## Classifying the segmentochora

Discussion of known convex regular-faced polytopes, including the Johnson solids in 3D, and higher dimensions; and the discovery of new ones.

### Re: Classifying the segmentochora

P.S. I should mention, that I did find patterns in the fundamental lace prisms with n-simplex symmetry; it's just that I haven't been able to rigorously prove them. Here's what I discovered so far:

- Just like 19D is a landmark in the lace prisms with n-cube symmetry, 8D is a special landmark in the lace prisms with n-simplex symmetry. In 8D, a large number of patterns from the previous dimensions all become sub-dimensional (planar, i.e., height zero), and only a relatively small number of fundamental lace prisms are left full-dimensioned. So it appears that many trends in earlier dimensions only go up to 7D.

- From 8D onwards, a fundamental lace prism A||B with n-simplex symmetry has at most two ringed nodes in either A or B, and for each ringed node in A at position i, B has a ringed node at (i+1).

- From 8D onwards, there are exactly 3 possible lace prism heights, one of which is 2 (just the prism of (n-1) uniform polytopes).

- There's also an interesting trend from 8D onwards in which the number of minimum-height lace prism in n dimensions is exactly the same as the number of planar lace prisms in (n+1) dimensions, suggesting that a certain class of lace prisms in n dimensions become subdimensional in (n+1) dimensions, but I haven't been able to discern the precise pattern for this.
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### Re: Classifying the segmentochora

I saw on the wiki that K4.8 hasn't been named yet. The way I always thought of K4.8 was as xxo2oxx&#xr. r means that it's a ring.
Because it's a ring, you can derive both xo2ox&#x || x2x aka tetrahedron || square, and xo2xx&#x || x2o aka triangular prism ||line out of it.
In this way, I think K4.8 should be called digonal gyrobicupolic ring, as xxoPoxx&#xr is called gyrobicupolic ring in every occasion.
The last way of looking at it (square pyramid || triangle) seems a bit less important to me, so I think it shouldn't get a completely new name because of this (the triangular prism doesn't get a special name just because it can be both written as ox2xx&#x and xx3oo&#x)
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### Re: Classifying the segmentochora

Yes, it well suits into this set of "gyrobicupolaic rings".
But because a triangle is not so much a ring, I usually tend to prefer that same set to be named "(2n-gon, n-antiprism)-wedge".
In the case of consideration ( K-4.8 ) we then have n=2.
(Similar wedge terms are generally well suited for all segmentotopes with a subdimensional base, which do not classify as a pyramid.)

It should be noted however, that this special instance also can be seen as a bidiminishing of the rectified pentachoron!

--- rk
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### Re: Classifying the segmentochora

But because a triangle is not so much a ring, I usually tend to prefer that same set to be named "(2n-gon, n-antiprism)-wedge".

As for wheather xxoPoxx&#xr is called magnabicupolic ring or a wedge, I will not discuss about, as I think it's just choosing between two equally good names (in my opinion a name is a loss of information, and therefore any not-well-known name is bad).
However, I think the fact that K4.8 is the bidiminishing of (xo2ox&#x) || (oxo2oxo&#xt) = ox3xo3oo&#x to (x2ox&#xt)||x2x is not very important for naming it, as we don't refer to the snub 24-cell by 24-diminished 600-cell, and thereby we do not underline the relation between the snub 24-cell and the 600-cell.
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### Re: Classifying the segmentochora

The "bicupolic ring" name came about because it was independently (re)discovered by Keiji, here in this forum. At the time, we didn't know that Klitzing had already included it in his list of segmentochora, so Keiji named it according to how he originally derived them.
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### Re: Classifying the segmentochora

but if we all agree on K4.8 being the gyrobicupolic ring, then why doesn't anyone change this in the wiki?
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student91
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### Re: Classifying the segmentochora

I've been calling the tetrahedron (disphenoid) || square "antiduowedge", I've encountered this shape as the vertex figure of xoxox - card, it has some interesting shaped facetings also.
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### Re: Classifying the segmentochora

student91 wrote:but if we all agree on K4.8 being the gyrobicupolic ring, then why doesn't anyone change this in the wiki?

We're waiting for you to sign up to the wiki and edit it yourself.

Frankly, though -- I've just been really busy with Real Life, and haven't had much time to do any 4D geometry recently. There's still a whole bunch of new CRF discoveries by Klitzing that I'd like to render someday, for example. And there's the "polytope of the month" on my website that hasn't been updated for the better part of the year; soon I'm going to have to rename it to "polytope of the year". If only there were a time shop where I can go and buy more spare time... so much geometry, so little time.
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### Re: Classifying the segmentochora

Can't create wiki account:
The specified invite key is invalid, already used or expired.

How easily one gives his confidence to persons who know how to give themselves the appearance of more knowledge, when this knowledge has been drawn from a foreign source.
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### Re: Classifying the segmentochora

student91 wrote:Can't create wiki account:
The specified invite key is invalid, already used or expired.

Hmm. PM Keiji and ask for help -- he's the one maintaining the server, so he's the right person to ask.
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### Re: Classifying the segmentochora

Most, but not all, of Richard's segmentochora are lace-towers or lace cities. A good deal fits on a tabular format, based on assuming that something like the ortho- and meta- and para- diminished icosahedron moves also to other symmetries.

The sole example of a non-lace construction is ike // cube, or "xo 3 * x% 2 oo" = so3so4ox&#x.
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### Re: Classifying the segmentochora

Well, all those diminished and gyrated ones are not lace prisms either.
--- rk
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### Re: Classifying the segmentochora

I counted them under lace prisms. They're not Wythoff lace prisms.

All that is needed for a lace prism is that there must be a clear progression between the elements. They were invented long before i figured out how the dynkin symbol worked. The actual idea sprang from an early map of the simplex product, and that the general face of a figure through non-axial coordinates, is a lace prism.

Post 1249: The base 18 number of this form is square, as is 249, 49 and 9 itself.
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### Re: Classifying the segmentochora

Hugh? How'd you derive e.g. a "line || triangle" (= squippy) as a lace prism (in that orientation)?
--- rk
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### Re: Classifying the segmentochora

A line on a triangle might be, eg ox&xo3oo&#x (where the line is perpendicular to the triangle), or oxx&#x (= square pyramid, line parallel to an edge).
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### Re: Classifying the segmentochora

Okay for that easy & tiny one. Should have noted that too.

But then, what about K-4.136 = "id || tedrid"?
or about K-4.168 = "bagydrid || tid"?
etc ...

(tedrid = tridiminished rhombicosidodecahedron = J83,
bagydrid = bigyrated diminished rhombicosidodecahedron = J79)

--- rk
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### Re: Classifying the segmentochora

All one needs in a lace prism is a clear progression. Wythoff figures have this, but there not alone. Simply implementing a pentagon at an icosahedral vertex, will do pretty much the same thing. You go from one pentagon-hack to another, but it's still a pentagon-face. Look at my previous post, where i divvy up all of theses under the base header.

viewtopic.php?f=25&t=1721&start=30#p18044

K-4.136 is the same as DDD K-1.31 and K.168 is of course, DGG K-159.

In short, they're like the ordinary K-1.31, and K4.158, except that they are variously diminished or gyrated in different ways. The active points match the same as the four ways of diminishing the icosahedron, but some have multiple operators (D vs G). All of the G's occur in just two figures, which are shown in the main table with *, **, and listed under each operator at the bottom, rather like the periodic table!

Actually the first riddle presented a bigger problem, because you have to realise that line || triangle makes no sense unless triangle is already line || point!

While ike // cube (which is between the dodecahedron and the f-icosahedron (ie rings 2 and 5, counting the vertex of 3,3,5 as ring 0), rather than between the first two. It's pretty straight-forward to put a cube on the dodecahedron.
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### Re: Classifying the segmentochora

To my understanding lace prisms so far were stacks of 2 Wythoffian bases with various edge lengths in their Dynkin symbols, laced by a further independent edge length.

Operations being added on those is secondary.

Segmentotopes OTOH do include all these by definition, but then restrict to having all unit edges.

This is why lace prisms and segmentotopes are closely related but not exactly the same. Either term does extend from their (great) common intersection into different directions.

If you now come to redefine "lace prisms" to include secondary operations, this is kind of unfair. At least WRT the former comparision.

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### Re: Classifying the segmentochora

Lace prisms have always been about 'progressions', the antiprism has always been part of the mix. The idea that they are exclusively WME figures is hardly correct, since the progressions are usually reckoned in terms of the alternate CF structure as well. Likewise, antitegums of non-regular figures must always involve a catalan, and not a WME. That is the wording used in my earlier post.

Still, the operations like DDG etc are not so much "operations done after the lace prism", since these are actually done before the lace-prism is applied. It's actually a sub-symmetry of the icosahedral group, in exactly the same way that tri-diminished icosahedron is subsymetric to the icosahedron. Most of them exist in Johnson's figures too. For example, K4.168 is DGG but then one of its bases is DGG too. There has never been any issue with admitting K4.168 as a lace-prism, because the icosahedron-vertex is variously diminished or gyrated.

On the other hand, the figures i have difficulty with calling lace-prisms, are things like ike || cube. The triangle || line is a valid segmentotope, but not a lace prism. Not in that form, because there is no progression from triangle to line. You have to deconstruct the triangle to something that does have a progression to a line: ie pt - line. The triangle || line never occurs where lace-prism faces might appear, but the pt.line.line can, because there are three bases. Writing, eg so3so4ox&#x, for example, does not make it a 'lace prism', but a parallel set of compounds laced together. The progression is not obvious. Even at "@3xo * xx 2 o% " which is the currently recieved orbifold, makes me no sense, since i have not untangled orbifold notation well enough to read lace towers. I can construct the thing, but the orbifold figures have lots of active regions, which i have not teased out.

The connection with WM figures seems to go no further back until the time when i was asked to describe the vertex-figures of the WME figures. Some experience and the use of 'vertex-nodes' and the radiant figures etc, allowed me to quickly grasp what was going on. They're lace prisms.

One should note that the lace-prism notation, as applied to segmentotopes, is an accommodation for the purpose. The unit edge do not occur in the vertices of uniform figures.

My earliest use of what we might call a lace prism comes from radiant space and the 'cuboctahedron product' that Jonathan played around with.

You imagine three coordinates, x,y,z. A point (1,0,0) represents the polytope X at the base size. You can scale it up and down accordingly. The point say (1,1,0) represents the prism-product of X,Y at their size. The prism and tegum products were defined as the figures defined by the sum(x)=1, and max(x)=1, which are their normal canonical vertices. The nature of the pyramid product was evaluated at first, from cosidering the figure stretched from (1,0,0) to (0,1,0) to (0,0,1), where the faces are indeed pyramid figures.

Of course, you can stick other figures in here too. Consider the triangle (1,1,0), (1,0,1), and (0,1,1). What does this mean? In modern parlence, we would write this as a lace-prism xxo2xox2xxo&#x. But there are elements of P(X, Pyr(Y,Z)) in there. The square faces at the end are P(x, T(y,z)), etc. But what is this exotic thing in the triangle?

Anyway, i decided that 'radiant space' leads to more dead ends than it solves, and come down to trying to hack Coxeter's exotic CD diagrams. The most reliable source of these is 'regular complex polytopes', which gives explicit examples. After many attempts (including reading x3o4x3o4z as a tiling of rhombic dodecahedra!), I introduced a thing called the vertex-node.
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### Re: Classifying the segmentochora

wendy wrote:Lace prisms have always been about 'progressions', the antiprism has always been part of the mix. The idea that they are exclusively WME figures is hardly correct, since the progressions are usually reckoned in terms of the alternate CF structure as well. Likewise, antitegums of non-regular figures must always involve a catalan, and not a WME. That is the wording used in my earlier post.

Admitted, catalans were within your scope too (or more generally: mirror margin structures). I just forgot to mention. But that does not Change my point at all.

Still, the operations like DDG etc are not so much "operations done after the lace prism", since these are actually done before the lace-prism is applied. It's actually a sub-symmetry of the icosahedral group, in exactly the same way that tri-diminished icosahedron is subsymetric to the icosahedron. Most of them exist in Johnson's figures too. For example, K4.168 is DGG but then one of its bases is DGG too. There has never been any issue with admitting K4.168 as a lace-prism, because the icosahedron-vertex is variously diminished or gyrated.

Sorry Wendy, this point is rather essential in here: whether operations will be applied before of after lace prism construction. E.g. for 3D lace prisms: whether operations apply to 2D polygons (bases) before those will be laced (as allowed for segmentohedra) or are applied onto the already assembled 3D object. - To my knowledge lace prisms so far did not include such additional operations, neither before, nor there-after.

On the other hand, the figures i have difficulty with calling lace-prisms, are things like ike || cube. The triangle || line is a valid segmentotope, but not a lace prism. ...

Hehe

... Not in that form, because there is no progression from triangle to line. You have to deconstruct the triangle to something that does have a progression to a line: ie pt - line. The triangle || line never occurs where lace-prism faces might appear, but the pt.line.line can, because there are three bases. ...

In fact even cube || ike is nothing different here, as it incorporates those line || triangle segmentohedra for lacing figures! So its just a related figure one dimension above ...

... Writing, eg so3so4ox&#x, for example, does not make it a 'lace prism', but a parallel set of compounds laced together. ...

Sorry Wendy once more! That description as so3so4ox&#x surely can be read as a (s3s4o = ike) || (o3o4x = cube), i.e. as first applying snubbing and there-after the lacing. But, and that is important, it well could be considered in the opposite order as well: first doing the lacing in the sense of xo3xo4ox&#y (= toe || cube, which by means of longer lacing edges y well can be made valid), then applying alternated faceting WRT alternate vertices of the toe only (i.e. a snubbing being applied to the full 4D lace prism, not to the 3D base only!), and finally (if needed) a relaxation to all unit edges (which additionally would have to resize those lacing y edges too!). - And, what is even more important, both points of view do result in the same structure! And only this, after all, does allow for the well-definiteness of this notation, i.e. of writing "so3so4ox&#x"!

... The progression is not obvious. Even at "@3xo * xx 2 o% " which is the currently recieved orbifold, makes me no sense, since i have not untangled orbifold notation well enough to read lace towers. I can construct the thing, but the orbifold figures have lots of active regions, which i have not teased out.

Don't ask me on them. I've even less ideas on them than you...

The connection with WM figures seems to go no further back until the time when i was asked to describe the vertex-figures of the WME figures. Some experience and the use of 'vertex-nodes' and the radiant figures etc, allowed me to quickly grasp what was going on. They're lace prisms.

One should note that the lace-prism notation, as applied to segmentotopes, is an accommodation for the purpose. The unit edge do not occur in the vertices of uniform figures.

My earliest use of what we might call a lace prism comes from radiant space and the 'cuboctahedron product' that Jonathan played around with.

You imagine three coordinates, x,y,z. A point (1,0,0) represents the polytope X at the base size. You can scale it up and down accordingly. The point say (1,1,0) represents the prism-product of X,Y at their size. The prism and tegum products were defined as the figures defined by the sum(x)=1, and max(x)=1, which are their normal canonical vertices. The nature of the pyramid product was evaluated at first, from cosidering the figure stretched from (1,0,0) to (0,1,0) to (0,0,1), where the faces are indeed pyramid figures.

Of course, you can stick other figures in here too. Consider the triangle (1,1,0), (1,0,1), and (0,1,1). What does this mean? In modern parlence, we would write this as a lace-prism xxo2xox2xxo&#x. But there are elements of P(X, Pyr(Y,Z)) in there. The square faces at the end are P(x, T(y,z)), etc. But what is this exotic thing in the triangle?

Nice nonapeton. Its incidence matrix should be as follows:
Code: Select all
`xxo2xox2oxx&#xo..2o..2o..    | 4 * * | 1 1 2 2 0 0 0 0 0 | 1 2 2 1 2 2 1 2 0 0 0 0 0 | 2 1 1 2 1 1 2 2 1 0 0 0 | 1 1 2 1 1 0.o.2.o.2.o.    | * 4 * | 0 0 2 0 1 1 2 0 0 | 0 2 1 2 0 0 0 2 1 2 1 2 0 | 1 2 1 0 0 0 2 1 2 1 2 1 | 1 0 1 2 1 1..o2..o2..o    | * * 4 | 0 0 0 2 0 0 2 1 1 | 0 0 0 0 1 2 2 2 0 1 2 2 1 | 0 0 0 1 1 2 1 2 2 1 1 2 | 0 1 1 1 2 1---------------+-------+-------------------+---------------------------+-------------------------+------------x.. ... ...    | 2 0 0 | 2 * * * * * * * * | 1 2 0 0 2 0 0 0 0 0 0 0 0 | 2 1 0 2 1 0 2 0 0 0 0 0 | 1 1 2 1 0 0... x.. ...    | 2 0 0 | * 2 * * * * * * * | 1 0 2 0 0 2 0 0 0 0 0 0 0 | 2 0 1 2 0 1 0 2 0 0 0 0 | 1 1 2 0 1 0oo.2oo.2oo.&#x | 1 1 0 | * * 8 * * * * * * | 0 1 1 1 0 0 0 1 0 0 0 0 0 | 1 1 1 0 0 0 1 1 1 0 0 0 | 1 0 1 1 1 0o.o2o.o2o.o&#x | 1 0 1 | * * * 8 * * * * * | 0 0 0 0 1 1 1 1 0 0 0 0 0 | 0 0 0 1 1 1 1 1 1 0 0 0 | 0 1 1 1 1 0.x. ... ...    | 0 2 0 | * * * * 2 * * * * | 0 2 0 0 0 0 0 0 1 2 0 0 0 | 1 2 0 0 0 0 2 0 0 1 2 0 | 1 0 1 2 0 1... ... .x.    | 0 2 0 | * * * * * 2 * * * | 0 0 0 2 0 0 0 0 1 0 0 2 0 | 0 2 1 0 0 0 0 0 2 0 2 1 | 1 0 0 2 1 1.oo2.oo2.oo&#x | 0 1 1 | * * * * * * 8 * * | 0 0 0 0 0 0 0 1 0 1 1 1 0 | 0 0 0 0 0 0 1 1 1 1 1 1 | 0 0 1 1 1 1... ..x ...    | 0 0 2 | * * * * * * * 2 * | 0 0 0 0 0 2 0 0 0 0 2 0 1 | 0 0 0 1 0 2 0 2 0 1 0 2 | 0 1 1 0 2 1... ... ..x    | 0 0 2 | * * * * * * * * 2 | 0 0 0 0 0 0 2 0 0 0 0 2 1 | 0 0 0 0 1 2 0 0 2 0 1 2 | 0 1 0 1 2 1---------------+-------+-------------------+---------------------------+-------------------------+------------x.. x.. ...    | 4 0 0 | 2 2 0 0 0 0 0 0 0 | 1 * * * * * * * * * * * * | 2 0 0 2 0 0 0 0 0 0 0 0 | 1 1 2 0 0 0xx. ... ...&#x | 2 2 0 | 1 0 2 0 1 0 0 0 0 | * 4 * * * * * * * * * * * | 1 1 0 0 0 0 1 0 0 0 0 0 | 1 0 1 1 0 0... xo. ...&#x | 2 1 0 | 0 1 2 0 0 0 0 0 0 | * * 4 * * * * * * * * * * | 1 0 1 0 0 0 0 1 0 0 0 0 | 1 0 1 0 1 0... ... ox.&#x | 1 2 0 | 0 0 2 0 0 1 0 0 0 | * * * 4 * * * * * * * * * | 0 1 1 0 0 0 0 0 1 0 0 0 | 1 0 0 1 1 0x.o ... ...&#x | 2 0 1 | 1 0 0 2 0 0 0 0 0 | * * * * 4 * * * * * * * * | 0 0 0 1 1 0 1 0 0 0 0 0 | 0 1 1 1 0 0... x.x ...&#x | 2 0 2 | 0 1 0 2 0 0 0 1 0 | * * * * * 4 * * * * * * * | 0 0 0 1 0 1 0 1 0 0 0 0 | 0 1 1 0 1 0... ... o.x&#x | 1 0 2 | 0 0 0 2 0 0 0 0 1 | * * * * * * 4 * * * * * * | 0 0 0 0 1 1 0 0 1 0 0 0 | 0 1 0 1 1 0ooo2ooo2ooo&#x | 1 1 1 | 0 0 1 1 0 0 1 0 0 | * * * * * * * 8 * * * * * | 0 0 0 0 0 0 1 1 1 0 0 0 | 0 0 1 1 1 0.x. ... .x.    | 0 4 0 | 0 0 0 0 2 2 0 0 0 | * * * * * * * * 1 * * * * | 0 2 0 0 0 0 0 0 0 0 2 0 | 1 0 0 2 0 1.xo ... ...&#x | 0 2 1 | 0 0 0 0 1 0 2 0 0 | * * * * * * * * * 4 * * * | 0 0 0 0 0 0 1 0 0 1 1 0 | 0 0 1 1 0 1... .ox ...&#x | 0 1 2 | 0 0 0 0 0 0 2 1 0 | * * * * * * * * * * 4 * * | 0 0 0 0 0 0 0 1 0 1 0 1 | 0 0 1 0 1 1... ... .xx&#x | 0 2 2 | 0 0 0 0 0 1 2 0 1 | * * * * * * * * * * * 4 * | 0 0 0 0 0 0 0 0 1 0 1 1 | 0 0 0 1 1 1... ..x ..x    | 0 0 4 | 0 0 0 0 0 0 0 2 2 | * * * * * * * * * * * * 1 | 0 0 0 0 0 2 0 0 0 0 0 2 | 0 1 0 0 2 1---------------+-------+-------------------+---------------------------+-------------------------+------------xx. xo. ...&#x | 4 2 0 | 2 2 4 0 1 0 0 0 0 | 1 2 2 0 0 0 0 0 0 0 0 0 0 | 2 * * * * * * * * * * * | 1 0 1 0 0 0  tripxx. ... ox.&#x | 2 4 0 | 1 0 4 0 2 2 0 0 0 | 0 2 0 2 0 0 0 0 1 0 0 0 0 | * 2 * * * * * * * * * * | 1 0 0 1 0 0  trip... xo. ox.&#x | 2 2 0 | 0 1 4 0 0 1 0 0 0 | 0 0 2 2 0 0 0 0 0 0 0 0 0 | * * 2 * * * * * * * * * | 1 0 0 0 1 0  tetx.o x.x ...&#x | 4 0 2 | 2 2 0 4 0 0 0 1 0 | 1 0 0 0 2 2 0 0 0 0 0 0 0 | * * * 2 * * * * * * * * | 0 1 1 0 0 0  tripx.o ... o.x&#x | 2 0 2 | 1 0 0 4 0 0 0 0 1 | 0 0 0 0 2 0 2 0 0 0 0 0 0 | * * * * 2 * * * * * * * | 0 1 0 1 0 0  tet... x.x o.x&#x | 2 0 4 | 0 1 0 4 0 0 0 2 2 | 0 0 0 0 0 2 2 0 0 0 0 0 1 | * * * * * 2 * * * * * * | 0 1 0 0 1 0  tripxxo ... ...&#x | 2 2 1 | 1 0 2 2 1 0 2 0 0 | 0 1 0 0 1 0 0 2 0 1 0 0 0 | * * * * * * 4 * * * * * | 0 0 1 1 0 0  squippy... xox ...&#x | 2 1 2 | 0 1 2 2 0 0 2 1 0 | 0 0 1 0 0 1 0 2 0 0 1 0 0 | * * * * * * * 4 * * * * | 0 0 1 0 1 0  squippy... ... oxx&#x | 1 2 2 | 0 0 2 2 0 1 2 0 1 | 0 0 0 1 0 0 1 2 0 0 0 1 0 | * * * * * * * * 4 * * * | 0 0 0 1 1 0  squippy.xo .ox ...&#x | 0 2 2 | 0 0 0 0 1 0 4 1 0 | 0 0 0 0 0 0 0 0 0 2 2 0 0 | * * * * * * * * * 2 * * | 0 0 1 0 0 1  tet.xo ... .xx&#x | 0 4 2 | 0 0 0 0 2 2 4 0 1 | 0 0 0 0 0 0 0 0 1 2 0 2 0 | * * * * * * * * * * 2 * | 0 0 0 1 0 1  trip... .ox .xx&#x | 0 2 4 | 0 0 0 0 0 1 4 2 2 | 0 0 0 0 0 0 0 0 0 0 2 2 1 | * * * * * * * * * * * 2 | 0 0 0 0 1 1  trip---------------+-------+-------------------+---------------------------+-------------------------+------------xx. xo. ox.&#x | 4 4 0 | 2 2 8 0 2 2 0 0 0 | 1 4 4 4 0 0 0 0 1 0 0 0 0 | 2 2 2 0 0 0 0 0 0 0 0 0 | 1 * * * * *  tepex.o x.x o.x&#x | 4 0 4 | 2 2 0 8 0 0 0 2 2 | 1 0 0 0 4 4 4 0 0 0 0 0 1 | 0 0 0 2 2 2 0 0 0 0 0 0 | * 1 * * * *  tepexxo xox ...&#x | 4 2 2 | 2 2 4 4 1 0 4 1 0 | 1 2 2 0 2 2 0 4 0 2 2 0 0 | 1 0 0 1 0 0 2 2 0 1 0 0 | * * 2 * * *  {4} || tetxxo ... oxx&#x | 2 4 2 | 1 0 4 4 2 2 4 0 1 | 0 2 0 2 2 0 2 4 1 2 0 2 0 | 0 1 0 0 1 0 2 0 2 0 1 0 | * * * 2 * *  {4} || tet... xox oxx&#x | 2 2 4 | 0 1 4 4 0 1 4 2 2 | 0 0 2 2 0 2 2 4 0 0 2 2 1 | 0 0 1 0 0 1 0 2 2 0 0 1 | * * * * 2 *  {4} || tet.xo .ox .xx&#x | 0 4 4 | 0 0 0 0 2 2 8 2 2 | 0 0 0 0 0 0 0 0 1 4 4 4 1 | 0 0 0 0 0 0 0 0 0 2 2 2 | * * * * * 1  tepe`

Anyway, i decided that 'radiant space' leads to more dead ends than it solves, and come down to trying to hack Coxeter's exotic CD diagrams. The most reliable source of these is 'regular complex polytopes', which gives explicit examples. ...

Sadly I'ven't got a sight of that one so far

... After many attempts (including reading x3o4x3o4z as a tiling of rhombic dodecahedra!), ...

what? rather of rhombic cuboctahedra and of cuboctahedra, cf:
Code: Select all
`x3o4x3o4*a   (N → ∞). . . .    | 12N |   4   4 |  2   4  2  2  2 | 2 1 2 1-----------+-----+---------+-----------------+--------x . . .    |   2 | 24N   * |  1   1  1  0  0 | 1 1 1 0. . x .    |   2 |   * 24N |  0   1  0  1  1 | 1 0 1 1-----------+-----+---------+-----------------+--------x3o . .    |   3 |   3   0 | 8N   *  *  *  * | 1 1 0 0x . x .    |   4 |   2   2 |  * 12N  *  *  * | 1 0 1 0x . . o4*a |   4 |   4   0 |  *   * 6N  *  * | 0 1 1 0. o4x .    |   4 |   0   4 |  *   *  * 6N  * | 1 0 0 1. . x3o    |   3 |   0   3 |  *   *  *  * 8N | 0 0 1 1-----------+-----+---------+-----------------+--------x3o4x .    |  24 |  24  24 |  8  12  0  6  0 | N * * *  sircox3o . o4*a |  12 |  24   0 |  8   0  6  0  0 | * N * *  cox . x3o4*a |  24 |  24  24 |  0  12  6  0  8 | * * N *  sirco. o4x3o    |  12 |   0  24 |  0   0  0  6  8 | * * * N  co`

... I introduced a thing called the vertex-node.

--- rk
Klitzing
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### Re: Classifying the segmentochora

I see what is going on here. It's a case of "A is B, therefore B is A".

It should be noted that lace prisms, like segmentotopes, represent "outcomes" (that is, one might test for them), while the stratic wythoff construction (and its matrices) are "constructions". Wythoff constructions of various forms construct all but one of the elliptical polytopes (except the GAP, which John Conway found). They do less well in euclidean space, where the families of laminate polytopes ellude them. The spreadsheet works on WLPs because they Wythoff figures are inherently position polytopes. It works on the cube on ike because i said to look for of3oo5xo&#f, and divide the edge by f: ie the ike works in much the same way that the K4.168 does: it's a subset of something that is a WLP.

But you can correctly enumerate from the rules for WLP the surtope of the figures like K4.168, but not for the cube||ike. This is because the progressions are clear in '168, but not in the cube//ike.

Lace prisms were set up with the notion of 'progressions' in mind. That is, there is an obvious change from X to Y. Then, putting X parallel to Y, and using these changes in sequence, gives a section through the LP on X, Y. Stott's addition is an eye-opener here: She found ways to hook up all of the figures of the same symmetry, except the snub.

If one uses the Dynkin symbol to write X and Y as WME or WMM, you will get lace prisms. But writing s3s4o or any of the other snubs, is neither WME or WMM. It's smoething else. However, lace-prisms are not restricted to WM figures, no more than any of the products are. You can have a pyramid based on a torus, for example, and a lace-prism that runs from a circle to an ellipse.

Writing a laced compound of non WME or WMM figures still gives access to what a laced compound does. So putting eg s3s4o || o3o4x is the same as writing so3so4ox&#x, but it does not mean that this is automatically going to be a lace prism, except in the broadest sense. One would have to necessarily show that using a non-edge node in the dynkin symbol (like s), would preserve the LP nature.

Wythoff and Stott give access to massive amounts of possible progressions, much more than kepler's corner-hacks. For that end, we see that one gets a better sense of what lace prisms might do through WLPs .
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### Re: Classifying the segmentochora

This is a "simple" regular complex polytope, by way of incidence-matrix. These are straight out of coxeter.

Code: Select all
`    pt          8    3                     3(3)3                 d0 = 1:60 = 3/2     3          3    8                     G=24                  d1 = 0:60 = 1/2    pt         27    8    8                 Hess Polyhedron      d0 =  2     3          3   72    3                                      d1 =  1   3(3)3        8    8   27              G=548 = dec 648         d2 =  0:60 = 1/2    pt       200    27   72    27         Whytting Polychoron    d0 = 2:80 = 8/3    3          3  1800    8     8          200 = dec 240         d1 = 1:80 = 5/3   3(3)3       8     8  1800    3         1800 = dec 2160        d2 = 1:20 = 7/6 3(3)3(3)3    27    72   27    200       G=10.9600 = dec 155520  d3 = 0:80 = 2/3.    pt        (1)   200  1800  1800  200     Tiling of WP     3         3    (80)  27    72    27      230 = dec 270   3(3)3       8      8  (230)   8     8 3(3)3(3)3    27     72   27   (80)    3  Wytting    200   1800  1800  200    (1)`
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### Re: Classifying the segmentochora

Dear Wendy (& for sure all other interested ones),

I think we should distinguish here better between your notion of lace prisms (kind a stack of bases in conjunction with some more or less obvious progression between pairs of either side elements) and your notational device introduced for represent those, based on more or less extended Dynkin symbols. It seems that you argues from the first POV, whereas I from the second.

With respect to using snub nodes, there are 2 very different POVs as well. Either lace(snubs) or snub(lace). And it is not obvious (at least to me) whether those constructions would commute in general. Even more surprising when they do! And e.g. so-2n/d-ox&#x would be such a (lesser dimensional) candidate: Either you consider s-2n/d-o, the 2D snub constructed by vertex alternation from x-2n/d-o (thus producing a regular n/d-gon), thereafter being placed atop of o-2n/d-x (regular n/d-gon), joined by with lacing edges of size x (which then imply restrictions on n/d in order to be valid). - Or you could consider the construction of the lace prism xo-2n/d-ox&#x first. And thereafter then alternating the top vertices, replacing those by the sectioning facets underneath (here considered as cuts, chopping of 3D chunks!), which here would have trapezium shapes. Relaxing this construction then back to all unit edges (as usual, if possible) again results in the same n/d-cupola as before.

The same holds true for os3os4xo&#x. When being considered as a corresponding snubbing of ox3ox4xo&#y (you'd need a different longer sized lacing edge here in order to make that latter one valid!), the sectioning facets underneath the omitted alternate vertices of the bottem base (only!) - in the sense of chopping off 4D-chunks! - would be right those oddy square pyramids (in the sense of line || triangle).

I even managed to find a similar construction for os3os4xo3oo&#x (rico || sadi) recently: not only in the sense of snubbing first (in 4D) and then stacking into 5D, but well also in stacking first into 5D and snubbing thereafter alternate vertices of the bottom base by chopping off 5D-chunks (i.e. by introducing 4D sectioning facets underneath as additional lacing elements - which here are trippies = pyramids ontop of trigonal prisms, here occuring in the sense of {3} || tet).

--- rk
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### Re: Classifying the segmentochora

It's possible to relax certain restrictions, which are more based on "whether i can see" a progression, as to "whether a progression exists". It's also useful to relax the requirement for a wythoff base.

A Wythoff lace prism has two or more WME and defined lacing. (ie the nodes are 'numeric', o v, x, q, f, h, &c) In short, the sort of thing that works in the spreadsheet.

A Dynkin lace prism is one written in generalised dynkin symbols (eg the nodes might include things like 's', 'm').

A Segmentotope is a figure whose vertices lie in a simplex of parallel spheres, being part of a common larger sphere.

A Subsymmetric group is derived from a group by diminishing, or gyration.
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### Re: Classifying the segmentochora

Yesterday I finally implemented a function in polyview to compute dichoral angles, and I was going through the bicupolic rings pages on the wiki to add dichoral angles to them, when I noticed something interesting: the n-gonal magnabicupolic rings are basically Stott expansions of the n-gonal prism pyramids, and the n-gonal orthobicupolic rings are basically Stott expansions of the n-gonal pyramid pyramids. All corresponding dichoral angles are the same, including the segmentochoron heights.

The digonal magnabicupolic ring would thus correspond with the square pyramid (i.e., digonal prism pyramid, since digonal prism = square), and the digonal orthobicupolic ring would correspond with the tetrahedron (i.e., digonal pyramid pyramid = triangle pyramid = tetrahedron).

I haven't quite worked out what corresponds with the gyrobicupolic rings, but I suspect it's a (2n/2)-gonal pyramid pyramid. (I.e., a disjoint star polygon pyramid pyramid.)
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### Re: Classifying the segmentochora

quickfur wrote:Yesterday I finally implemented a function in polyview to compute dichoral angles, and I was going through the bicupolic rings pages on the wiki to add dichoral angles to them, when I noticed something interesting: the n-gonal magnabicupolic rings are basically Stott expansions of the n-gonal prism pyramids, ...

Yep, just cf.:
Code: Select all
`    x-n-o                           o-n-o                   = Stott exp. of               x-n-x   x-n-x                   o-n-x   o-n-x`
i.e. n-g||2n-p = Stott exp. of pt||n-p.

... and the n-gonal orthobicupolic rings are basically Stott expansions of the n-gonal pyramid pyramids. All corresponding dichoral angles are the same, including the segmentochoron heights.

Cf.:
Code: Select all
`    x-n-x                           o-n-x                   = Stott exp. of               x-n-o   x-n-o                   o-n-o   o-n-o`
i.e. 2n-g||n-p = Stott exp. of n-g||ortho line

Both immediate and known.

--- rk
Klitzing
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### Re: Classifying the segmentochora

Klitzing wrote:[...]Both immediate and known.

--- rk

Yes. But I was more interested in the gyrobicupolic ring case. Does a Stott-contracted version exist? and if so, what would it be?
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### Re: Classifying the segmentochora

for a general polygon, you could write just 'N' eg oNo or oNx, where M and N are generic polygons. P is different in that you can give an explicit polygon, eg P8 for the octagon, or xP1728000o for the centionshot. You can use it to separate numbers, eg p18P3o is the vertex-figure of x18o3o, here the 3 is converted to a generic polygon P3, to separate it from the generic polygon shortchord p18. Brackets also work, but are best used sparingly.

Note that lace prisms were first used with the vertex figures of wythoff polytopes, and thus can have any number of bases, arranged perpendicular to an 'altitude', in the form of a simplex. Lacing is of the forms with q, h, qq, ff, hh (meaning p8, p10, p12). The implementation to other uses is a design feature of extensibility i try to build in.

With Stott expansion, one supposes a node, completely, goes to completely full. So eg ox3oo&#x 'triangular pyramid' goes to ox3xx 'triangular cupola'.

I'm still wondering if one can do a row expansion, but sweet dreams are made of this. Still, every little bit helps.
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wendy
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### Re: Classifying the segmentochora

quickfur wrote:
Klitzing wrote:[...]Both immediate and known.

--- rk

Yes. But I was more interested in the gyrobicupolic ring case. Does a Stott-contracted version exist? and if so, what would it be?

Well, first: what is the "gyrobicupolic ring"? - It ought to be
Code: Select all
`     x-n-x                    x-n-o     o-n-x`

Then, you might like to Stott contract the left nodes each, resulting in
Code: Select all
`     o-n-x                      o-n-o  (-x)-n-x`

But then, what "is" (-x)-n-x? - It should obviously have 2n sides. But instead bending always 1/n of the straight angle (= of 180 degrees) to the right, it still does so, but instead going then into that direction, it procedes into the opposite one (according to the minus sign). Thus you well could say, you'd bend (n-1)/n of the straight angle to the left. That is, (-x)-n-x "=" x-(n-1)/n-x.

You might want to apply a further Stott contraction to the right nodes each, resulting in
Code: Select all
`      o-n-o                       o-n-(-x) (-x)-n-o`

Again, (-x)-n-o "=" x-(n-1)/n-o. Thus the latter contraction results in the pyramid ontop of the (n-1)/n-antiprism.

--- rk
Klitzing
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### Re: Classifying the segmentochora

Recently I considered by coincidence some segmentochora with non-convex bases.
Among those there were also the conjugate pair
• sissid || did = oo5ox5/2xo&#x
• gad || did = xo5ox5/2oo&#x

When I calculated the respective circumradii I got really surprized:
Both do have the same value here: R = sqrt((8+2 sqrt(5))/11) = 1.064815

That is, we not only can consider the stack of those two into a lace tower
• sissid || (pseudo) did || gad = oox5oxo5/2xoo&#xt
but that figure moreover would still be orbiform (i.e. have unit edges throughout and all vertices lie on a unique circumsphere)!

Anyone would have expected that?

Btw., the cells of that bistratic stack then clearly are 1 sissid o5o5/2x, 12 peppies oo5ox&#x, 12 staps ox5/2xo&#x, 12 paps ox5xo&#x, 12 stappies xo5/2oo&#x, and 1 gad x5o5/2o.

--- rk
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