Publishing an article on BTP polytopes

[Klitzing]

The fast stroke of discoveries of some special CRFs (convex regular faced polytopes, the extrapolation of "Johnson solids" to higher dimensions, which sticks to regular elements of 2D rather than (d-1)D), esp. in last February (Quickfur called that then CRFebruary), which is still ongoing, mostly within thread Construction of BT-polytopes via partial Stott-expansion, already settled on the need to publish these finds somehow. We also had some preliminary ideas on that article. - These shall be extracted / refrased here for more ease of access, esp. as this became a topic within other posts (in other threads) already. (Welcome back PWrong )

1. we would need some basics on Coxeter Dynkin symbols and Wythoff's kaleidoscopical construction to be reviewed for the ease of the reader and also for some selfcontainedness of the article.
2. we need to outline Wendy's skew coordinates (mainly the ability to scale the different edge types obtained in the Wythoff construction freely and independently - in its effect to the represented polytopes this already is known since the days of Max Brückner as "Varietäten" (varieties), but as notational concept applied to (ASCII linearizations of) Coxeter Dynkin diagrams it still remains unpublished.
3. we need some review on Alicia Boole-Stott's definition on what is now called Stott expansion and its translation to Coxeter Dynkin diagrams. - That part in retrospective kind of seems trivial, but in fact her paper was written way before the notational concepts of Dynkin and Coxeter. It rather contributed to the setup of the latter.
4. we then will have to go beyond the scope of her article and at least outline the different paths of generalizations of her concept, e.g. as already being done on my corresponding webpage (also no printed publication on that so far).
5. we then need to outline the concept of edge flips introduced by student91. Here not only the notational techniques ought to be explained, but also its relation to the kaleidoscopical construction with seed points lying outside the fundamental domain. - That topic then surely has been investigated earlier, some references should be spottable, perhaps even one or two more general pics deduced therefrom. - But here we restrict to those special positions at the outside, which occur as constructed vertex set of some given (other) seed point within. A further genuinity of the ideas in consideration amount in not to translate that outside seed point into one within some larger fundamental domain with multiplicity (order, index) greater than 1, as usually being done in the past, but still to use the original (elementary) fundamental domain plus an extrapolation of Wendy's concept (cf. list item 2) towards negative edge lengths.
6. the to be described BTP polytopes then result in some specific applications of the last 2 list items.

The idea behind that last list item then always runs as follows (as meanwhile being boiled down to):

• start with some specific polytope, e.g. the icosahedron.
• define some specific subsymmetry to be chosen, e.g. the brique symmetry o2o2o.
• decompose the vertex set of your polytope into subsets with the chosen subsymmetry
  Here: x3o5o -> x2f2o + f2o2x + o2x2f
• apply one or more edge flip(s) to one or more component(s)
  E.g.: (-x)2f2o + f2o2x + o2x2f
  (the thus described vertex set as a whole will have to be equivalent to that of the starting figure. In more detail, in the sense of a kaleidoscopical constructed polytope, it just defines an edge faceting of the starting polytope.)
• now use a correspondingly subsymmetric Stott expansion uniformly to all the components, which then lead back to prograde edges only
  Here: (x-x=o)2f2o + (x+f=F)2o2x + (x+o=x)2x2f
• in order to result in a CRF polytope some restrictions have to be met here in general (the chosen example surely will pass)
  The outcome in our chosen example then is the bilunabirotunda (J91).

Why do I refer to "BTP" here?

B represents bilbiro (bilunabirotunda, J91). T represents thawro (triangular hebesphenorotunda, J92). P represents pocuro (pentagonal orthocupolarotunda, J32). All three can be derived as just outlined from ike (icosahedron), when dealing with o2o2o, o2o3o, resp. o2o5o subsymmetry.

In fact, x3o5o = x3o || o3f || f3o || o3x -> (-x)3x || o3f || f3o || o3x -> (x-x=o)3x || (x+o=x)3f || (x+f=F)3o || (x+o=x)3x = thawro. Respectively x3o5o = o5o || x5o || o5x || o5o -> o5o || (-x)5f || o5x || o5o -> (x+o=x)5o || (x-x=o)5f || (x+o=x)5x || (x+o=x)5o = pocuro.

The most, if not all of the above mentioned thread on BT polytopes then is about similar applications to polychora, i.e. within 4D. Mainly ex = hexacosachoron = 600-cell = x3o3o5o is being used for starting figure.

--- rk

[student91]

I think it is a great idea to start this topic. Here we can discuss the structure/layout of the article, and the other topic then is about the discoveries themselves. Great .
About this list:

[list with contents, from 1 to 6]

I think 2 and 5 are highly related, and thus I think they might be seen as a single chapter with two parts. Then 3 and 4 are clearly related, and then 6 is a conclusion. I think you did very well on lining out what the article needs.
Also, you are referring to things which I am highly interested to know. As you know, I highly like free sources which everyone can reach . If anyone finds such sources, I suggest he posts them in this topic. The things I would like to know are:

[point 2] Max Brückner as "Varietäten" (varieties)

[point 5]- That topic then surely has been investigated earlier, some references should be spottable, perhaps even one or two more general pics deduced therefrom.

If you know some titles of articles or articles that refer to them I'd be glad to know these, so I can search the web. Also it seems a lot on this topic has been written in Coxeters book about regular and semiregular polytopes. I myself do not own a copy, nor am I planning to buy one. This means that if we want to use it as a reference (and I guess we have to, because it is such important) someone else will have to include citations on what came from the book.
I hope you don't mind my aversion to paid sources, but I think mathematical knowledge should be in the public domain.

[Marek14]

(Quickfur called that then CRFebruary)

I think it was actually me -- first use of the term on forums seems to be my post (viewtopic.php?f=32&t=1927&p=20646&hilit=crfebruary#p20646). Just a note

[Klitzing]

I think it is a great idea to start this topic. Here we can discuss the structure/layout of the article, and the other topic then is about the discoveries themselves. Great .

This was why.

The things I would like to know are:

[point 2] Max Brückner as "Varietäten" (varieties)

...

Brückner, Max (1900). Vielecke und Vielflache: Theorie und Geschichte. Leipzig: B.G. Treubner. ISBN 978-1-4181-6590-1. (German) WorldCat English: Polygons and Polyhedra: Theory and History. (Online accessible here.)

...

[point 5]- That topic then surely has been investigated earlier, some references should be spottable, perhaps even one or two more general pics deduced therefrom.

I had in mind here: Vladimir L. Bulatov (2000). Isogonal Kaleidoscopical Polyhedra Families. Mosaic2000, Millenial Open Symposium on the Arts and Interdisciplinary Computing, Seatle, WA. (Online accessible here.)



And wrt. CRFebruary:

I think it was actually me ... Just a note

Sorry for wrong attribution. (Just having been called from mind only.)

--- rk

[Polyhedron Dude]

Talk about publishing articles in peer review journals, I think its also time to get something done with the uniform star polytopes (something I've been familiar with for over 20 years now). I suspect nearly all of the 4-D ones are found (there could be a few oddballs hiding in the unknown though). Norman Johnson has written an article on the uniform polychora (including star ones) over 15 years ago, but it seemed to have vanished from view, I was hoping to see his article get published so the 4-D star polytopes would finally be in the domain of mathematical publishings. So any suggestion what would be the best course of action here? I have thought about getting articles written on smaller groups of them, like the pentachoric ones, then the tesseractic ones.

[wendy]

Ye would have to deal with the notion of 'lace prisms' as well. Something like xo2ox&#xt is something ye just don't see in the literature. It's utterly new, whereas ye can see what Klitzing refers to as 'varieties', and even Coxeter writes a $\sqrt{2}$ over the top of a dot to show a 'q' node. What i did is to try to linearise the notation for ascii input, without superscript or subscript.

I could produce some TpX diagrams showing how the lace-prisms might work, and then from that, lace towers.

Coxeter's Kaleidoscope Notation

There is an awful lot of stuff out there on this, but most people regard it as a group-thing. We really need to brush a lot of this off, and apart from getting it to show numbers of things, really leave Coxeter's view out of it. It's group theory, not polytope construction.

Stott-Wythoff

Stott's discovery procedes Coxeter. The notation of writing t_0,1,2 is Wythoff's modification of Stott's form (which is e_1,2, since t_0 = 1 in her scale). But neither Stott nor Wythoff had a scale. The whole idea here is that the node-marks are actually the pedal nodes of mirror-edge constructions.

When you use vertex-nodes, the actual count becomes very simple, since every surtope is a vertex-node and mirror-nodes in various connections. The same holds true for lace prisms.

A lace prism simply has multiple vertex nodes. The dimensionality is still N-1, where N is the number of nodes, but if you have 2 vertices, ye get a symmetry of n-1 dimensions (the remaining symmetry is the lacing-height).

You can still find the surtope and consist with the additional construction.

Position-polytope

The key notion under my scale is 'position-polytope'. Whereas group theory is about the patterns of the symmetry cells of one group, against another, and their mathematical abstraction, what we're doing here is making up an image to look in the kaleidoscope in one cell.

The coordinate system represents vectors that produce a point unit distance from the opposite mirror, so (1,0,0) is measured not from the origin, but that (1,x,y) represents a plane that is unit distance from (0,x,y). The general point (a,b,c) then represents a point whose reflections in the mirrors give a polytope a(5)b(3)c. In much the same way, the notional coordinate l,b,h in the rectangular symmetry [2,2] gives a cartesian point (l,b,h), which is reflected by change of sign to give (\pm l, \pm b, \pm h) of a rectangular blob centred on the origin.

It probably would be useful to include here the finding of stott-matrices and the matrix-dot method (which is the magic behind the spreadsheet). The general theory is that if you have an oblique coordinate system of non-unit vectors, you can still do the dot product, by pre-multiplying one vector by $[x_i \cdot x_j]_{ij}$.

Lace-Coordinates

The image ye could use here is one of those christmas decorations, you buy flat, and when you pick them up, make a kind of 3d thing.

In essence one can take a point (a,b,c) at h=0, and then a second point (A,B,C) at some larger h. But instead of measuring h direct, we measure the length of the rod that connects (a,b,c,0) to (A,B,C,h).

The relevant height is given by the right angle triangle: matrix-norm of (a-A, b-B, c-C, 0) : h : 4E

[Klitzing]

Ye would have to deal with the notion of 'lace prisms' as well. Something like xo2ox&#xt is something ye just don't see in the literature. It's utterly new, whereas ye can see what Klitzing refers to as 'varieties', and even Coxeter writes a $\sqrt{2}$ over the top of a dot to show a 'q' node. What i did is to try to linearise the notation for ascii input, without superscript or subscript.

I could produce some TpX diagrams showing how the lace-prisms might work, and then from that, lace towers.

Yes, we might include all that &#xt and &#zx stuff as well. It surely is worth publishing some time. But I fear that the paper might get too overloaded by that further concept, and even less readable because of further mathematical unusual notions which have to be reverse translated in each readers mind. I thought we could bypass this surely more compact notational concept just by taking refuge to A || B || C ... resp. A + B + C ... within this article.
- What would you think?

On the other hand, if some of the smaller incidence matrices will have to be included, the shorter description of the subelements clearly is to be preferred...

--- rk

[wendy]

On the other hand, if we use some sort of diagram i saw student91 floating around on x6o + o6x = x6x, and use this to show how this worked (as a position polytope), the second diagram might be to draw a wedge of the kaleidoscope, and show the lacing-edge and its images to make an x6o || o6x, which we could directly connect to $---o--6--o----$.

We would probably need to develop the stott-matrix and vector thing, but one does not need deep theory to do this. A section suffices.

I'll draw some pictures up tomorrow on the gadget, so ye could have a stickybeak at what i mean.

[student91]

[...] i saw student91 floating around on x6o + o6x = x6x, [...]

I don't think I've ever used something like this. I rather sometimes use +'s for &#zx's, or sometimes use it like x6o + .6(+x) =x6x. This is something I thought I saw Klitzing do, and thus I might have copied it wrongly/ambiguously.
The use of + for &#zx's is much easier: You take some representation of ex, e.g. [[5,2,5]] oxofox5ooxofx xofoxo5xfoxoo&#zx oxofox5ooxofx xofxoo5xfooxo&#zx, when you write it like o5o x5x + x5o o5f + o5x f5o + f5o o5x + o5f x5o f5o x5o + o5f o5x + x5x o5o, you can spot the individual parts much easier, while you practically have only lost the information &#zx. this might even be appended to the "addition." + reads here more as "and" than as "add."

[student91]

[...]
Brückner, Max (1900). Vielecke und Vielflache: Theorie und Geschichte. Leipzig: B.G. Treubner. ISBN 978-1-4181-6590-1. (German) WorldCat English: Polygons and Polyhedra: Theory and History. (Online accessible here.)

Thanks!! Unfortunately my german isn't very good, but I think I can get it through.

I had in mind here: Vladimir L. Bulatov (2000). Isogonal Kaleidoscopical Polyhedra Families. Mosaic2000, Millenial Open Symposium on the Arts and Interdisciplinary Computing, Seatle, WA. (Online accessible here.)

This article is great! Not only does it mention the "varieties" and vertices out of the domain, it also is very interesting on its own .

About the clear duality in our article between 1. wendy's notation and 2. BTP-polytopes, could we write two articles, each about one of these subjects, and then ask the guy of the journal if he can place them consecutively, or submit them as one article with two titles and two abstracts etc. which have to be read "first 1, then 2", or is this an idea which the journal will not be happy with?

[Klitzing]

... You take some representation of ex, e.g. [[5,2,5]] oxofox5ooxofx xofoxo5xfoxoo&#zx, ...

The mentioned representation seems to be wrong, I fear.
Just calculated the inter-layer lacing edge lengths. For your representation I'd get:

Code: Select all

1-2 = 5-6 = 1
1-3 = 4-6 = 1
1-4 = 3-6 = f
1-5 = 2-6 = f
1-6 =       fq          = 2.288246
2-3 = 4-5 = 1
2-4 = 3-5 = rt[1-1/rt5] = 0.743496
2-5 =       rt[1+1/rt5] = 1.203002
3-4 =       rt[1+1/rt5]

Esp. because we were looking onto a convex unit-edged polychoron here (ex), lacing "edges" (or vertex distances) of (absolute) size <1 cannot be allowed.

--- rk


Edit: this post alread got an answer, even so it was placed within an other thread. For reference cf. here.

[Klitzing]

I once was calling the idea of student91 with all that (-x) stuff - in a rather kidding mood - simply "quirks".
That one so far served well while sticking to this forum, and as long we all know it to be a working definition only.

But with respect to a future publication we surely need a better naming.

I already was thinking about

• IEE (inverted edge(s) expansions),
• FEE (flipped edge(s) expansions),
• KFE (kaleido-flipped expansions).

But still am open to other / better namings. My particular invitation goes to student91 himself: how would you like your "baby" being called?

We clearly might well manage to omit this question and restrict our paper to a title along the lines of this very thread. But we already know the concept is much more general than its restriction to an applications on ex only. In its very beginning we also have found examples within 3D (applications on ike). And probably loads of others might be lurking to be discovered as well.

--- rk

[student91]

I once was calling the idea of student91 with all that (-x) stuff - in a rather kidding mood - simply "quirks".
That one so far served well while sticking to this forum, and as long we all know it to be a working definition only.

But with respect to a future publication we surely need a better naming.

I already was thinking about

• IEE (inverted edge(s) expansions),
• FEE (flipped edge(s) expansions),
• KFE (kaleido-flipped expansions).

But still am open to other / better namings. My particular invitation goes to student91 himself: how would you like your "baby" being called?

Are you searching for a name for the node-flipping, or for an expansion which is done using node-changings? I guess it doesn't make that much difference, as you can just postfix "expansion" to whatever the name will be.
Anyway, (bullshit alert ) I don't like the flipping suggestion, as I don't think it is a flip, its more something which walks slowly along all vertices of a polytope. Flip would rather suggest that there are few states between which you can flip. The contrary is true, as most polytopes have quite some vertices. I'd rather like something like mirrored or endeepened if that is a word. Maybe pseudo-faceted? (the bullshit alert isn't there for nothing ) After all the changing of something from e.g. x5o3o into (-x)5f3o looks a bit like taking a (weird) faceting, where you have f-triangles at places comparable with those of sidtid. (the differnce here is that the pentagons are also connected to each other, but it is comparable, not equal) Therefore, the node-changing might be interpreted as the taking of a "weird" faceting. Furthermore, you also interpreted the expansions of ike as expansions of facteings of ike. Therefore I think something like "partly faceted expansion" or "pseudo-faceted expansion" might be best. If you don't like to compare the node-changing with faceting, I think we could call it "kaleido-mirrored expansion" or "kaleido-endeepened expansion" or maybe this without the kaleido-prefix.

We clearly might well manage to omit this question and restrict our paper to a title along the lines of this very thread. But we already know the concept is much more general than its restriction to an applications on ex only. In its very beginning we also have found examples within 3D (applications on ike). And probably loads of others might be lurking to be discovered as well.

--- rk

Unfortunately, There is very little to be discovered besides expansions of ex and maybe rox (and it might be, but it is very unlikely, that tex has some surprises), and of course the other symmetry families, such as ico, hex, and pen-symmetries. Also in 3D there is no other expansion from [5,3]-symmetry than the already discovered ones. This is because whenever you try to expand a different polytope from these families, the taking of a subsymmetry always has a part that looks like ofx&#xt which must be expanded. When this is a true pentagon, this won't stay RF, and thus this is impossible.

There is an exception though, as id can be expanded to epgybro. This is an annoying exception, which isn't based on a reflective symmetry, and also doesn't have any similar expansions in 4D. (even ex doesn't allow a not-faceted axial expansion while retaining convexity or producing u-edges)

So do we want to call it pseudo-faceted expansion, partly-faceted expansion or kaleido-mirrored expansion. My favorite is partly-faceted expansion , though I also like kaleido-mirrored expansion .

[wendy]

The paper of mine is up to eight pages, but i spent today rewriting some sections, rather than writing new stuff.

Currently, it covers names, the regular products, as surtope products, and then as radiant products.

[Klitzing]

Just for compilation:

So far we have the following systematical EKFs (extended kaleido-facetings), which evaluate to be CRFs:

A - of the icosahedron (ike, x3o5o)
A.1 - wrt. o2o2o => ike = xof 2 fxo 2 ofx &#zx = xofox 2 ofxfo &#xt
A.1.1 - oxF 2 fxo 2 ofx &#zx = oxFxo 2 ofxfo &#xt (bilbiro, J91)
A.2 - wrt. o2o3o => ike = xofo 3 ofox &#xt
A.2.1 - oxFx 3 xfox &#xt (thawro, J92)
A.3 - wrt. o2o5o => ike = oxoo 5 ooxo &#xt
A.3.1 - xoxx 5 ofxo &#xt (pocuro, J32)

B - of the hexacosachoron (ex, 600-cell, x3o3o5o)
B.1 - wrt. o2o3o5o => ex = VFfxo 2 oxofo 3 oooox 5 ooxoo &#zx = oxofofoxo 3 ooooxoooo 5 ooxoooxoo &#xt
B.1.1 - BAFox 2 oxofo 3 oooox 5 ooxoo &#zx = oxoofooxo 3 oooxoxooo 5 ooxoooxoo &#xt (with 24 ikes, 60 squippies, 180 tets, 20 trips)
B.1.2 - VFfxo 2 xoxFo 3 oxooo 5 ooxof &#zx = xoxFoFxox 3 oxoooooxo 5 ooxofoxoo &#xt (with 30 bilbiroes, 26 ikes, 80 octs, 60 squippies, 40 tets)
B.1.3 - VFfxo 2 oxofx 3 xxxxo 5 ooxof &#zx = oxofxfoxo 3 xxxxoxxxx 5 ooxofoxoo &#xt (with 2 ids, 30 ikes, 40 octs, 60 pips, 180 squippies, 180 tets, 80 tricues, 120 trips)
B.1.4 - VFfxo 2 ooofx 3 xoxxo 5 ofxof &#zx = ooofxfooo 3 xoxxoxxox 5 ofxofoxfo &#xt (with 2 ids, 40 ikes, 40 octs, 12 pips, 24 pocuroes, 180 squippies, 80 tets)
B.1.5 - VFfxo 2 oxofo 3 oofox 5 xxoxx &#zx = oxofofoxo 3 oofoxofoo 5 xxoxxxoxx &#xt (with 2 does, 40 ikes, 60 pips, 300 squippies, 100 tets, 120 trips)
B.1.6 - BAFox 2 xoxFx 3 oxoox 5 ooxoo &#zx = xoxxFxxox 3 oxoxoxoxo 5 ooxoooxoo &#xt (with 48 gyepips, 20 hips, 2 ikes, 80 octs, 120 squippies, 40 tricues, 60 trips)
B.1.7 - BAFox 2 oxofx 3 xxxxo 5 ooxof &#zx = oxoxfxoxo 3 xxxoxoxxx 5 ooxfofxoo &#xt (with 30 bilbiroes, 2 ids, 40 octs, 24 pips, 24 pocuroes, 120 tets, 80 tricues, 20 trips)
B.1.8 - BAFox 2 ooofx 3 xoxxo 5 ofxof &#zx = oooxfxooo 3 xoxoxoxox 5 ofxfofxfo &#xt (with 30 bilbiroes, 2 ids, 40 octs, 48 peroes, 140 tets, 20 trips)
B.1.9 - BAFox 2 oxofo 3 oofox 5 xxoxx &#zx = oxoofooxo 3 oofxoxfoo 5 xxoxxxoxx &#xt (with 2 does, 24 pips, 24 pocuroes, 120 squippies, 40 teddies, 40 tets, 140 trips)
B.1.10 - VFfxo 2 xoxFx 3 oxfox 5 xxoxx &#zx = xoxFxFxox 3 oxfoxofxo 5 xxoxxxoxx &#xt (with 40 octs, 12 pips, 24 pocuroes, 120 squippies, 2 srids, 120 tets, 40 thawroes, 180 trips)
B.1.11 - VFfxo 2 oxofx 3 xxFxo 5 xxoxF &#zx = oxofxfoxo 3 xxFxoxFxx 5 xxoxFxoxx &#xt (with 30 bilbiroes, 60 Dips, 240 squippies, 40 thawroes, 2 tids, 40 tricues, 60 trips)
B.1.12 - oxo|fxf|ooo 3 xxx|xox|xox 5 oox|ofo|xfo &#xt (= mixture of B1.3 + B1.4, with 2 ids, 30 ikes, 40 octs, 24 pips, 12 pocuroes, 180 squippies, 130 tets, 40 tricues, 60 trips)
B.1.13 - oxo|xfx|ooo 3 xxx|oxo|xox 5 oox|fof|xfo &#xt (= mixture of B1.7 + B1.8, with 30 bilbiroes, 2 ids, 40 octs, 24 peroes, 12 pips, 12 pocuroes, 130 tets, 40 tricues, 20 trips)
B.2 - wrt. o3o3o3o => ex = xffoo 3 oxoof 3 fooxo 3 ooffx &#zx
B.2.1 - oFFxx 3 xxoof 3 fooxo 3 ooffx &#zx (with 5 coes, 30 ikes, 20 octs, 90 squippies, 125 tets, 40 trips)
B.2.2 - xFfoo 3 xoxxF 3 fxoxo 3 ooffx &#zx (with 20 ikes, 25 octs, 60 squippies, 270 tets, 40 tricues, 60 trips, 5 tuts)
B.2.3 - oFfoo 3 ooxxF 3 Fxoxo 3 ooffx &#zx (with 30 bilbiroes, 25 octs, 20 teddies, 80 tets, 20 tricues, 5 tuts)
B.2.4 - xAFxx 3 ooxxF 3 Fxoxo 3 ooffx &#zx (with 30 bilbiroes, 25 octs, 30 pips, 20 teddies, 60 tets, 5 toes, 40 tricues, 40 trips)
B.2.5 - xFfxo 3 xoxoF 3 fxooo 3 oofFx &#zx (with 20 ikes, 25 octs, 60 squippies, 55 tets, 20 thawroes)
B.2.6 - oFFxx 3 xxoxf 3 Fxxox 3 oofFx &#zx (with 10 hips, 20 octs, 30 mibdies, 90 squippies, 20 thawroes, 40 tricues, 90 trips, 10 tuts)
B.2.7 - oFFxx 3 xxoof 3 fooxx 3 xxFFo &#zx (with 60 bilbiroes, 10 coes, 40 octs, 70 tets, 20 trips)
B.2.8 - xFfxo 3 xxxoF 3 Foxxx 3 oxfFx &#zx (with 10 coes, 60 pips, 30 tets, 40 thawroes, 40 tricues)
B.3 - wrt. o3o3o *b3o => ex = foxo 3 ooof 3 xfoo *b3 oxfo &#zx
B.3.1 - foxo 3 xxxF 3 xfoo *b3 oxfo &#zx (= plain subsymmetrical Stott expansion, with 480 tets, 96 tricues, 96 trips, 24 tuts)
B.3.2 - Fxox 3 ooxf 3 xfoo *b3 oxfo &#zx (with 8 coes, 32 ikes, 40 octs, 96 squippies, 136 tets, 48 trips)
B.3.3 - Foxo 3 oxxF 3 ofoo *b3 xxfo &#zx (with 136 tets, 32 thawroes, 32 tricues, 16 tuts)
B.3.4 - Foxo 3 oxxF 3 xFxx *b3 xxfo &#zx (with 8 coes, 48 pips, 96 tets, 32 thawroes, 8 toes, 32 tricues, 48 trips, 8 tuts)
B.3.5 - Fxox 3 xoxf 3 oFxx *b3 oxfo &#zx (with 8 coes, 40 octs, 96 squippies, 32 teddies, 32 thawroes, 96 trips, 8 tuts)
B.4 - wrt. o5o2o5o => ex = xfooxo 5 xofxoo 2 oxofox 5 ooxofx &#zx
B.4.1 - oFxxox 5 Fofxfo 2 oxofox 5 ooxofx &#zx (with 10 gyepips, 25 ikes, 10 pips, 150 squippies, 75 tets, 50 trips)
B.4.2 - oFxFox 5 AxFoFx 2 oxofox 5 ooxofx &#zx (with 50 bilbiroes, 10 dips, 10 gyepips, 5 pips, 75 squippies)

where generally I used F=ff=f+x, V=F+v=2f, A=F+x=f+2x, B=V+x=2f+x
(Other combinatorical cases each either provide non-unit inter-layer or surviving non-unit inner-layer edges.)
For all those above ones incidence matrices could also be provided, if needed. (But I'm in the run to provide these too in my next homepage update, for sure. There even will be an extra subpage for those. - Stay tuned.)

B.5 - wrt. o3o2o3o
B.6 - wrt. o2o2o2o
B.7 - wrt. o2o3o3o
remain to be evaluated systematically - as far as I know.

Other individuals (or even systematicals) known so far?

--- rk

[Klitzing]

... and for sure:

C - of the icosahedral pyramid (ikepy, ox3oo5oo&#x)
C.1 - wrt. o2o2o2o
C.1.1 - line || bilbiro (with 1 bilbiro, 4 peppies, 4 squippies, 4 tets, 2 trips)
C.2 - wrt. o2o2o3o
C.2.1 - {3} || thawro (with 1 oct, 3 peppies, 3 squippies, 9 tets, 1 thawro, 1 tricu, 3 trips)
C.3 - wrt. o2o2o5o
C.3.1 - {5} || pocuro (with 5 peppies, 2 pips, 1 pocuro, 10 squippies, 5 tets, 5 trips)


Btw., the latter 2 symmetries likewise will be open at case B.

--- rk

[Klitzing]

Wow, nearly one year nothing was posted in this thread, ...


Thought of we should mention not only the direct production of CRFs (convex regular faceted ones) by means of the EKF (expanded kaleido facetings) techniques, but also at least some of ist possible derivatives.

Just consider one of those for example:

We all know that the 600-cell (ex) vertex-wise can be described as a layer stack of point || icosahedron || dodecahedron || f-scaled icosahedron || icosidodecahedron || f-scaled icosahedron || dodecahedron || icosahedron || point (or alternatively written as oxofofoxo-3-ooooxoooo-5-ooxoooxoo-&#xt). By means of the additional prismatic top-down symmetry we even can rewrite that as VFfxo-2-oxofo-3-oooox-5-ooxoo-&#zx. (Here we use "edge"-length qualifiers f = 1.618, F = f+x = ff = 2.618, V = 2f = 3.236. - Even so, the ex for sure is assumed all unit-edged.)

I think, if I remember correctly, it was one of the first examples found, that we considered the tele-elongated 600-cell. - Tele-elongation here is a bit different from usual normal elongation. - I'll remind here for brevety how. We clearly could apply normal elongation to the 600-cell. Then we cut the ex in halves, cutting at the equatorial icosidodecahedronal section plane. But note, the triangles here indeed are triangles of ex, but the pentagons of the icosidodecahedron are just equatorial sections of clusters of 5 rosettewise adjoined tetrahedra around some perpendicular edges (w.r.t. to that cutting plane). Thus by elongation we not only have to fill in triangular prisms and pentagonal prisms between those slightly pulled appart halves, but we also have to erase those dissected tetrahedra, and replace each such halved rosette by a pentagonal pyramid. Accordingly this figure will have as total cell count (600 - 12*5 = 540) tetrahedra + 20 triangular prisms + 12 pentagonal prisms + (2*12 = 24) pentagonal pyramids.

For tele-elongation o.t.o.h. we consider the vertex planes, neighbouring to the equatorial one, to belong to the opposite halve each. If we now pull these "halves" appart, then those neighbouring layers each will move into the opposite direction. And finally, when having reached one edge-length distance, these 2 formerly neighbouring layers will exactly coincide. - I will further visualize this by the lace city displays of the 600-cell

Code: Select all

                    o2o                   
                                          
          o2x f2o   x2f   f2o o2x         
                                          
                                          
    x2o   f2f o2F   F2x   o2F f2f   x2o   
                                          
    o2f   F2o       f2F       F2o   o2f   
                                          
                                          
o2o f2x   x2F F2f  Vo2oV  F2f x2F   f2x o2o
                                          
                                          
    o2f   F2o       f2F       F2o   o2f   
                                          
    x2o   f2f o2F   F2x   o2F f2f   x2o   
                                          
                                          
          o2x f2o   x2f   f2o o2x         
                                          
                    o2o                   


and of the tele-elongated 600-cell

Code: Select all

                    o2o         o2o                   
                                                      
          o2x       x2f   f2o   x2f       o2x         
                                                      
                                                      
    x2o   f2f       F2x   o2F   F2x       f2f   x2o   
                                                      
    o2f   F2o       f2F         f2F       F2o   o2f   
                                                      
                                                      
o2o f2x   x2F      Vo2oV  F2f  Vo2oV      x2F   f2x o2o
                                                      
                                                      
    o2f   F2o       f2F         f2F       F2o   o2f   
                                                      
    x2o   f2f       F2x   o2F   F2x       f2f   x2o   
                                                      
                                                      
          o2x       x2f   f2o   x2f       o2x         
                                                      
                    o2o         o2o                   


In the sequel in these days it was proven that this resulting figure still is convex and thus qualifies as CRF. That figure then has a total cell count of 180 tetrahedra (still) + 24 icosahedra + 20 triangular prisms + 60 square pyramids.

In fact, tele-elongation is a very specific application of an EKF-construction: you just consider those edges, which formerly connected the 2 neighbouring layers directly, in inverted orientation (applying an x to (-x) "quirk"). This would result, when starting from ex = VFfxo-2-oxofo-3-oooox-5-ooxoo-&#zx accordingly in VFf(-x)o-2-oxofo-3-oooox-5-ooxoo-&#zx. And then we did pull those "halves" apart, i.e. we applied a (partial = subsymmetric) Stott expansion w.r.t. to that first node position (each). Thus we get BAFox-2-oxofo-3-oooox-5-ooxoo-&#zx as the description of the final figure: B = V+x (= 2f+x = fff), A = F+x (= f+2x), F = f+x, o = (-x)+x, x = o+x.

Now getting back from my digression. We all know that the 600-cell has plenty of possible diminishings. For instance we can cut off just a single vertex. This would replace that cap by ist vertex figure, which happens to be an icosahedron here. There already a quite a lot of possibilities to use multiple such diminishings. But when those diminishing spots come too close to each other, then those vertex figures would intersect. Thus, when considering the CRF solutions, we will have to take the corresponding cut offs of those, i.e. the vertex figures themselves then get diminished. They thus might become gyroelongated pentagonal pyramids, metabidiminished icosahedra, pentagonal antiprisms, and tridiminished icosahedra. (Pentagonal pyramids would not occure, as the circumcentrum will always belong to the diminished figure here.) This opens quite a Pandorra's box of possibilities. And this even does only describe the monostratic diminishings.

The same aspect now also can be considered for our new findings. To any EKF in general, and especially to our specific one, the tele-elongated 600 cell. To that purpose we would have to have a sneek preview into the vertex figures of that fellow. Esp. the ones, which are 3D CRFs for that purpose. Here we get icosahedra (again), gyroelongated pentagonal pyramids, tridiminished icosahedra, and pentagonal prisms (yes, upright prisms, not antiprisms!). But there are other, still monostratic diminishings possible here: Looking at its above displayed lace city, one recognizes that the top-most layer consists of just 2 vertices and thus describes a single edge. And the layer underneath can be recognized to describe just a bilunabirotunda! (In fact, that monostratic cap underneath those edges just are bilunabirotundaic wedges: line || bilunabirotunda - a figure which is quite close to lace prisms and segmentochora, but misses both descriptions just because the bilunabirotunda cannot be given in a (classical) Coxeter-Dynkin diagram (thus not a lace prism) nor does it have an unique circumsphere (thus not a segmentochoron).) The combination of such here possible diminishings, together with also possible deeper cuts, provides plenty of further CRFs to be derived therefrom!

Just wanted to point that out.

--- rk

PS: as an aside to Jonathan: how about "telex" as potential OBSA for that back then derived EKF-figure?

[Polyhedron Dude]

Telex sounds good, kinda sounds like a telephone company .