "truncated cuboctahedron" then the consistently-derived scheme will definitely be shown to be superior. However, consider this: from the side of representation, we are interested not in the pronuncibility of the label, but a compact, mathematically-accurate representation that unambiguously designates a specific shape. But we already have such a representation: the Coxeter-Dynkin symbol. When we write x4o3o (using Wendy's notation), for example, it unambiguously designates the cube, and furthermore provides full information about its symmetry group, the shapes of its surface elements, its construction, etc..

So it seems that we're still stuck with the sucky traditional terminology for communication purposes, and Coxeter-Dynkin symbols for representation purposes (which, really, doesn't have any significant drawbacks save being unpronunciable).

Coming back to Klitzing's original comment, his point was that we seem to be too trigger-happy with coining new terms and reinventing existing names, when doing so doesn't really help clarify the essentials of the subject, but only adds more noise to an already noisy, inconsistent, traditional terminology. I'm guilty of this too -- it is rather fun to invent new names, after all: it's the thrill of being the first discoverer of some shape and having bragging rights to name it -- but let's take the name "rotunda", for example. Back when Mrrl first discovered a non-monostratic shape containing pentagonal rotunda cells, we didn't know of the existence of other non-trivial polystratic shapes (non-trivial meaning not made by simply gluing two segmentochora together). At the time, we only knew of Klitzing's monostratic segmentochora, so this was an exciting discovery for us. The fact that it was bowl-shaped (or cap-shaped, depending on which way you look at it), and had pentagonal rotunda cells, inspired us to name it a 4D "rotunda".

In retrospect, however, this choice of name was premature. As it turns out, subsequent research has revealed lots of other polystratic bowl-like (cap-like) shapes. Should they all be called rotundae now, just because they aren't monostratic? Should the term "rotunda" merely mean "not monostratic"? Seems like a waste of such a good term for something that could've been called, oh, a "polystratic cap"? Not to mention that Wendy has already anticipated them by defining the notion of lace prism and lace city long before Mrrl made his discovery. Klitzing's definition of "rotunda" makes much more sense: a hemispherical CRF, generalizing the 3D pentagonal rotunda in a slightly different, and arguably more useful manner. It lets us conveniently designate the pseudo-bisected 600-cell as an augmented icosahedral/icosidodecahedral rotunda, for example, without needing to count just how many layers of vertices it has (in order to know whether it's a bistratic, tristratic, or tetrastratic polychoron, e.g.).

Furthermore, there's the issue of, do we really need to invent a new set of terminology to cover every possible variation of every possible class of shapes that we can find? There's only so many short, pronunciable names available in English, no matter how cleverly you try to derive them from Greek/Latin/whatever-else, we should reserve them for the important things. Like, shapes with special properties that we may want to refer to frequently. Obscure shapes that are just another entry in the big list of CRF polychora (and now we know that there are lots and lots and lots of them) are OK to be named something less convenient to pronounce. Like 1,(1,2,1',2'),0,1-hexadiminished 24-cell, one of the 19 diminishings of the 24-cell that we probably won't be referring to over and over again.

Perhaps the way to proceed is for all of us to lay all currently existing terminology / naming schemes on the table, and see if we can distill it into an IUPAC-style consistent set of terms. I don't think we can ever completely get rid of different naming schemes -- in any sufficiently interesting subject, there's always more than one way of looking at the same thing, which leads to different ways of generalizing / fitting things into patterns. But there should at least be a "lingua franca" of terminology that lets us communicate without being completely frustrated by inconsistent use of terms.

[Keiji]

On the side of representation, though, one has to concede that a consistently-derived name like "geohedron" is far better than an arbitrary, unanalysable term like "cube". In this case the benefit may not be obvious, but once we start talking about "truncated cuboctahedron" then the consistently-derived scheme will definitely be shown to be superior. However, consider this: from the side of representation, we are interested not in the pronuncibility of the label, but a compact, mathematically-accurate representation that unambiguously designates a specific shape. But we already have such a representation: the Coxeter-Dynkin symbol. When we write x4o3o (using Wendy's notation), for example, it unambiguously designates the cube, and furthermore provides full information about its symmetry group, the shapes of its surface elements, its construction, etc..

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

Dynkin symbols are great for identifying something in a list or table or expression or when you otherwise need a compact symbol, but they're not very useful in the middle of a sentence. It's similar to why we like to use numbers and single letters in algebra, but spell out numbers (like "four" instead of "4") and write out quantities when we're putting them in sentences. Both are necessary.

[quickfur]

Re: an IUPAC-style nomenclature, I agree this would be very useful, and I wouldn't even begin to suggest that my "Tamfang-inspired" naming system be a part of this, mainly because it's restricted to uniform polytopes, and a generalised naming convention needs to work for anything that one might want to describe. Taking that to the extreme, it would have to include figures with less symmetry, maybe non-convex or self intersecting, having curved facets, being non-orientable, or even fractals.

Some of you may remember FLD, which was my attempt to come up with something similar to SMILES, but that only does convex figures with flat facets, and it does have a few undesirable quirks. Then there are incidence matrices, where my two main points of interest lie (finding a way to represent curved facets in incidence matrices, and finding an algorithm for the generalised brick product operating on incidence matrices). I do also very much like Wendy's lace city idea, and I would use it, if only I could properly understand it. But this probably too has its own limits.

At the end of the day I think coming up with one be all and end all notation or naming convention is an impossible task. These things are only there if they're useful, so it's often more appropriate to use a different convention depending on what your goal is.

I agree that there will never be a one-size-fits-all terminology that covers everything. I mean, we haven't discovered everything yet, which is the prerequisite for inventing a consistent naming scheme that covers all possible cases!

But I do believe it's possible to distill existing naming schemes into consistent classes of names. Some objects may fit into more than one class, but that's OK. For example, in 4D, the alternated tesseract is the same as its dual, but that doesn't mean alternation and dual-taking should be conflated. Before the Coxeter-Dynkin symbol was discovered, for example, the relationship between various uniform polytopes was unclear -- indeed, it was unclear whether to generalize the Archimedean polyhedra by forcing their facets ((n-1)-dimensional surtopes) to be regular, or to allow their surtopes to be themselves Archimedean (i.e., recursively uniform). But in retrospect, the latter definition allows for a more consistent naming scheme: it seems that the so-called "semi-regular" polytopes, which are the former generalization of the Archimedeans, are sporadic: they don't have any simple pattern of occurrence that leads to nice, aesthetically-pleasing designations. The uniform polytopes, OTOH, can be seen as resulting from all possible ringings of the nodes in their underlying symmetry group's Coxeter-Dynkin symbol, which is a far more general and consistent way of looking at these shapes.

Incidentally, this is why I've begun preferring to use Wendy's CD notation (x4x3o) for referring to uniform polytopes: they are compact, precise, and do not require remembering convoluted Greek/Latin stems to understand what they refer to.

[quickfur]

On the side of representation, though, one has to concede that a consistently-derived name like "geohedron" is far better than an arbitrary, unanalysable term like "cube". In this case the benefit may not be obvious, but once we start talking about "truncated cuboctahedron" then the consistently-derived scheme will definitely be shown to be superior. However, consider this: from the side of representation, we are interested not in the pronuncibility of the label, but a compact, mathematically-accurate representation that unambiguously designates a specific shape. But we already have such a representation: the Coxeter-Dynkin symbol. When we write x4o3o (using Wendy's notation), for example, it unambiguously designates the cube, and furthermore provides full information about its symmetry group, the shapes of its surface elements, its construction, etc..

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

Ah, but this is exactly the point I was driving at: when someone reads "cube", they immediately know what it is because it's so entrenched in current terminology. Had the word "cube" been written "x4o3o" since day one, we would have no problem whatsoever in recognizing it on sight. But it isn't, so we have a conflict between communication (using terms that people are already familiar with) vs. representation (existing terms are not always consistent with the underlying mathematical derivation of the object).

Dynkin symbols are great for identifying something in a list or table or expression or when you otherwise need a compact symbol, but they're not very useful in the middle of a sentence. It's similar to why we like to use numbers and single letters in algebra, but spell out numbers (like "four" instead of "4") and write out quantities when we're putting them in sentences. Both are necessary.

I agree. So what we need is a way to derive pronunciable names from their compact representations. Hence my allusion to the IUPAC -- one could just draw the chemical structure out explicitly and it would serve just as well, but having a semi-pronunciable equivalent of the drawn-out diagram allows us to put it in sentences without breaking up the paragraph (imagine, for example, if you needed to refer to a section of DNA in the middle of a sentence without having any pronunciable name for it -- you'd have to somehow insert a diagram of what you're referring to into the middle of the paragraph without breaking up the flow of the text).

This, of course, is predicated upon having a compact representation that's consistent and useful, to begin with. It wouldn't help if, for example, my terminology is of the form "polytope 1", "polytope 2", "polytope 3", ..., where the numbers refer to a list of polytopes I put together in arbitrary order. Klitzing's original complaint, in my understanding, was that we're far too trigger-happy to invent names when we haven't fully sorted out a consistent underlying representation yet (e.g. the whole deal with "rotunda", and many other such terms). The resulting names, because they're based on rather shaky grounds, tend to cause more confusion than help.

[Keiji]

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

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