higher-dimensional CRFs

[Klitzing]

Well, it's known that the number of CRF polychora would be enormous (if not infinite). And in higher dimensions this wouldn't become much better. But still, there are already some individual items or even families known therein, which deserve a special place to be re-found again later, instead of being scattered in different threads.

To start with, we have the easiest ones, ...

• the prisms of subdimensional CRFs. These then are clearly CRFs too.
• Similar would count in the multiprisms of CRF components.
• And also for all subdimensional CRFs with a well-defined circumradius, which happens to be less than unity (in edge scale), there would be its corresponding pyramid, its dipyramid, and even its elongated pyramid and elongated dipyramid.

But on the other hand there are already known some families too, which come in as a cross-dimensional application of the very same idea. A quite early example here was the family based on Wendy's ursatopes, cf. http://hi.gher.space/wiki/Ursatope.

I'd like to open and to recommend this thread both for a collection of according ideas and for the discussion of individuals.
Even cross-links to older posts from other threads ought be desirable, I think.

But most of all, I'm very curious on what will appear herein ...

--- rk

[Klitzing]

Just to start with posting in here myself, I'd like to re-post within this forum, what I already posted in the polyhedron list archive nearly one month ago. - Sadly it didn't get any reply there so far. Hope it creats more interest in here then...

On the "Infinite serie of extreme Delaunay polytope"

Recently I stumbled across an article of Mathieu Dutour Sikirić from 2013: "Infinite serie of extreme Delaunay polytope", as can be viewed at http://arxiv.org/pdf/math/0305196v1.pdf. It describes a quite interesting set of polytopes, even so the article is quite error prone. (Eg. neither the circumcenter nor the upper lacing edge length is calculated correctly - at least wrt. the therein provided coordinates. Even the symmetry assertion (for n=6) of the main theorem is wrong when using those coordinates, because the layer distance there is not conform to an circumscribing hypersphere (as for Delaunay cells needed), rather it is given fixed.)

The main idea of that article is to consider a series of some special type of lace towers polytopes with dimensionally analoguous layers, running through all applicable dimensions. There he considers especially the stack of a single point (top) atop a (pseudo) demihypercube (equatorial) atop a crosspolytope (bottom). - In Wendy Krieger's lace notation that fellow would generally be described as oxo3ooo3ooo *b3ooo...ooo3oox&#yt (where the lacing edge y quite generally could be considered as independent from the equal layerwise edge sizes x). Clearly the 4D representant then ought be compressed into oxo3oox3ooo&#yt and the 3D one then even could be described as oxx oox&#yt.

In terms of the layer distance d the lacing edge length y = y(d) can be given only depending on the dimension value n as: y = sqrt((n-1)/8 + d²), at least when using unit edge lengths x here. - In the crystallographic context of Delaunay cells it ought be essential now to provide a further formula d = d(n), which asserts the existance of an unique circumradius. - In the mathematical context of polytopists it might be more interesting however to provide a different such formula d = d*(n), which provides equal sized edges throughout, i.e. y = x. This simply can be achieved by using d = sqrt((9-n)/8). This latter choice then clearly provides the there proclaimed coincidence of the 6D (n=6) case with the well-known Schläfli-Gosset polytope 2_2,1 (or "jak" in Jonathan Bowers' terms).

We even could - again generally (but then n>3) - consider this fellow as the following lace city.

Code: Select all

     P           where:
                  P = o3o3o *b3o...o3o
   H   h          H = x3o3o *b3o...o3o
                  h = o3o3x *b3o...o3o
 P   C   P        C = o3o3o *b3o...o3x 


Refering to that picture the coordinates can be given like this (again using unit edge length x and arbitrary layer distance d):

P (top layer):
  (0; 0, 0, 0, ...; +d)
H (equatorial left):
  (-1/sqrt(8); -1/sqrt(8), 1/sqrt(8), 1/sqrt(8), ...; 0) & all even changes of sign in the medial block of coordinates
h (equatorial right):
  (1/sqrt(8); 1/sqrt(8), 1/sqrt(8), 1/sqrt(8), ...; 0) & all even changes of sign in the medial block of coordinates
P (bottom left/right):
  (+-1/sqrt(2); 0, 0, 0, ...; -d)
C (bottom central):
  (0; 1/sqrt(2), 0, 0, ...; -d) & all permutations and changes of sign in the medial block of coordinates

For the remainder I will consider equal edge lengths throughout now, i.e. assuming the above provided formula for d.

The case n=3 is not too interesting here. Here the medial block of above coordinates has just the count 1. Thus neither even changes of sign nor permutations are possible there. The bottom layer degenerates into a mere square, the equatorial one into a single unit edge. The bottom segment becomes a trigonal prism (trip). Whereas the top segment even becomes a mere (dimensionally degenerate) triangle. Moreover it is evident, that the derived "polyhedron" is concave.

The case n=4 is a bit more intricate: The equatorial pseudo layer here is the tetrahedron (tet). Thus the upper segment represents the segmentochoron K-4.1 (pen), the lower one is K-4.5 (rap). But as in this external blend the respective lacing cells tet (upper segment) and oct (lower segment - those are the only ones, which are fully connected to the equatorial layer) happen to become corealmic, they can be united. But then their lacing faces in turn are coplanar, resulting in 60°-rhombs.
Thus, even so this polychoron is convex, it still is not a CRF.

The case n=5 then becomes interesting. The equatorial pseudo layer here is the hexadecachoron (hex), while the bottom layer is the same solid, they simply are in different orientations. Thus the upper segment represents the segmentoteron hexpy. The lower one is just the hemipenteract (hin). The lacing elements of the upper segment clearly are all pentachora (pen). The diteral angle between those and the base of that pyramid equals arccos(1/sqrt(5)) = 63.434949°.
The lacing elements of the lower segment, which are fully connected to the equatorial layer, are hexes and pens. The diteral angle between the hexes here is 90°, while that between the equatorial hex and the lacing pens is arccos[-1/sqrt(5)] = 116.565051°. Thus the upper lacing pens and the directly connected lower pens are corealmic and connect into tetrahedral (line)tegums (aka. tetrahedral dipyramids, tete).
Accordingly the total facet count here becomes (8+8) pens + (8+1) hexes + 8 tetes. This polyteron then clearly is CRF.

As already stated, n=6 is uniform, the even higher-symmetrical jak. It clearly is a convex polyexon. The equatorial pseudo facet here is hin. That one dissects the lacing triacontaditera (tac), which strech through both segments, into pairs of (thus corealmic) hexpies.
The total facet count is 27 tacs + 72 hixes (hexatera).

I'll skip some further dimensions here. More interesting then is again the case n=9. The above equation for d freely provides that the 8D unit-edged demicube (hocto) and the 8D unit-edged crosspolytope (ek) can be symmetrically superimposed as a compound and their hull then still will be a unit-edged 8D polytope with 128+16=144 vertices.
And on the other hand, the 8D unit-edged demicube (hocto) itself is decomposable into all unit-edged centri-pyramids underneath all of its facets, i.e. into 128 decayotta (day) plus 16 hemihepteractic pyramids. Esp. hocto thus needs to have a circumradius of 1.

Today I now considered that first of those degenerate n=9 cases in more detail. By its description it is given as .xo3.oo3.oo *b3.oo3.oo3.oo3.oo3.ox&#zx, where the dots just refer to the here omitted "opposite" single point (top P - and therefore these freely could be neglected as well) and the trailing &#zx just states that the height here equals zero, while the lacing edges have the same size x as the "layers". (Note that mutual 8D circumradii of the 2 individual "layers" differ according to our choice of d=0.) By little of research one recognizes that these 16 ek vertices under that hull operation define just simply shallow pyramids attached to any of the 16 hesa facets of that other "layer", hocto.

Accordingly this here considered new polyzetton consists out of the 128 octaexa (oca) of hocto plus 16x the sloping facets of those hesa-pyramids, which are 14 hemihexeractic pyramids (haxpy) plus 64 ocas each. Thus giving rise to a total of 224 haxpies + 1152 ocas. - And, by construction, that polyzetton is still a biform CRF!

Btw., Dutour's restriction, of n to be even and >5, here nowhere had to be taken account for.

--- rk

--- rk

[Klitzing]

Code: Select all

     P           where:
                  P = o3o3o *b3o...o3o
   H   h          H = x3o3o *b3o...o3o
                  h = o3o3x *b3o...o3o
 P   C   P        C = o3o3o *b3o...o3x 


It might be interesting that the lower part of these Dutour polytopes (of the first kind - there are further ones too, which then might become topics for subsequent post, perhaps), i.e. the mere segmentopes of (H h) || (p C p) aka demihypercube || crosspolytope clearly can be given as lace prisms too. Then those are just xo3oo3oo *b3oo....ooo3ox&#x (in general - where applicable, and esp. for 4D: xo3ox3oo&#x resp. for 3D: xx2ox&#x). And it happens generally that those lace prisms all are known otherwise as well! For those are always the crosspolytope-first monostratic segments of o3o3o3o *c3o...o3x (n<9, where n=4 becomes o3o3x3o and n=3 becomes o3x . x).

In detail:

• line||{4} = trip
• tet||oct = rap
• hex||gyro hex = hin
• (pt||)hin||tac = jak
• (gee||)hax||gee = naq
• (zee||hesa||rez||)alt hesa||zee = fy

--- rk

[Klitzing]

...

In terms of the layer distance d the lacing edge length y = y(d) can be given only depending on the dimension value n as: y = sqrt((n-1)/8 + d²), at least when using unit edge lengths x here. - In the crystallographic context of Delaunay cells it ought be essential now to provide a further formula d = d(n), which asserts the existance of an unique circumradius. - In the mathematical context of polytopists it might be more interesting however to provide a different such formula d = d*(n), which provides equal sized edges throughout, i.e. y = x. This simply can be achieved by using d = sqrt((9-n)/8). This latter choice then clearly provides the there proclaimed coincidence of the 6D (n=6) case with the well-known Schläfli-Gosset polytope 2_2,1 (or "jak" in Jonathan Bowers' terms).

We even could - again generally (but then n>3) - consider this fellow as the following lace city.

Code: Select all

     P           where:
                  P = o3o3o *b3o...o3o
   H   h          H = x3o3o *b3o...o3o
                  h = o3o3x *b3o...o3o
 P   C   P        C = o3o3o *b3o...o3x 


Refering to that picture the coordinates can be given like this (again using unit edge length x and arbitrary layer distance d):

P (top layer):
  (0; 0, 0, 0, ...; +d)
H (equatorial left):
  (-1/sqrt(8); -1/sqrt(8), 1/sqrt(8), 1/sqrt(8), ...; 0) & all even changes of sign in the medial block of coordinates
h (equatorial right):
  (1/sqrt(8); 1/sqrt(8), 1/sqrt(8), 1/sqrt(8), ...; 0) & all even changes of sign in the medial block of coordinates
P (bottom left/right):
  (+-1/sqrt(2); 0, 0, 0, ...; -d)
C (bottom central):
  (0; 1/sqrt(2), 0, 0, ...; -d) & all permutations and changes of sign in the medial block of coordinates

...

The concrete value d=d*(n) for all unit edges already was derived above as d = sqrt((9-n)/8).

Just for completeness - and more as an aside tribute to Dutour's original aim - I'll derive the corresponding crystallographic d=d(n) here as well. That one asks generally to have y different from x. Instead the corresponding restriction is given by the existance of a well-defined circumradius, running through all vertices. (This then also is the definition of being a Delaunay cell.)


By consideration of the 3 P's of the lace city display, it becomes evident that the according circumcenter M of these 3 points ought be some (0; 0, 0, 0, ...; -m) and then r = MP_top = d+m as well as r = MP_bottom = sqrt((d-m)² + 1/2) in either of the 2 cases. Thus one derives d² + 2dm + m² = d² - 2dm + m² + 1/2 or equivalentely m = 1/(8d). And thus for the circumradius itself r = d+m = (8d² + 1)/(8d).

As P||C||P in the bottom layer defines a uniform crosspolytope, it becomes evident that our circumsphere also connects to all the vertices of C already. Whereas, in order to connect as well to the vertices of H (and then also of h) of the medial demihypercube H||h, we further have to evaluate the distance M ver(H) = sqrt((n-1)/8 + m²). Thus we get from r = M ver(H), using the above derived value for m, the equation (8d² + 1)²/(8d)² = (n-1)/8 + 1/(8d)² or equivalently 64d⁴ + 16d² + 1 = 8(n-1)d² + 1. Whenever d>0 this runs down to 8d² = n-3 or the final result d = sqrt((n-3)/8).

We even could evaluate now the corresponding (crystallographic) radius formula from r=r(d)=r(d(n)) as r = (8d² + 1)/(8d) = (n-2)/sqrt(8(n-3)). Or we could also provide the circumcenter M=M(d)=M(d(n)) explicitely as M = (0; 0, 0, 0, ...; -1/sqrt(8(n-3))). Further we can evaluate also the lacing edge size y=y(d)=y(d(n)) to y = sqrt((n-1)/8 + d²) = sqrt((n-1)/8 + (n-3)/8) = sqrt(n-2)/2. (As throughout, all provided absolute values were scaled according to layerwise unit edges.)


It is obvious, that the single n, which solves both equations to the same value d=d(n)=d*(n) ought have the same distance from 9 and 3. Thence it is n=6, in agreement to our previous knowledge that the n=6 fellow here is the well-known uniform figure jak = 2_2,1.

Our previously already provided explicite description of the case n=3 however looks a bit strange with this today result. But when looking at the above also derived radius formula r=r(d(n)) one recognizes that in this case we'd get r = infinite, the sphere degenerates into a euclidean plane. And this then is indeed in conformance to the derived distance d=d(n) of zero (which latter one only was accessed here as limiting case then). This then too would be in accordance to the lacing length of y = 1/2.

On the other hand, true (spherical) Delaunay polytopes are derived whenever n>3, even beyond our limit of n=9 for the all unit edge consideration of the former posts.

--- rk

[Klitzing]

It might be added furthermore, that the skipped 7D Dutour polytope of the first kind - in its variant with all unit edges throughout - consists out of a total of   12 hints + 224 hops + 60 hexascs + 33 gees.


- If you might ask what these Bowers acronyms might represent in turn, then I'll give some clue here below:

"hin" is the hemipenteract, x3o3o *b3o3o. "hint" then is the (hin,line)-tegum or equivalently the dipyramid with "hin" as its equatorial section.
"hop" is just the heptapeton, the 6D simplex.
"hex" is the hemitesseract, x3o3o *b3o, or equivalently the regular hexadecachoron, the 4D crosspolytope. "hexasc" then is the hexadecachoric scalene or equivalently the pyramid of a hexadecachorical pyramid. It also could be considered as the segmentopeton   line || perp hex.
"gee" finally represents the 6D crosspolytope, x3o3o3o3o4o, also known as hexacontitetrapeton.


The "hints" there run clearly through all 3 layers: from the single point at the top through the hin cells of the equatorial "hax" section ("hax" is the hemihexeract, x3o3o *b3o3o3o), down to the vertices of the bottom "gee". As can be seen from the lace city of some earlier post, these towers P || H || P are straight, i.e. without a bend at H.
The other cells of the upper segment then are 32 of those "hops".
One single "gee" clearly represents the bottom layer itself. Thus all the remaining cells all are lacing ones of the bottom segment (H || h) || (P || C || P).
The "hexascs" span between their "hex" at the equatorial layer and an opposing edge at the bottom layer.
The remaining 32 "gees" are implemented as "hix" antiprisms (where "hix" is the hexateron, the 5D simplex).
And the remaining "hops" fall into 2 separate classes. One of them functions as segmentopetons   tet || perp {3} (where "tet" clearly is the tetrahedron, the 3D simplex). There the "tets" stick to the equatorial layer, and the triangles belong to the bottom layer. That first class counts 160 items. The other class (with again 32 items) functions as "hix" pyramids. There the base then belongs to the bottom layer.

--- rk

[Klitzing]

Recently I pointed out a second series of Dutour polytopes as well. Here they come just for completeness. But despite of the title of this thread except of a single example those will lack to be CRFs in general.

He also consideres a further infinite series of polytopes, which he constructs again as a lace tower according to P || D || inv D || P, where P again is a mere point, while D is the respective Dutour polytope of the first series of one dimension less. Accordingly here he then states that n≥7 odd.

Just as in the first series, this definition here too is guided by n=7, where this definition describes nothing but the uniform polyexon naq (Gosset polytope 3_2,1, i.e. o3o3o3o *c3o3o3x), the Delone cell of lattice E₇.

Using the lace tower description of D from the former section, we can provide here an lace city display too:

Code: Select all

      P   
                 where (n≥4):
P   H   C        P = o3o3o *b3o...o3o
                 H = x3o3o *b3o...o3o
  C   h   P      h = o3o3x *b3o...o3o
                 C = o3o3o *b3o...o3x
    P     


Within this display the extremal segments show that this provides interesting polytopes only, when all lacing edges y between the layers of any of these segments are of the same size. But as those are pyramids with a Dutour polytope of the first series as its base, we ask for a circumsphere of the base. And this in turn then implies that the crystallographical spacings are to be used! According to the previous section on Dutour polytopes of the first kind in their crystallographical variants, which occure here as sectioning layers, in general 2 different edge lengths ought be involved. The unit edge size x, as used within C and H (as well as its inverted version h), and y = sqrt((n-1)-2)/2 = sqrt(n-3)/2 as lacing edge size of the Dutour polytopal sections. (Note that we here had to use n-1 instead of n, as we now have gone one dimension up.) But because of the mirror symmetry of the above lace city of these Dutour polytopes of the second kind along the H - h hyperplane, it is obvious that the new segmental lacings in here all ought be the same y. Therefore, outside of n=7 these polytopes generally need 2 different edge lengths and therefore never become CRFs again.

--- rk

[Klitzing]

I've to state now, that after having been in direct contact to Mathieu Dutour Sikirić in the meantime, that some of my harsh critics to that preprint - which already has been published as Inhomogeneous extreme forms together with Schürmann and Valentin within the Annales de l'Institut Fourier, 62, 6 (2012), 2227-2255 (and is online available here) - can be viewed a bit more differentiated at least.

...

...

Recently I stumbled across an article of Mathieu Dutour Sikirić [in its version] from 2013: "Infinite serie of extreme Delaunay polytope", as can be viewed at http://arxiv.org/pdf/math/0305196v1.pdf. It describes a quite interesting set of polytopes, even so the article is quite error prone. (Eg. neither the circumcenter nor the upper lacing edge length is calculated correctly - at least wrt. the therein provided coordinates. Even the symmetry assertion (for n=6) of the main theorem is wrong when using those coordinates, because the layer distance there is not conform to an circumscribing hypersphere (as for Delaunay cells needed), rather it is given fixed.)

...

...

Within the published version, the false coordinates of the circumcenter no longer are contained explicitely. Thus that error is no longer there. Further it now dawned to me, that his other coordinates where given - even so not being addressed as such - within lattice coordinates rather than in homogeneous euclidean coordinates of embedding space. That is, within the sectioning subspaces of vertex layers he defines point (P), hemihypercube (H), and crosspolytope (C) as to be stacked. All these are described by the lattice D_n and thus describing coordinates are (up to a global scaling and perhaps a global orientation) nothing but the corresponding hypercubical, i.e. euclidean coordinates. But for the axial stacking on the other hand he uses lattice coordinates as well, that is the layer distance is not to be calculated according to any constraint, rather it is given just as 1. - But here I still have to point some minor critics. Esp. in view of my provided distance formulas of d = d(n) it becomes evident, that this very distance is truely a function of the dimension of the embedding space. And, when applying the constraint of throughout equal sized edges, it happens that this series comes to an end at n=9. Therefore this dependency is neither mentioned nor it is explicitely shown that his intended series in fact is infinite!

Moreover, in the publication he just states - without any further comment - that for the series of D1(n) = P||H(n-1)||C(n-1) this dimensional number n has to be chosen even and greater or equal to 6, resp. that for D2(n) = P||D(n-1)||-D(n-1)||P that very n has to be chosen odd and greater or equal to 7. Again he relates D1(6) = jak, D2(7) = naq (in Bower's terms). Whereas in the firstly cited article he at least comments on that constraint with the words "It is easy to see, that ... if and only if n is even." relating thereby to the lattice embedding. - As I already showed, the polytopal definition as such is completely independent of even or odd. Neither it is restricted to n being greater or equal to 6. - In one of his recent mails to me he now points out that the hemihypercube has the (dimensionwise) alternating property of being point-inversive or not. E.g. tet = o3o4s is not, while hex = o3o3o4s is. - I'm still unclear whether this restriction is a mere constraint to a lattice embedding only, as it simply relates to the how of the laminate stacking. But at least the restriction to the lower bounds clearly relates to his extremality attribution only...

At least the today mentioned published work is more explicit on the second type of Dutour polytopes, and moreover mentions all the other Delone cells of these (dimensionally) alternate laminated lattices as well.

--- rk

[Klitzing]

I'd like to open and to recommend this thread both for a collection of according ideas and for the discussion of individuals.
Even cross-links to older posts from other threads ought be desirable, I think.

Just remembered myself on a small correspondance within this forum of last August, which is fitting in here as well. Even so this was a quite general description on polytopes in general (by concept), it happened to come down to just 4 CRF polypeta (i.e. polytopes in 5D). But at least those even are all scaliforms!

For further details on those gyrotrigonisms cf.

• tedjak = hexgyt
• pex tedjak = thexgyt
• ritgyt
• tahgyt

--- rk

[Klitzing]

It might be added furthermore, that the skipped 7D Dutour polytope of the first kind - in its variant with all unit edges throughout - consists out of a total of   12 hints + 224 hops + 60 hexascs + 33 gees.

Now I worked out the final one of this series too, the 8D Dutour polytope of the first kind, when considered in its all unit edges variant. The used 7D boundaries here are:

Code: Select all

             14 haxts
64+560+64 = 688 ocas
             84 hinscs
            280 hexetes
      64+1 = 65 zees


- If you might ask what these Bowers acronyms might represent in turn, then I'll give some clue here below:

"hax" is the hemihexeract, x3o3o *b3o3o3o. "haxt" then is the (hax,line)-tegum or equivalently the bipyramid with "hax" as equatorial section.
"oca" is just the octaexon, the 7D simplex.
"hin" is the hemipenteract, x3o3o *b3o3o. "hinsc" then is the hemipenteractic scalene or equivalently the pyramid of a hemipenteractic pyramid. It also could be considered as the segmentoexon   line || perp hin.
"hex" is the hemitesseract, x3o3o *b3o, or equivalently the regular hexadecachoron, the 4D crosspolytope. "hexete" then is the hexadecachoric tettene or equivalently the scalene of the hexadecachoric pyramid resp. the pyramid of the hexadecachoric scalene. It also could be considered as the segmentoexon   triangle || perp hex.
"zee" finally represents the 7D crosspolytope, x3o3o3o3o3o4o, also known as hecatonicosoctaexon.


The "haxts" there run clearly through all 3 layers: from the single point at the top through the "hax" cells of the equatorial "hesa" section ("hesa" is the hemihepteract, x3o3o *b3o3o3o3o), down to the vertices of the bottom "zee". As can be seen from the lace city of some earlier post, these towers P || H || P are straight, i.e. without a bend at H.
The other cells of the upper segment then are 64 of those "ocas".
One single "zee" clearly represents the bottom layer itself. Thus all the remaining cells all are lacing ones of the bottom segment (H || h) || (P || C || P).
The "hinscs" span between their "hin" at the equatorial layer and an opposing perpendicular edge at the bottom layer.
The "hexetes" have their "hex" in the equatorial layer and the opposing perpendicular triangle at the bottom layer.
The remaining 64 "zees" are implemented as "hop" antiprisms (where "hop" is the hexateron, the 6D simplex).
And the remaining "ocas" fall into 2 separate classes. One of them functions as segmentoexon tet || perp tet (where "tet" clearly is the tetrahedron, the 3D simplex). That first class counts 560 items. The other class (with again 64 items) functions as "hop" pyramids. There the base then belongs to the bottom layer.



(As was mentioned already somewhere above: The 9D Dutour polytope of the first kind happens to become degenerate, when considered with all unit edges. I.e. its height then would be zero. Therefore, despite of Dutour's original claim ("infinte serie"), the according set of herein to be considered CRFs (convex polytopes with regular faces, i.e. polygons) terminates here.)

--- rk

[quickfur]

Not sure if this is really relevant here, but back when I was still active, I did some calculations on higher dimensional CRF monostratics of the form A || B where A and B are uniform polytopes from the n-cube family. My first foray into this topic was described in this post. A little later in that same topic I expanded my findings, as detailed in this post.

Executive summary of findings: 19D is a special boundary, past which most of the families of these monostratics stop, leaving only 3 families from 20D and onwards, whose heights (from A to B) are constrained to be either 1 or 1/sqrt(2) (assuming unit edge). Those of height 1 are just the uniform n-cube prisms; the other two families are the bisected n-crosses and their Stott expansions, and a class of n-cube uniforms laced with another n-cube uniform with the marked node displaced by one position, along with the corresponding Stott expansions.

Unfinished work on analogous monostratics derived from the n-simplices suggest a special boundary at 8D, but I've lost track of my notes since, and haven't had the time to revisit this, so I don't have any proofs for this. Perhaps interested parties can finish this research? (hint, hint )

Other related avenues of research include exploring the possibilities of the n-demicube series and the monostratics derived from them.

[Klitzing]

Not sure if this is really relevant here, ...

For sure it does. Thanx for reminding me! - And welcome back Quickfur!

... Other related avenues of research include exploring the possibilities of the n-demicube series and the monostratics derived from them.

Dutour's polytopes belong therein, or rather their monostratic segments do. For those are ox3oo3oo *b3oo...3oo&#x (pyramids) and xo3oo3oo *b3oo...3ox&#x (demihypercube || crosspolytope) respectively.

--- rk

[Klitzing]

Not sure if this is really relevant here, but back when I was still active, I did some calculations on higher dimensional CRF monostratics of the form A || B where A and B are uniform polytopes from the n-cube family. My first foray into this topic was described in this post. A little later in that same topic I expanded my findings, as detailed in this post.

Executive summary of findings: 19D is a special boundary, past which most of the families of these monostratics stop, leaving only 3 families from 20D and onwards, whose heights (from A to B) are constrained to be either 1 or 1/sqrt(2) (assuming unit edge). Those of height 1 are just the uniform n-cube prisms; the other two families are the bisected n-crosses and their Stott expansions, and a class of n-cube uniforms laced with another n-cube uniform with the marked node displaced by one position, along with the corresponding Stott expansions.

Unfinished work on analogous monostratics derived from the n-simplices suggest a special boundary at 8D, but I've lost track of my notes since, and haven't had the time to revisit this, so I don't have any proofs for this. Perhaps interested parties can finish this research? (hint, hint )

Other related avenues of research include exploring the possibilities of the n-demicube series and the monostratics derived from them.

Especially within the second of these linked posts of Quickfur he outlines an important reduction of research on the lace prisms. I'll recap it here (for being self-reliant) and, moreover, will confine that even deeper with some additional observations!

• We note, that for mere (partial) Stott expansions (or contractions) wrt. to the overall axial symmetry only the 3 relevant circumradii would change (i.e. that of the bases and thus also the overall one), whereas the height always would remain the same. From this observation it will be enough to abandon in here all those lace prisms, which would have an x node simultanuously at any single position within both layers. Instead we can restrict to the fundamental cases with positional node pairs oo, ox, and xo only!
• A further restriction on to be considered cases can be achieved – at least for fully connected diagrams (i.e. whenever the axial symmetry is not reducible) – that whenever a connected substring of the diagram bears only o nodes at one layer, then at the corresponding part on other layer there cannot be more than one x node. (Else you would ask at best for lacing elements which are hippies (ox3ox&#x), copies (ox3oo3ox&#x), spidpies (ox3oo3oo3ox&#x), etc., which all are degenerate themselves (zero height) and so the overall height is forced to be zero as well.) That is, according to this restriction the ringed nodes x only can alternate between the layers, i.e. after some ox we first need an xo (and the other way round) - with possible node ommissions oo inserted anywhere! (And, as already stated above, the non-fundamental remainder then only are the Stott expansions of those more basic shapes, replacing some of those oo nodes into xx nodes.)
• The same observation furthermore holds true as well for bifurcation points of the Dynkin diagram of the axial symmetry. This simply is because any (sub-)polytope of one according layer, with at least 2 nodes being ringed, already has a circumradius which is larger than 1. Thus then an according all unit-edged pyramid, which would be needed as some lacing (sub-)element in the respective lace prism whenever the corresponding part of the other layer will be completely unringed, ought belong to hyperbolic geometry at least. - But from this here likewise valid restriction it becomes clear additionally, that only 2 legs emmanating off from the bifurcation point are allowed to bear ringed nodes at all! Thus, for not necessarily fundamental lace prisms, the other legs either have to be completely unringed (only oo nodes) or are Stott expansions thereof (possibly some xx nodes).

The first 2 of these points especially apply both for the simplical and the hypercubic/crosspolytopic symmetries (around the lace prismal axis). The addition of the 3rd point then applies for the 3rd infinite series of demihypercubic symmetries (around that axis).

--- rk

Edit: the last bullet point is erronious in its proof, while correct in its claim. For a better proof see here. --- rk

[Klitzing]

Quickfur in his linked posts prooved that convex unit-edged lace prisms with axially hypercubic symmetry generally cease to exist, except for few inter-dimensionally series. When looking into these posts, it becomes clear that his approach fully runs thru coordinates, which happen to be readily accessable in that case. But then he states, that he struggled with the axially simplectic symmetry.

But when considering the numbers given, i.e. 19D for the hypercubical one versus just 8D/9D for the simplectic one, and moreover taking into account, that the Dynkin diagrams there will have an additional right-left symmetry, it becomes possible freely to omit all that coordinate fiddling. Just list all possible cases up to 10D, say, and calculate the respective (squared) heights by that spead sheet which Wendy once provided. This then can be done rather fast.

When looking at the results, one can easily extract the here desired results:
Beyond 9D there survive just 8 dimensional series in the realm of axially simplectical symmetry. These are:

h = 1

• oo((3oo)*)&#x (prisms)

h = sqrt[(d+1)/(2d)]

• ox((3oo)*)&#x   (simplexes)
• ((oo3)*)xo3ox((3oo)*)&#x   (n-rectified simplex || (n+1)-rectified simplex)

h = 2/sqrt(2d)

• oo3ox((3oo)*)&#x   (pyramids on 1-rectified simplexes)
• xo((3oo)*)3ox&#x   (cross-polytopes)
• ((oo3)*)xo3oo3ox((3oo)*)&#x   (n-rectified simplex || (n+2)-rectified simplex)
• ox((3oo)*)3xo3ox((3oo)*)&#x
• ((oo3)*)xo3ox((3oo)*)3xo3ox((3oo)*)&#x

(where the notational ((...)*) here is used informal in the reg-ex sense, stating that the contained subdiagramm may be repeated any number of times, from 0 to infinity; and d represents the number of node positions + 1, i.e. the embedding dimensional number).

Note that this approach in contrast to a mere abstract theorem additionally provides access to all the individual cases underneath that threshold.

Moreover it can be deduced also that the heights in fact happen to be quantized. Within this axial simplectical symmetry we have

Code: Select all

 d         |  2   3   4   5   6   7   8   9  10  11  12    | +1
-----------+-----------------------------------------------+-----
2d*(h(d))² |  4   6   8  10  12  14  16  18  20  22  24    | +2
           |  3   4   5   6   7   8   9  10  11  12  13    | +1
           |          4   4   4   4   4   4   4   4   4    |  0
           |              .   3   2   1   0  -1  -2  -3    | -1
           |                  .       0  -2  -4  -6  -8    | -2
           |                      .          -5  -8 -11    | -3
           |                          .             -12    | -4


--- rk

[quickfur]

Excellent! So 9D is a kind of special boundary for simplicial (simplectic) lace prisms. I find it quite interesting that between 8D and 9D the series gradually die off, as opposed to having a sharp cut-off point. It's also interesting that of the remaining series, their heights vary with dimension, unlike the hypercubic family where the quantized heights of the remaining series are constant.

[Klitzing]

Excellent!

Thanx

... It's also interesting that of the remaining series, their heights vary with dimension, unlike the hypercubic family where the quantized heights of the remaining series are constant.

This already shows up when considering the simple series of simplex(d+1) = point || simplex(d) versus half of crosspolytope(d+1) = point || crosspolytope(d), the first obviously belongs to the axially simplectic symmetry and has varying heights (by dimension), the latter belongs to axially crosspolytopical symmetry (= axially hypercubical symmetry) and has constant height sqrt(2)/2 (when all unit-edged).

--- rk

[Klitzing]

hummm,
found some holes in my claim wrt. bifurcated axial symmetries here.
None the less, the assertion as such was right. Today I came around to proove it better. - Thus please restate that part rather by:

When taking into account bifurcation points of the Dynkin diagram of the axial symmetry, then the same alternation rule holds true there as well. (Again at least when arguing for cases with truely positive heights.) But for prooving this claim for such symmetries we have to distinguish 3 cases now. – The first one assumes that the bifurcation point itself and its 3 nearest neighbours all are unringed in at least one layer. Now simply consider any (sub-)polytope of that other layer, which opposes this (possibly larger) unringed region, and additionally has at least 2 ringed nodes. Any such polytope then already has a (layer-wise) circumradius which is larger than 1. Therefore then an according all unit-edged pyramid, which here would be needed for some lacing (sub-)element in the respective lace prism, ought belong to hyperbolic geometry already. But from this restriction it becomes clear additionally, that only 2 legs, emmanating off from the bifurcation point, are allowed to bear ringed nodes at all! I.e., for not necessarily fundamental lace prisms, the third leg either has to be completely unringed (only oo nodes) or is a Stott expansion thereof (possibly some xx nodes).

Next consider the bifurcation node itself of some reduced fundamental lace prism to be ringed. Then to all 3 emanating legs the above consideration wrt. axially simplectic symmetries applies (with final inclusion of this bifurcation point). Thus again that alternation condition does apply here too. (Sadly, in this case nothing can be said about fully empty legs.)

In the complemental final case (of reduced fundamental lace prisms with bifurcation nodes) we thus can assume that this bifucation point itself is of type oo and that in either layer at least one of its neighbouring nodes is ringed. Thus the here possible (sub-)polytopes are either xo3oo3ox *b3oo&#x (hin = hex || alt. hex, which has height h = 1/sqrt(2) = 0.707107) or xo3oo3ox *b3ox (hexarit = hex || rit, which itself is degenerate with zero height). Therefore just the former of these 2 cases has to be considered further here. But that one then again requires that the third leg has to be fully unringed. This here would follow again by mere application of the axially simplectic result to the first-to-third substring as well as to the second-to-third substring. These elsewise would contradict. But thereby our claim is proved in that final case as well.

--- rk

[Klitzing]

haha,

instead of considering ANY bifurcated graph (in the last post of mine of this thread), I could have restricted to the demihypercubic graphs only, i.e. those with 2 legs of size 1 and just the third being arbitrary. All link marks being 3. Under that precondition I further could have restricted to the cases where those 2 (first) end-nodes are marked either complementarily or one being completely empty ("oo"). For else we would have been back to hypercubical symmetry (by identification of these 2 end-nodes). Thus for fundamental cases (no "xx"-nodes in these lace prisms), this assumes these end-nodes to be either "xo" and "ox" respectively or "xo" and "oo".

But when doing a short search of the low-dimensional examples here, we see that the only possible ones of the latter kind would have for height formula: h²(d) = (9-d)/8, i.e. would die off for d beyond 8. - And when considering the former case, the at-most alternation-condition from the axially simplectic symmetries implies that the bifurcation node itself as well as all other nodes too have to be "oo", whenever we assume a strictly positive height and only consider the fundamental cases. Thus we are just left with xo3oo3ox *b((3oo))&#x, where the part within double parantheses can be repeated as many times as wanted. That fellow then has always height 1/sqrt(2). Moreover it is nothing but the demihypercube of the next dimension!

--- rk

[quickfur]

So in other words past 8-D the only remaining segmentotopes with demihypercubic symmetry are those derived via Stott expansions of the demihypercube itself?

[Klitzing]

Speaking of the demihypercubes, when considered as dimensional series, we might look at these here as well in more detail. Not that this is all new stuff. But it might be overseen sometimes. So I'd like to recap it here, even so the demihypercubics are way too special to be considered within the CRF thread elsewise.

For that purpose let's start with the hypercubes first. We want to consider those vertex first as axial stacks. Then we are within axially simplectic symmetry, isn't it? Thus we have:

Code: Select all

3D: cube
=  o3o || q3o || o3q || o3o 
=  oqoo-3-ooqo-&#xt
4D: tes 
=  o3o3o || q3o3o || o3q3o || o3o3q || o3o3o 
=  oqooo-3-ooqoo-3-oooqo-&#xt
5D: pent
=  o3o3o3o || q3o3o3o || o3q3o3o || o3o3q3o || o3o3o3q || o3o3o3o
=  oqoooo-3-ooqooo-3-oooqoo-3-ooooqo-&#xt
6D: ax   
=  o3o3o3o3o || q3o3o3o3o || o3q3o3o3o || o3o3q3o3o || o3o3o3q3o || o3o3o3o3q || o3o3o3o3o
=  oqooooo-3-ooqoooo-3-oooqooo-3-ooooqoo-3-oooooqo-&#xt
7D: hept
=  o3o3o3o3o3o || q3o3o3o3o3o || o3q3o3o3o3o || o3o3q3o3o3o || o3o3o3q3o3o || o3o3o3o3q3o || o3o3o3o3o3q || o3o3o3o3o3o
=  oqoooooo-3-ooqooooo-3-oooqoooo-3-ooooqooo-3-oooooqoo-3-ooooooqo-&#xt
8D: octo
=  o3o3o3o3o3o3o || q3o3o3o3o3o3o || o3q3o3o3o3o3o || o3o3q3o3o3o3o || o3o3o3q3o3o3o || o3o3o3o3q3o3o || o3o3o3o3o3q3o || o3o3o3o3o3o3q || o3o3o3o3o3o3o
=  oqooooooo-3-ooqoooooo-3-oooqooooo-3-ooooqoooo-3-oooooqooo-3-ooooooqoo-3-oooooooqo-&#xt


Next we consider the demihypercube as being the vertex alternated form thereof. As the above was taken vertex first, that one now consist just from every alternate vertex layer. Therefore we have:

Code: Select all

3D: q-tet
=  o3o || o3q (q-laced)
=  oo-3-oq-&#q
4D: q-hex (alternation 1)
=  o3o3o || o3q3o || o3o3o (q-laced)
=  ooo-3-oqo-3-ooo-&#qt
q-hex (alternation 2)
=  q3o3o || o3o3q (q-laced)
=  qo-3-oo-3-oq-&#q
5D: q-hin
=  o3o3o3o || o3q3o3o || o3o3o3q (q-laced)
=  ooo-3-oqo-3-ooo-3-ooq-&#qt
6D: q-hax (alternation 1)
=  o3o3o3o3o || o3q3o3o3o || o3o3o3q3o || o3o3o3o3o (q-laced)
=  oooo-3-oqoo-3-oooo-3-ooqo-3-oooo-&#qt
q-hax (alternation 2)
=  q3o3o3o3o || o3o3q3o3o || o3o3o3o3q (q-laced)
=  qoo-3-ooo-3-oqo-3-ooo-3-ooq-&#qt
7D: q-hesa
=  o3o3o3o3o3o || o3q3o3o3o3o || o3o3o3q3o3o || o3o3o3o3o3q (q-laced)
=  ooooo-3-oqooo-3-ooooo-3-ooqoo-3-ooooo-3-ooooq-&#qt
8D: q-hocto (alternation 1)
=  o3o3o3o3o3o3o || o3q3o3o3o3o3o || o3o3o3q3o3o3o || o3o3o3o3o3q3o || o3o3o3o3o3o3o  (q-laced)
=  oooooo-3-oqoooo-3-oooooo-3-ooqooo-3-oooooo-3-ooooqo-3-oooooo-&#qt
q-hocto (alternation 2)
=  q3o3o3o3o3o3o || o3o3q3o3o3o3o || o3o3o3o3q3o3o || o3o3o3o3o3o3q  (q-laced)
=  qooo-3-oooo-3-oqoo-3-oooo-3-ooqo-3-oooo-3-oooq-&#qt


--- rk

[Klitzing]

So in other words past 8-D the only remaining segmentotopes with demihypercubic symmetry are those derived via Stott expansions of the demihypercube itself?

Correct
- provided the ones with axially (full) hypercubical symmetry are blended out from consideration here.

Cf.:  A3B3A *b3...   "="   ...3B3A4o .

--- rk

[Klitzing]

Thought of posting onto this thread a new find:

Within the CRF research (of any dimension) clearly the orbiform ones (which additionally have a well defined circumradius, i.e. all vertices happen to lie on an according dimensional hypersphere) are rather few - when sewing out the already uniform ones resp. their diminishings. Also and esp. when looking at axially stacked segmentotopes this same observation holds, even when skipping the convexity condition.

In fact, so far just a single 4D orbiform polychoron of axially stacked segmentochora was known: sidrepcu = xoo5/2oxo5oox&#xt, which then clearly was a concave one.

Today I stumbled upon a second such polytope, this time within 6D, which then even happens to be convex, and therefore indeed qualifies as (higher dimensional) CRF as well!

Just consider ox3oo xx3oo3oo&#x = tet || (para) tratet. That one is just a different representation of tetdip (= x3o3o x3o3o). Within that given orientation it has the height = sqrt(2/3) = 0.816497 and (generally) the circumradius = sqrt(3)/2 = 0.866025.

Today now I came across to consider ox3oo ox3oo3xo&#x = tet || inv tratet - thus having Bowers style acronym "tetal tratet". Within this orientation it happens to have the height = 1/sqrt(6) = 0.408248 (just half of the former!) and furthermore the same circumradius!

Accordingly we could consider the bistratic stack of those two: oxo3ooo oxx3ooo3xoo&#xt, which then is nothing but a (mono-) "tetal tratet" augmented tetdip.

Because tetdip itself was uniform, this new blend thus would set the vertices of those two layers into different classes. And I guess that the opposite extremal tet's vertices would make up a third one. So I'd strongly believe that this new orbiform CRF won't happen to be obtainable as a diminishing of some uniform polypeton (and clearly itself is not).

--- rk

[quickfur]

Speaking of which, did we ever prove whether or not the teddi series continues through all dimensions, or whether they only go up to some finite dimension before becoming degenerate (zero height) or non-existent? And also, what about the other teddi analogues, like the ones based on the tridiminished x5o3x, and perhaps other similarly "expanded" analogues in 5D and beyond?

[Klitzing]

Hi quickfur,
yes you are right, I really overlooked that one. Thanks for reminding me!

You were talking about T(n) = ofx3xoo(3ooo)*&#xt, the top layer of which then is R(n-1) = o..3x..(3o..)* and the bottom layer is S(n-1) = ..x3..o3(..o3)*.

In fact, both its find and the orbiformity already was found by Andrew Weimholt way back in 2004, cf. http://lists.mathconsult.ch/mailman/private/polyhedron/2004b/msg00668.html
- or, if you've no access to the polyhedron mailing list archive - which I could convey, btw. - that respective mail was as follows:

From: Andrew Weimholt
Subject: An infinite series of Johnson polytopes
Date: Mon, 20 Sep 2004 03:09:48 -0700

I've been exploring a series of Johnson polytopes beginning with
the Tridiminished Icosahedron in 3 dimensions, continuing in 4
dimensions with a decachoron (which is the vertex section
of Gosset's non-Wythoffian tetracomb), and apparently going on
forever.

Let T(n) be the n-dimensional polytope in the series
Let S(n) denote a regular n-simplex,
Let R(n) denote a rectified n-simplex

Let T(2) be a regular pentagon, and define T(n) for n>2 as follows...

The base of T(n) is a S(n-1).
Connect n T(n-1) cells by their bases to this S(n-1)
Complete the polytope with n more S(n-1) cells and a R(n-1) cell
opposite from the base.

3D:

T3 is the tridiminished icosahedron (T3 = J63), with
3 pentagons (T2s) and 5 triangles (4 S2s and 1 R2). The R2
on top is of course just another S2.

4D:

for T4 we get a decachoron with a tetrahedron (S3) at its base,
then 4 tridiminished icosahedra (T3) connected around it, and
completed with 4 more tetrahedra, and an octahedron (R3).

  10 polyhedra (5 tetrahedra, 4 tridiminished icosahedra, 1 octahedron)
  30 polygons (24 triangles, 6 pentagons)
  34 edges
  14 vertices

5D:

For T5, the base is a pentachoron (S4), surrounded by 5 T4s, and completed
with 5 more S4s and an R4 (also known as RAP, or rectified pentachoron).

  12 polychora (6 S4s, 5 T4s, 1 R4)
  45 polyhedra (30 tetrahedra, 10 tridiminished icosahedra, 5 octahedra)
  80 polygons (70 triangles, 10 pentagons)
  65 edges
  20 vertices

6D:

T6 has

  14 polytera (7 S5s, 6 T5s, 1 R5)
  63 polychora (42 S4s, 15 T4s, 6 R4s)
  140 polyhedra (105 tetrahedra, 20 tridiminished icosahedra, 15 octahedra)
  175 polygons (160 triangles, 15 pentagons)
  111 edges
  27 vertices

I continued these calculations for several more dimensions.

Another remarkable property of this series is that all of its members
have all of their vertices on a hypersphere.

There are only two types of vertex sections for each member of this series.
One is a simplex-spike (to use George Olshevsky's terminology), in which the
base is a regular (n-2)-simplex with unit edge lengths, and the lateral edge lengths
are all tau. The second vertex figure is a little harder to describe...It has a base
which is a prism of an (n-3)-simplex. Opposite this base is an edge of length tau.
Two (n-2)-simplexes attach to either end of the prism, and have their opposite vertex
at on end of the length-tau edge (all other edges are unit length). In three dimensions,
these two vertex figures reduce to an isosceles triangle and a trapezoid.

I was initially expecting this series to terminate somewhere, but from my dihedral
angle calculations, it appears this series is infinite.

Andrew

Wendy then later was extending that find to ofx3xoo(3ooo)*Pooo&#xt with P=4 or 5. (The existance of those within dimensions > 4 I have not checked really.)
Note that we have: teddi = ofx3xoo&#xt itself is a diminishing of ike, ofx3xoo5ooo&#xt is a diminishing of ex, thus their orbiformity is trivial.
Orbiformity beyond was never checked, I think.

And I then was pointing out subsequently also the according Stott expansions in across symmetry (any "ooo" -> "xxx").

--- rk

[username5243]

So, the xfo3oox3ooo3ooo...3ooo&#xt series goes on forever?

Have you checket the xfo3oox3ooo...3ooo4ooo&#xt series? How about the ico and ex ursatera (xfo3oox4ooo3ooo&#xt and xfo3oox3ooo5ooo&#xt respectively)? I wouldn't be too surprised if that lastt one is degenerate - especially if it continues the pattern to be a diminishing of x3o3o3o5o.

Another way to expand the concept is things like oox3xfo3oox3ooo&#xt - is that kind of construction valid? (What I am trying is an ursaqron based on the rectified pentachoron)
I've also came up with a series that is similar to the hexagon in the same way ursatopes resemble the pentagon - it looks like xux3oox...&#xt. The n-simplex based ones are just the truncated (n+1)-simplexes, while the n-cross-polytope based ones are formed from cutting a truncated (n+1)-cross-polytope in half.

Also, how do I get access to that "polyhedron mailing list", if even possible?

[Mercurial, the Spectre]

It appears that the series of ursatopes is infinite and orbiform, although only ones based on the simplex and the orthoplex.
To consider, look at them as vertex figures.
The 2D ursachoron is simply a pentagon, derived from a 3D icosahedron which is an alternation of the truncated octahedron (x3x4o). x3x4o's verf is an isosceles triangle which represents a mirror-subsymmetry of the pentagon.
The 3D ursachoron is the tridiminished icosahedron, derived from a snub 24-cell which is an alternation of tico (x3x4o3o). x3x4o3o's verf is an equilateral-triangular pyramid with x3x4o's verf as lateral sides.
The 4D ursachoron is the tetrahedral ursachoron, derived from a snub 24-cell honeycomb which is an alternation of ticot (x3x4o3o3o). x3x4o3o3o's verf is a regular-tetrahedral pyramid with x3x4o3o's verf as lateral sides. Note that x3x4o3o3o is not a polyteron, it is a tiling of the 4D euclidean plane.
The 5D ursachoron likewise could be derived from the alternation of x3x4o3o3o3o which is a paracompact hyperbolic tessellation. x3x4o3o3o3o's verf is a regular 5-cell pyramid, with x3x4o3o3o's verf as lateral sides.

Since there is no better notation for the general extension of the toe/tico/ticot family, I'll denote them as t[x] and their alternations as s[x], where [x] starts with 2 or the hexagon. t[1] is a line segment.
t[x] can always be shown to have these two facets in general: t[x-1] and a hypercube of dimension x-1. Verf is a regular (x-2)-simplex pyramid. The (x-2)-simplex pyramid has a regular (x-2) simplex of length sqrt(2), corresponding to the hypercubic facet. Lateral sides have length sqrt(3), and lateral cells are (x-3)-simplex pyramids. This has 2 degrees of freedom and 2 degrees of variation under symmetry.
s[x] then can always be made uniform, with facets: s[x-1], a demihypercube of dimension x-1, and an (x-1)-simplex. Verf has these cells: rectified (x-2) simplex corresponding to the demihypercubic verf, s[x-1] verf for lateral sides, and an (n-2) simplex as its common intersection at the top.

For the hyperoctahedral ursatopes, as a verf of s4[x], t4[x]'s verf then becomes an (x-2)-orthoplex pyramid, the (x-2)-orthoplex having length sqrt(2), and the lateral sides (x-3)-simplex pyramids. Lateral sides are always of length sqrt(3). Again, s4[x] can be made uniform, as it again has the same degrees of freedom and variation. Though its verf is CRF for x equal or greater than 5, s4[4]'s verf is not CRF since t4[4]'s verf is a 2-orthoplex pyramid or square pyramid, which has a 3.3.4 triangle configuration. Again, the verf follows s[x]'s verf, except that the rectified (x-2) simplex becomes a rectified (x-2) orthoplex, with (x-3) orthoplex pyramids replacing all of the tets.

Conclusion:
s[x]'s verf happens to be the (x-1)-dimensional ursachoron. Its infinitude (and orbiformity) is due to it being represented as a verf of s[x], which can be derived from t[x], from which t[x]'s verf can always be deformed to equal lengths! Though its circumradius does not hold, it keeps on increasing so that at 4D, you'd have an ursatope with a circumradius equal to its edge length, and in 5D, it is more than 1, so it can only be represented as a verf of a hyperbolic honeycomb. Same applies to the hyperoctahedral ursatopes.

I think all of you missed a simple trick, haha!
But that's ok , I'm contributing to your efforts.

Cheers,
Mercurial

[Klitzing]

So you say t[N] = x3x4o(3o)* (N nodes) and s[N] = s3s4o(3o)*. Then ofx3xoo(3ooo)*&#xt (N-1 nodes) is the vertex figure of s[N]. - So far I could follow.

Then you speak of t4[n]. - I cannot find a clear definition thereof.
s4[N] then probably is again its snub. And ofx3xoo(3oo)*4ooo&#xt then ought to be the vertex figure of s4[N]. - Right?

--- rk

[wendy]

In this class of figure, the pentagon has three layers of edge 1, f, 2. The verticies of the base must lie in the mid-points of the top, of edge 2. This is possible if the faces are triangles. The teddi, for example, is a xfu3ooo&#xt, with the feet cut in the base layer, so the triangle is reversed. It gives then xfo3oox3&xt.

The second thing that governs how big the thing can be, is that the top level has to be smaller than an decagon. Pentagons would stand flat in the plane of the decagon, the triangles would make icosahedral pyramids at the far end, so xfo3oox3ooo5ooo&#xt is flat.

So as long as your base consists of surhedra entirely of triangles, and the figure has a circumdiameter less than the decagon (ie 2f, or D2 = 4f+4=10.472, then it can form a teddi-class figure. The simplex and its rectate would form examples, the rectate is oox3xfo3oox[3ooo]&xt, the cross polytope.

You get these figures with the 2_21, 3_21 and 4_21 too. I should imagine that the CO class figure from 4D onwards (ie o3x3o[3o]4o) would also pass.

[Mercurial, the Spectre]

So you say t[N] = x3x4o(3o)* (N nodes) and s[N] = s3s4o(3o)*. Then ofx3xoo(3ooo)*&#xt (N-1 nodes) is the vertex figure of s[N]. - So far I could follow.

Then you speak of t4[n]. - I cannot find a clear definition thereof.
s4[N] then probably is again its snub. And ofx3xoo(3oo)*4ooo&#xt then ought to be the vertex figure of s4[N]. - Right?

--- rk

Sorry if I was ambiguous, but I'll give examples:
t4[2] - x3x (hexagon)
t4[3] - x3x4o (truncated octahedron, same as t[3] since t[2] and t4[2] are congruent) - verf is a 1-orthoplex pyramid (isosceles triangle)
t4[4] - x3x4o4o (truncated order-4 octahedral tessellation, cells are truncated octahedra and square tilings) - verf is a 2-orthoplex pyramid (square pyramid)
t4[5] - x3x4o3o4o (facets are x3x4o3o (tico or t[4]) and x4o3o4o (cubic honeycomb)) - verf is a 3-orthoplex pyramid (octahedral pyramid)
t4[6] - x3x4o3o3o4o (facets are x3x4o3o3o (ticot or t[5]) and x4o3o3o4o (tesseractic honeycomb)) - verf is a 4-orthoplex pyramid (16-cell pyramid)

s4[2] - s3s (triangle)
s4[3] - s3s4o (pyritohedral icosahedron, same as s[3]) - verf is a regular pentagon
s4[4] - s3s4o4o (snub order-4 octahedral tessellation, cells are icosahedra, square tilings, and square pyramids) - verf is (square|f-square|(45° q-square)), basically an analogue of teddi. It is the square ursahedron with 2 square faces (one of length 1, and a 45° copy of length sqrt(2)), 4 regular pentagons, and 2 isosceles triangles of lengths 1, 1, and sqrt(2).
s4[5] - s3s4o3o4o (facets are sadi, s4o3o4o (tetoct), and octahedral pyramids) - verf is ofx3xoo4ooo&#xt (oct|f-oct|co)
s4[6] - s3s4o3o3o4o (facets are s3s4o3o3o (sadit), s4o3o3o4o (hext), and 16-cell pyramids) - verf is (hex|f-hex|ico) and its facets contain hex, ico, octpy, and ofx3xoo3ooo&#xt.

Basically t4[x] is similar to t[x] except that their verfs have orthoplex bases instead of simplex bases.

Facets of t4[x] are t[x-1] and the (x-2)-dimensional hypercubic honeycomb.
Facets of s4[x] are s[x-1], (x-2)-dimensional alternated hypercubic honeycomb, and (x-2)-orthoplex pyramids. Verf is the (x-1)-dimensional hyperoctahedral ursatope.

[quickfur]

Wow, that's a sudden flurry of activity!

From what I can remember, we have (at least) 3 classes of ursatopes that are currently known:

1) The "basic" ursatope, which is a straight application of Wendy's general scheme (Edit: fixed wrong notation) xfo3oox3ooo...Nooo&#xt. For 2D, the only member is the pentagon itself. For 3D, the only CRF member is the tridiminished icosahedron. For 4D, we found 3 CRF members: the tetrahedral (N=3), octahedral (N=4), and icosahedral (N=5) ursachora, of which the last is nothing but a particular diminishing of the 600-cell. We surmised that in 5D, there ought to be, in addition to the pentachoric ursateron, the 16-cell and 24-cell ursatera. But these were never checked for convexity / non-degeneracy. I think somebody has already proven that the 600-cell ursateron is degenerate (height zero). At the time we didn't really think about going beyond 5D, so I'm happy to hear that this series does generalize to higher dimensions unboundedly.

2) The Stott expansions of (1), a few of which I had built models for, in 4D. Basically, this involves a straight Stott expansion along the symmetries of the base facet, so for example the octahedral ursachoron would be expanded into something with x4o3x as base, and pentagonal prisms and triangular prisms filling the gaps between the lacing teddies, and square cupola bridging to the bottom cell. To my knowledge, no one has checked whether analogous constructions in 5D and above are CRF.

3) A related construction wherein the lacing teddi cells are replaced with tridiminished x5o3x's. I had successfully built a CRF model of such a construction, but when I tried to generalize it to its Stott expansion (i.e., insert decagonal prisms between the tridiminished x5o3x's), I could not find a way to close it up in a CRF way. So it's still unknown whether Stott-expanded analogues exist, though it's clear to me that if such an analogue exists, it's no longer a straightforward Stott expansion, but would require (AFAICT) arbitrary configurations of cells to close up the bottom part (and possibly the top part as well). But more pertinently, it's not clear whether this construction generalizes to higher dimensions, or even how such a generalization would be carried out. Seems to be a fertile area for more research.

[Klitzing]

Wow, that's a sudden flurry of activity!
...

2) The Stott expansions of (1), a few of which I had built models for, in 4D. Basically, this involves a straight Stott expansion along the symmetries of the base facet, so for example the octahedral ursachoron would be expanded into something with x4o3x as base, and pentagonal prisms and triangular prisms filling the gaps between the lacing teddies, and square cupola bridging to the bottom cell. To my knowledge, no one has checked whether analogous constructions in 5D and above are CRF.

...

Well, whenever a Wythoffian figure with some o nodes exists, then you can apply a Stott expansion in order to make that specific o into an x. Moreover all cells of the latter, which already existed in the former and already there had a common face, would keep the same dihedral angle.

Same applies to axial lace towers too. Whenever there is some tower with all layerwise nodes at a common position throughout being o, then there would be a Stott expansion within across symmetry, which just makes all these o's simultanuously into x'es. And again we have the same observation for the dihedral angles again.

--- rk