New lace term?

[Klitzing]

I'd like to call this new 6D figur with its hexagonal lace city "oddimo", as a Bowers style acronym for "octadeca-diminished mo".

I finally mannaged to derive its incidence matrix too. Here it is:

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xo3ox xo3ox xo3ox&#zx

o.3o. o.3o. o.3o.    & | 54 |   6   8 |  3  12  36 |  12  12  30  24 |  3  36  18 |  6 12
-----------------------+----+---------+------------+-----------------+------------+------
x. .. .. .. .. ..    & |  2 | 162   * |  1   4   4 |   6   4   8   4 |  2  16   5 |  4  6
oo3oo oo3oo oo3oo&#x   |  2 |   * 216 |  0   0   6 |   0   3   6   6 |  0  12   6 |  3  6
-----------------------+----+---------+------------+-----------------+------------+------
x.3o. .. .. .. ..    & |  3 |   3   0 | 54   *   * |   4   4   0   0 |  2  12   0 |  4  4
x. .. x. .. .. ..    & |  4 |   4   0 |  * 162   * |   2   0   2   0 |  1   4   1 |  2  2
xo .. .. .. .. ..&#x & |  3 |   1   2 |  *   * 648 |   0   1   2   2 |  0   6   3 |  2  4
-----------------------+----+---------+------------+-----------------+------------+------
x.3o. x. .. .. ..    & |  6 |   9   0 |  2   3   0 | 108   *   *   * |  1   2   0 |  2  1  trip
xo3ox .. .. .. ..&#x & |  6 |   6   6 |  2   0   6 |   * 108   *   * |  0   4   0 |  2  2  oct
xo .. xo .. .. ..&#x & |  5 |   4   4 |  0   1   4 |   *   * 324   * |  0   2   1 |  1  2  squippy
xo .. .. ox .. ..&#x & |  4 |   2   4 |  0   0   4 |   *   *   * 324 |  0   2   2 |  1  3  tet
-----------------------+----+---------+------------+-----------------+------------+------
x.3o. x.3o. .. ..    & |  9 |  18   0 |  6   9   0 |   6   0   0   0 | 18   *   * |  2  0  triddip
xo3ox xo .. .. ..&#x & |  9 |  12  12 |  3   3  18 |   1   2   3   3 |  * 216   * |  1  1  traf
xo .. xo .. .. ox&#x & |  6 |   5   8 |  0   1  12 |   0   0   2   4 |  *   * 162 |  0  2  squasc
-----------------------+----+---------+------------+-----------------+------------+------
xo3ox xo3ox .. ..&#x & | 18 |  36  36 | 12  18  72 |  12  12  18  18 |  2  12   0 | 18  *  tridafup
xo3ox xo .. .. ox&#x & | 12 |  18  24 |  4   6  48 |   2   4  12  18 |  0   4   6 |  * 54  triddaf


Obviously this figure happens to be a scaliform polypeton. In fact, there is just a single orbit of vertices. Moreover it is CRF. But it is not uniform (as there are squippies (J1) for 3D cells being used, for example).

Its 5D facets are 18 tridafups and 54 triddafs. Both are scaliform segmentotera. The tridafups are the stack of 2 bidually arranged triddips (x3o x3o || o3x o3x). While the triddafs are the stack of 2 bidually arranged trips (x3o x2o || o3x o2x). Here the triddafs occur as the remainders from the 54 5D facets of mo (o3o3o3o3o *c3x), i.e. of the hins (o3o3x *b3o3o), when the diminishing was applied. In fact, there is a lace city display for hin which looks like

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o   3        3 = o o3x (triangle)
  P   l      P = x x3o (gyro trip)
o   3        l = x o3o (line)
             o = o o3o (point)


and there the central but still vertex inscribed triddaf bit can be spotted as

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.   3        3 = o o3x (triangle)
  P   .      P = x x3o (gyro trip)
.   3        l = x o3o (line)
             o = o o3o (point)


clearly.

On the other hand, the 18 tridafups occur as the facets underneath the choped off vertices of mo. The vertex figure of mo is known to be dot (o3o3x3o3o). Thus tridafup happens to be a central part of dot. In order to see this, we represent dot by its lace city display

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  O   t      t = o3o3x = o3o||o3x
             O = o3x3o = o3x||x3o
T   O        T = x3o3o = x3o||o3o


That lace city by purpose is drawn slightly slanted. In fact, the tetrahedra are connected to the octahedra, and the octahedra too are connected to each other. But the 2 tetrahedra are not connected directly. In this lace city we further represent each spot by the given segmentohedron. Now let's maintain from this dot representation at the left lace city parts each left segmentohedron bases and at the right lace city parts each the right segmentohedral bases. This then provides us in the top layer some o3x||o3x (trip) and at the bottom layer of the lace city some x3o||x3o (trip again, triangles mutually in dual orientation). But additionally we have different slopes for the lacing edges of those trips each. In the lace city display those just are projected parallelly. But in fact the lacings of the top trip and those of the bottom trip are mutually perpendicular too. Therefore, in sum, this diminishing comes out to be tridafup, the stack of a pair of bidually arranged trips.

The relation to trittip (x3o x3o x3o) was already mentioned last time and once more is reminescent in the provided Dynkin style description above. (Actually, oddimo happens to be the hull of the compound of a pair of tridually arranged trittips. Or, in other words, it is the tegum sum of those 2 tridual trittips.)

As Wendy pointed out in a different thread, oddimo is related to mo in quite a similar fashion -within 6D- as was gap (the grand antiprism) related to ex (x3o3o5o) -within 4D. In fact, there one chops off 2 perpendicular great circles of 10 consecutive vertices each, while in this case we chop off 3 mutually perpendicular great circles of 6 vertices each. There likewise the vertex figures (ikes) had been pairwise dissected into pap cells only. But the remaining tet cells of ex had been already small enough not to be further diminished there. Here in contrast the original facets of mo, the hins, are much larger. Thus those get dissected likewise here.

--- rk

[Klitzing]

Today I got a new idea onto generating further CRFs!

The theme of this very thread once led us to the definition of the EKFs, the expanded kaleideo facetings.
Infact the idea for those was to derive some subsymmetrical representation of a given polytope and to represent that polytope accordingly as a tegum sum of the inscribed components or as lace tower of the layers. Then some of that list of individual Dynkin diagrams get represented in an alternate way, using retrograde edges. And finally to the whole polytope wrt. the given subsymmetry a partial Stott expansion was applied, i.e. any diagram of the list gets the same Stott expansion individually. - In order to become a CRF, the retrograde edges for sure are forced to become unringed nodes again by means of that expansion, and also the required lacings are bound to be of unit size too. Also all polygonal faces are bound to come out regular and the cells to come out as Johnson solids at least. (This was always the hard part to check.)

But so far we considered here as starting figures only regular polychora and some of the Wythoffian ones. So far we neglected all their CRF diminishings! Sure, the presumption was mostly right, that the lesser the symmetry, the fewer finds. - But then what about those cases where the application of the kaleido faceting takes place at one hemiglome, but the diminishing would take place completely independently at the other, while the afterwards expansion would not affect the local diminished figures? Or what about reversing the sequence, i.e. first applying EKF and only thereafter trying to apply any diminishings?

So far I have not looked deeper than that, esp. cannot provide concrete examples so far. But the sheer mass of already known EKFs makes it not too unlikely to come up with new CRFs, I suppose.

(BTW, we never did a systematic (eg. computer based) research on EKFs so far. All those listed ones only where found through manual listings of possibilities and individual considerations of cases. So there might be so far unconsidered ones or even overlooked ones out there as well.)

--- rk

[Klitzing]

Just to provide a first (rather trivial) set of examples here:

Ex = 600-cell can be given as lace tower as
oxofofoxo-3-ooooxoooo-5-ooxoooxoo-&#xt = VFfxo-2-oxofo-3-oooox-5-ooxoo-&#zx.

When inverting the 1st node of the 4th layer in the right representation we get as according kaleido faceting
VFf(-x)o-2-oxofo-3-oooox-5-ooxoo-&#zx.

That on now can be expanded into
BAFox-2-oxofo-3-oooox-5-ooxoo-&#zx = oxoofooxo-3-oooxoxooo-5-ooxoooxoo-&#xt
(being already known and called "telex", the tele-elongated ex).
- I used distance characters as B = 2f+x = fff, A = f+2x, V = 2f, F = f+x = ff here.

But then, the final tower representation exhibits that it can be diminished at either pole independently either by 1, 2, or 3 strati, i.e. providing e.g. as symmetrical paradiminishings:
xoofoox-3-ooxoxoo-5-oxoooxo-&#xt
oofoo-3-oxoxo-5-xooox-&#xt
ofo-3-xox-5-ooo-&#xt
But, for sure, we here have in total already 10 different combinations!

And as the central bistratic layer ofo-3-xox-5-ooo-&#xt just happens to be the 12-augmented iddip (icosidodecahedral prism), any number (and relative geometry thereof) of those augmentations could be neglected independently in turn, thus boosting up the count accordingly.

--- rk

[wendy]

My problem here is that if you use negative coordinates, which is fine, then the lacing varies outside of the symmetry, and potentially crosses itself, since the lacing-tower describes at least a lacing thread. If this actually crosses the mirror, rather than simply run along it, it would cross itself, and produce a non-convex form.

I still have not got my mind around Student91's EKF thingie.

[Klitzing]

Starting with that very same telex, I finally found a much more interesting new CRF.
Its cell count totals are
   2 does
  24 mibdies (J62)
   6 bilbiroes (J91)
   8 trips
  36 squippies (J1)
  16 tets

In fact it could be obtained from telex by removing the bistratic caps at the poles, thus freeing the does, and by additionally chopping off 6 axis-parallel equatorial edges in octahedral across symmetry, giving birth to those bilbiroes underneath.

It should be mentioned here, that the common usage of dodecahedra at the poles and octahedral diminishings at the equator clearly does lead to pyrithohedral across symmetry only.

In the sense of tegum sums this new find can be described as: oFx|xFf|fFo|f-2-xoF|fxF|ofF|f-2-Fxo|Ffx|Fof|f-2-FFF|xxx|ooo|F-&#zx. (Here the pipe symbols are used only for visual grouping purpose and elsewise could be removed.)

Constructively it can be described like 2 dodecahedra at top and bottom, the 6 octahedrally smmetrical edges of which get connected by bilbiroes. It should be pointed out here, that doe and bilbiro only attach edgewise. Nonetheless those cells happen to be orthogonal. All the pentagons then get matched by mibdies, thereby attaching one each towards a dodecahedron and the other towards a bilbiro. The pairs of cubical vertices of the doedecahedra get connected by ind a 3D lune out of tet + trip + tet each. And finally to all these squares (so far still un-matched) 2 equivalence classes of squippies are to be added (one attaching to bilbiroes, one to those trips).

Quite a nice new CRF find, ain't it?
To my knowledge that one was not encountered before, correct?

--- rk

[Klitzing]

My problem here is that if you use negative coordinates, which is fine, then the lacing varies outside of the symmetry, and potentially crosses itself, since the lacing-tower describes at least a lacing thread. If this actually crosses the mirror, rather than simply run along it, it would cross itself, and produce a non-convex form.

I still have not got my mind around Student91's EKF thingie.

Yes indeed, Wendy.

The first idea indeed is to consider some subsymmetrical non-convex faceting figure.
This is what can be called a kaleido faceting.

The second step then applies expansions (if possible) in such a way that all retrograde edges get zero (or poisitve).

Student91's idea is that simple.
Whether such an expanded kaleido faceting (EKF) truely results in a CRF has to be checked individually.
It just describes the techniques to be applied. And it is that these techniques can be handled at a very high, i.e. formal level as mere operations on Dynkin diagrams, what makes the EKF idea so attractive.

You also find some description of his technique in his recently reposted paper (which we once aimed to publish together, hehe), or in my according webpage on EKFs. It is recommended to look there at the quite illustrative 3D applications, i.e. the derivations of bilbiro (J91), thawro (J92), and pocuro (J32) as expansions from according (clearly non-convex) kaleido facetings of ike.

--- rk

[Klitzing]

Ah, one thing still to add to my description of his mere kaleido faceting part:

It is not that Student91 simply inverts some edges. He recaptures that always by an appropriate enlargement of the neighbouring node in such a way, that the vertex set as such remains unchanged!
So, when starting with x-P-y, then you'll have (-x)-P-(y+x*cos(pi/p)) for sure!

Esp.
x2y -> (-x)2(y+o)
x3y -> (-x)3(y+x)
x4y -> (-x)4(y+q)
x5y -> (-x)5(y+f)
etc.

--- rk

[student91]

To my knowledge that one was not encountered before, correct?

Some posts around that link I come up with the telex-expansion exactly because it has hidden bilbiros, which can be shown by removing bilbiro-pseudopyramids.

[Klitzing]

To my knowledge that one was not encountered before, correct?

Some posts around that link I come up with the telex-expansion exactly because it has hidden bilbiros, which can be shown by removing bilbiro-pseudopyramids.

Okay, there back in May 2014 you derived what later became named telex, and even pointed out clearly, that it hides bilbiroes underneath its axis-parallel equatorial edges. None the same the according actual diminishing never was done explicitely, thereby deriving all its cells etc. Esp. not my above octa-diminishing of telex (the 2 from medial parabidiminishing down from the pole vertices to does plus the 6 from ocahedrally across diminishing of such axis-parallel edges down to these bilbiroes).

Here then comes its full incidence matrix - including all its Dynkin derived element structurings:

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oFx|xFf|fFo|f-2-xoF|fxF|ofF|f-2-Fxo|Ffx|Fof|f-2-FFF|xxx|ooo|F-&#zx

o.. ... ... . 2 o.. ... ... . 2 o.. ... ... . 2 o.. ... ... .      & | 24  *  *  * |  1  2  2  0  0  0  0  0  0 |  2  3  1  2  0  0  0  0  0  0  0  0 | 1  3 1 0  0  0  0
... o.. ... . 2 ... o.. ... . 2 ... o.. ... . 2 ... o.. ... .      & |  * 48  *  * |  0  1  0  1  1  2  1  1  1 |  1  0  1  1  1  2  1  1  1  1  2  2 | 0  2 1 1  2  1  1
... ... o.. . 2 ... ... o.. . 2 ... ... o.. . 2 ... ... o.. .      & |  *  * 12  * |  0  0  0  0  0  0  4  4  0 |  2  0  0  0  0  0  0  2  2  2  4  0 | 0  2 1 0  2  1  0
... ... ... o 2 ... ... ... o 2 ... ... ... o 2 ... ... ... o        |  *  *  * 16 |  0  0  3  0  0  0  0  0  3 |  0  3  0  3  0  0  0  0  0  0  0  3 | 1  3 0 0  0  0  1
---------------------------------------------------------------------+-------------+----------------------------+-------------------------------------+------------------
... ... ... .   x.. ... ... .   ... ... ... .   ... ... ... .      & |  2  0  0  0 | 12  *  *  *  *  *  *  *  * |  2  2  0  0  0  0  0  0  0  0  0  0 | 1  2 1 0  0  0  0
o.. o.. ... . 2 o.. o.. ... . 2 o.. o.. ... . 2 o.. o.. ... . &#x  & |  1  1  0  0 |  * 48  *  *  *  *  *  *  * |  1  0  1  1  0  0  0  0  0  0  0  0 | 0  2 1 0  0  0  0
o.. ... ... o 2 o.. ... ... o 2 o.. ... ... o 2 o.. ... ... o &#x  & |  1  0  0  1 |  *  * 48  *  *  *  *  *  * |  0  2  0  1  0  0  0  0  0  0  0  0 | 1  2 0 0  0  0  0
... x.. ... .   ... ... ... .   ... ... ... .   ... ... ... .      & |  0  2  0  0 |  *  *  * 24  *  *  *  *  * |  0  0  1  0  1  0  0  0  1  0  0  0 | 0  1 1 0  0  1  0  (ortho)
... ... ... .   ... ... ... .   ... ... ... .   ... x.. ... .      & |  0  2  0  0 |  *  *  *  * 24  *  *  *  * |  0  0  0  0  1  2  0  1  0  1  0  0 | 0  0 1 1  2  1  0  (para)
... oo. ... . 2 ... oo. ... . 2 ... oo. ... . 2 ... oo. ... . &#x  & |  0  2  0  0 |  *  *  *  *  * 48  *  *  * |  0  0  0  0  0  1  1  0  0  0  1  1 | 0  1 0 1  1  0  1
... o.. o.. . 2 ... o.. o.. . 2 ... o.. o.. . 2 ... o.. o.. . &#x  & |  0  1  1  0 |  *  *  *  *  *  * 48  *  * |  1  0  0  0  0  0  0  1  0  0  1  0 | 0  1 1 0  1  0  0
... o.. ..o . 2 ... o.. ..o . 2 ... o.. ..o . 2 ... o.. ..o . &#x  & |  0  1  1  0 |  *  *  *  *  *  *  * 48  * |  0  0  0  0  0  0  0  0  1  1  1  0 | 0  1 0 0  1  1  0
... o.. ... o 2 ... o.. ... o 2 ... o.. ... o 2 ... o.. ... o &#x  & |  0  1  0  1 |  *  *  *  *  *  *  *  * 48 |  0  0  0  1  0  0  0  0  0  0  0  2 | 0  2 0 0  0  0  1
---------------------------------------------------------------------+-------------+----------------------------+-------------------------------------+------------------
... ... ... .   x.. f.. o.. .   ... ... ... .   ... ... ... . &#xt & |  2  2  1  0 |  1  2  0  0  0  0  2  0  0 | 24  *  *  *  *  *  *  *  *  *  *  * | 0  1 1 0  0  0  0
... ... ... .   x.o ... ... f   ... ... ... .   ... ... ... . &#xt & |  3  0  0  2 |  1  0  4  0  0  0  0  0  0 |  * 24  *  *  *  *  *  *  *  *  *  * | 1  1 0 0  0  0  0
o.. x.. ... .   ... ... ... .   ... ... ... .   ... ... ... . &#x  & |  1  2  0  0 |  0  2  0  1  0  0  0  0  0 |  *  * 24  *  *  *  *  *  *  *  *  * | 0  1 1 0  0  0  0
o.. o.. ... o 2 o.. o.. ... o 2 o.. o.. ... o 2 o.. o.. ... o &#x  & |  1  1  0  1 |  0  1  1  0  0  0  0  0  1 |  *  *  * 48  *  *  *  *  *  *  *  * | 0  2 0 0  0  0  0
... x.. ... .   ... ... ... .   ... ... ... .   ... x.. ... .      & |  0  4  0  0 |  0  0  0  2  2  0  0  0  0 |  *  *  *  * 12  *  *  *  *  *  *  * | 0  0 1 0  0  1  0
... ... ... .   ... ... ... .   ... ... ... .   ... xx. ... . &#x  & |  0  4  0  0 |  0  0  0  0  2  2  0  0  0 |  *  *  *  *  * 24  *  *  *  *  *  * | 0  0 0 1  1  0  0  (para)
... ooo ... . 2 ... ooo ... . 2 ... ooo ... . 2 ... ooo ... . &#x    |  0  3  0  0 |  0  0  0  0  0  3  0  0  0 |  *  *  *  *  *  * 16  *  *  *  *  * | 0  0 0 1  0  0  1
... ... ... .   ... ... ... .   ... ... ... .   ... x.. o.. . &#x  & |  0  2  1  0 |  0  0  0  0  1  0  2  0  0 |  *  *  *  *  *  *  * 24  *  *  *  * | 0  0 1 0  1  0  0  (para)
... x.. ..o .   ... ... ... .   ... ... ... .   ... ... ... . &#x  & |  0  2  1  0 |  0  0  0  1  0  0  0  2  0 |  *  *  *  *  *  *  *  * 24  *  *  * | 0  1 0 0  0  1  0  (ortho)
... ... ... .   ... ... ... .   ... ... ... .   ... x.. ..o . &#x  & |  0  2  1  0 |  0  0  0  0  1  0  0  2  0 |  *  *  *  *  *  *  *  *  * 24  *  * | 0  0 0 0  1  1  0  (para)
... oo. o.. . 2 ... oo. o.. . 2 ... oo. o.. . 2 ... oo. o.. . &#x  & |  0  2  1  0 |  0  0  0  0  0  1  1  1  0 |  *  *  *  *  *  *  *  *  *  * 48  * | 0  1 0 0  1  0  0
... oo. ... o 2 ... oo. ... o 2 ... oo. ... o 2 ... oo. ... o &#x  & |  0  2  0  1 |  0  0  0  0  0  1  0  0  2 |  *  *  *  *  *  *  *  *  *  *  * 48 | 0  1 0 0  0  0  1
---------------------------------------------------------------------+-------------+----------------------------+-------------------------------------+------------------
oFx ... ... f 2 xoF ... ... f 2 Fxo ... ... f   ... ... ... . &#zx   | 12  0  0  8 |  6  0 24  0  0  0  0  0  0 |  0 12  0  0  0  0  0  0  0  0  0  0 | 2  * * *  *  *  * doe
o.x x.f ..o f   ... ... ... .   ... ... ... .   ... ... ... . &#xr & |  3  4  1  2 |  1  4  4  1  0  2  2  2  4 |  1  1  1  4  0  0  0  0  1  0  2  2 | * 24 * *  *  *  * mibdi
o.. x.. f.. . 2 x.. f.. o.. .   ... ... ... .   F.. x.. o.. . &#zx & |  4  8  2  0 |  2  8  0  4  4  0  8  0  0 |  4  0  4  0  2  0  0  4  0  0  0  0 | *  * 6 *  *  *  * bilbiro
... ... ... .   ... ... ... .   ... ... ... .   ... xxx ... . &#x    |  0  6  0  0 |  0  0  0  0  3  6  0  0  0 |  0  0  0  0  0  3  2  0  0  0  0  0 | *  * * 8  *  *  * trip
... ... ... .   ... ... ... .   ... ... ... .   ... xx. o.. . &#x  & |  0  4  1  0 |  0  0  0  0  2  2  2  2  0 |  0  0  0  0  0  1  0  1  0  1  2  0 | *  * * * 24  *  * squippy
... x.. ..o .   ... ... ... .   ... ... ... .   ... x.. ..o . &#x  & |  0  4  1  0 |  0  0  0  2  2  0  0  4  0 |  0  0  0  0  1  0  0  0  2  2  0  0 | *  * * *  * 12  * squippy
... ooo ... o 2 ... ooo ... o 2 ... ooo ... o 2 ... ooo ... o &#x    |  0  3  0  1 |  0  0  0  0  0  3  0  0  3 |  0  0  0  0  0  0  1  0  0  0  0  3 | *  * * *  *  * 16 tet


And this is the according lace city:

Code: Select all

o2x       x2f   f2o   x2f       o2x
                                   
                                   
f2f       F2x   o2F   F2x       f2f
                                   
F2o       f2F         f2F       F2o
                                   
                                   
x2F             F2f             x2F
                                   
                                   
F2o       f2F         f2F       F2o
                                   
f2f       F2x   o2F   F2x       f2f
                                   
                                   
o2x       x2f   f2o   x2f       o2x


--- rk

[Klitzing]

And here comes a further interesting EKF diminishing.

Let's start with that EKF itself. It is derived once more from ex. That one - this time - wrt. pentic subsymmetry can be given as xffoo3oxoof3fooxo3ooffx&#zx. Here we apply a kaleido faceting to the first level at the first node: (-x)ffoo3xxoof3fooxo3ooffx&#zx. Then one applies a Stott expansion within across symmetry at every level wrt. that first node. This then results in the (already known) EKF oFFxx3xxoof3fooxo3ooffx&#zx.

When considering the sub-figure of second and fourth nodes, one has ..... xxoof ..... ooffx&#zx, which is an ike. Thus it looks promizing to eliminate either the third or the fourth level. (In either case the ike would transform into a mibdi.) - The fourth level so does not work, asking for an other figure to match the thereby generated pentagons, which is not CRF itself.

But the elimination of the third level works fine: oF.xx3xx.of3fo.xo3oo.fx&#zx is indeed possible. - The according total cell counts there are
 30 mibdi
 20 oct
 90 squippy
 25 tet
 20 teddi
  5 co
 40 trip

Moreover, these processes here happen to be commutative! One could well apply the 20-diminishing (deleting the 3rd level) to ex first, and only thereafter the EKF construction then onto that diminishing. The result here would be the same. In fact the ikes, occuring while diminishing, then would be transformed into the teddies.

--- rk

[Klitzing]

In fact, within the last 2 lines of my previous post, I take reference to some 20-diminishing of ex.

That one occurs thereby when one writes down ex as tegum sum in pentic symmetry: xffoo3oxoof3fooxo3ooffx&#zx. Then, as in the commutative process above as well, one deletes the 3rd layer, which describes the vertex set of an inscribed f-scaled spid, i.e. selects 20 out of the ex-vertices which are to be chopped off.

Find below additionally the full incidence matrix description of that 20-diminished ex:

Code: Select all

xfoo3oxof3foxo3oofx&#zx   → all heights = 0 – except those of the not existing lacing(2,3)

o...3o...3o...3o...     & | 60  * |  2   2   2   4  0 |  1  1   4   6   8  0 |  2  2  2   6  0 verf: mibdi
.o..3.o..3.o..3.o..     & |  * 40 |  0   3   3   0  3 |  0  3   3   0   6  3 |  1  0  3   3  1 verf: teddi
--------------------------+-------+-------------------+----------------------+----------------
x... .... .... ....     & |  2  0 | 60   *   *   *  * |  1  0   2   2   0  0 |  2  1  0   2  0
oo..3oo..3oo..3oo..&#x  & |  1  1 |  * 120   *   *  * |  0  1   0   0   2  0 |  0  0  2   1  0
o.o.3o.o.3o.o.3o.o.&#x  & |  1  1 |  *   * 120   *  * |  0  0   2   0   2  0 |  1  0  1   2  0
o..o3o..o3o..o3o..o&#x    |  2  0 |  *   *   * 120  * |  0  0   0   2   2  0 |  0  1  1   2  0
.... .x.. .... ....     & |  0  2 |  *   *   *   * 60 |  0  1   0   0   0  2 |  0  0  2   0  1
--------------------------+-------+-------------------+----------------------+----------------
x...3o... .... ....     & |  3  0 |  3   0   0   0  0 | 20  *   *   *   *  * |  2  0  0   0  0
.... ox.. .... ....&#x  & |  1  2 |  0   2   0   0  1 |  * 60   *   *   *  * |  0  0  2   0  0
x.o. .... .... ....&#x  & |  2  1 |  1   0   2   0  0 |  *  * 120   *   *  * |  1  0  0   1  0
x..o .... .... ....&#x  & |  3  0 |  1   0   0   2  0 |  *  *   * 120   *  * |  0  1  0   1  0
oo.o3oo.o3oo.o3oo.o&#x  & |  2  1 |  0   1   1   1  0 |  *  *   *   * 240  * |  0  0  1   1  0
.... .x..3.o.. ....     & |  0  3 |  0   0   0   0  3 |  *  *   *   *   * 40 |  0  0  1   0  1
--------------------------+-------+-------------------+----------------------+----------------
x.o.3o.o. .... ....&#x  & |  3  1 |  3   0   3   0  0 |  1  0   3   0   0  0 | 40  *  *   *  *
x..o .... .... o..x&#x    |  4  0 |  2   0   0   4  0 |  0  0   0   4   0  0 |  * 30  *   *  *
.... oxof3foxo ....&#xt   |  6  6 |  0  12   6   6  6 |  0  6   0   0  12  2 |  *  * 20   *  * ike, lace tower as b-a-d-c
.... .... .... oo.x&#x  & |  3  1 |  1   1   2   2  0 |  0  0   1   1   2  0 |  *  *  * 120  *
.... .x..3.o..3.o..     & |  0  4 |  0   0   0   0  6 |  0  0   0   0   0  4 |  *  *  *   * 10


--- rk

[Klitzing]

It just occured to me that the "spid-diminished ex" of the previous post happens to have regular cells only. Therefore it is one of the millions of Blind polychora for sure. Further it happens to have mibdies (J62) and teddies (J63) for vertex figures. Therefore it surely can be truncated like any of the regular polytopes: then using a cell count of 20 ties, 200 tuts, 60 mibdies, and 40 teddies. - Or it could be rectified, resulting then in a total cell count of 20 ids, 200 octs, 60 mibdies, and 40 teddies.

Note that this is quite similar to sadi, aka the ico-diminished ex. That one too allowed for truncation and rectification.

--- rk

[username5243]

It just occured to me that the "spid-diminished ex" of the previous post happens to have regular cells only. Therefore it is one of the millions of Blind polychora for sure. Further it happens to have mibdies (J62) and teddies (J63) for vertex figures. Therefore it surely can be truncated like any of the regular polytopes: then using a cell count of 20 ties, 200 tuts, 60 mibdies, and 40 teddies. - Or it could be rectified, resulting then in a total cell count of 20 ids, 200 octs, 60 mibdies, and 40 teddies.

Note that this is quite similar to sadi, aka the ico-diminished ex. That one too allowed for truncation and rectification.

--- rk

Interesting one here. I wonder, could this diminishing apply this kind of pennic-symmetric diminishing applied to other uniform polychora with ex symmetry, just like how the ico-diminishing can be applied to various truncations of ex? An example of what I mean would be cutting 20 doe||srid from sidpixhi (x5o3o3x) in spid based pattern.

[Klitzing]

Yes, username5243, these all exist (according CRF-cut subsumed).

In fact my truncated version is nothing but the bistratically spid-diminished tex.
And my rectified version is nothing but the monostratically spid-diminished rox.

--- rk

[Klitzing]

Polyhedron Dude privately mailed me that he just coined acronyms for those as well:

• idimex = icosidiminished hexacosachoron
• tidimex = truncated icosidiminished hexacosachoron
• ridimex = rectified icosidiminished hexacosachoron

• sadi = snub disicositetrachoron = icositetradiminished hexacosachoron
• tisadi = truncated snub disicositetrachoron = truncated icositetradiminished hexacosachoron
• risadi = rectified snub disicositetrachoron = rectified icositetradiminished hexacosachoron

--- rk

[Klitzing]

in the run of thinking about those 20-diminished-ex relates, I revisited the 24-diminished-ex (=sadi) relates again. I even digged out my own post on idsrahi (24-diminished srahi) and once more appreciated the following commentary thereon:

...
I think this polytope is the coolest of the 24-diminished .5.3.3.'s. I mean, it has J83's!!

Then I found that in those days I considered monostratic diminishings only (except of tisadi, for sure). So I found a new one of that old family, idprahi = the (bistratically) 24-diminished prahi. Here we chop off 24 bistratic srid-first caps of prahi. These, in turn, recently were considered in a different thread, described as srid-atop-grid-tutsatope or "srida gridtum".

The details of that new one, idprahi, then can be obtained in the very same way as they were derived in the case of idsrahi (re-linked above), just that the srid, to be inserted into the fundamental domain, here will not be incident to its boundary, like for idsrahi, rather it would be a bit smaller, thus giving rise for the pips of prahi as inter domain connectors. And that, right because those srid vertices no longer are boundary incident, the then applied vertex class equivalences here now no longer apply. This is because of the then still "coincident pairs of" vertices now become truely separated by means of the to be applied Stott expansion from srahi to prahi.

The incidence matrix has been calculated already, but is way to huge to be displayed here. (Stay tuned for the next update of my website.)
The mere cell counts ought to be given here none the less:

96 tedrids (J83)
120 tuts
480 hips
432 pips
24 grids

Thus idprahi is a quite huge CRF with 1152 cells and 4320 vertices!

--- rk

[Klitzing]

Yesterday then I returned to the 20-diminishings again. I chose the 20-diminished srahi to be investigated first.

I applied there the same techniques as for the 24-diminished srahi, as outlined in that recently cited post. Halas, sadi was uniform but idimex is only biform. Accordingly there we had a single fundamental domain into which a single vertex for the ex-diminishing resp. a full srid for the srahi-diminishing was to be placed, but now we have 2 such fundamental domains. The domains themselves are the duals of teddi and mibdi. As these are diminishings of ike thus those therefore are according (subsymmetrical) stellations of doe. Their faces then in turn are (subsymmetrical) stellations of the regular pentagon: either obtuse golden triangles (+2 apices), kites (+1 apex), or the pentagon itself (no additional apex). Accordingly, in correlation to the symmetry of those fundamental domains, the inscribed srids have to be considered in that implied subsymmetry only, resulting in vertex types A-L for the 3fold an vertex types a-q for the 2fold case. Next we have to omit the vertices C,E,G resp. b,d,g from these sets, as those would be the bitten off ones by virtue of the diminishings. Leading thereby from those srids towards the tedrids (J83) resp. the mabidrids (J81). Finally one has to consider the inter-domain correlations as well, which here adds the additional vertex type identifications F=a, H=c, J=e, L=p, K=m, I=j, as well as f=i and h=l.

The resulting figure idimsrahi (20-diminished srahi) then is even cooler than idsrahi (24-diminished srahi), as it incorporates not only tedrids (J83), it additionally uses mabidrids (J81) too! In fact, its total cell count runs like this:

 20 tids
 40 tedrids (J83)
 60 mabidrids (J81)
200 octs
600 trips

--- rk

[Klitzing]

The last days I considered idimsrix, the (spid-wise) 20-diminished srix. Kind of "x3o3x5o - 20 ox3xx5oo&#x".

That one clearly is quite intricate, even more than idimsrahi: Again we use 2 different types of fundamental domains. Within idimex a single vertex was contained therein. (In fact this is the definition of those domains.) Whereas, in idimsrahi a complete srid with 60 vertices was contained. But all those vertices where placed on the domain boundaries. Thus several vertex class identifications (by virtue of domain connections) took place. Further the diminishings, situated at the sharp domain apices, in that case were deep enough to bite off bits of those srids. This is how tedrids and mabidrids occured there. - Here now we have a complete id with 30 vertices contained within each of those 2 types of domains. The id thereby is fully contained, i.e. its vertices do not lie on the boundary. Instead, the pentagons give rise here to extra pips, which then do cross those domain boundaries. Therefore external symmetry derived vertex class identifications do not take place. Moreover the according diminishings at those domain apices are not as deep as there, they just are tangential to those inscribed ids. Therefore none of those vertex classes can be ruled out.

On the other hand idimsrix looks a bit less outstanding than idimsrahi. It neither does provide tedrid nor mabidrid cells. The only non-uniform used cells here will be tricues (J3). Those derive from dissected coes. Whereas the ids all remain intact here.

The total cell count of idimsrix will be:

200 coes
100 ids
20 ties
480 pips
400 tricues (J3)

Thus idimsrix is again a quite huge new CRF with 1200 cells and 3000 vertices. The according incidence matrix meanwhile has been calculated, but again is way to huge to be displayed here. (Stay tuned for the next update of my website.)

--- rk

[Klitzing]

An example of what I mean would be cutting 20 doe||srid from sidpixhi (x5o3o3x) in spid based pattern.

That one I considered now finally too - right in the same way as the already mentioned other such 20-diminishings.
This new CRF polychoron happens to be quite special as it still has just uniform cells only!

In fact, its cells are:
100 does
480 pips
600 trips
200 tets
20 srids

Thus it has a total of 1400 cells and 2000 vertices.

--- rk

[Klitzing]

After that excess on subsymmetrical diminishings - derived as application of a symmetry pattern onto higher Wythoffian relatives (more nodes being ringed), while considering the mere fundamental domain(s) of the lesser ones - I'll come back to the Topic of this thread, resp. to my original idea:

Today I got a new idea onto generating further CRFs!

The theme of this very thread once led us to the definition of the EKFs, the expanded kaleideo facetings.
[...]

But so far we considered here as starting figures only regular polychora and some of the Wythoffian ones. So far we neglected all their CRF diminishings! Sure, the presumption was mostly right, that the lesser the symmetry, the fewer finds. - But then what about those cases where the application of the kaleido faceting takes place at one hemiglome, but the diminishing would take place completely independently at the other, while the afterwards expansion would not affect the local diminished figures? Or what about reversing the sequence, i.e. first applying EKF and only thereafter trying to apply any diminishings?
[...]

Today I'll like to speak about a specific x3o3o5o based EKF with o5o2o5o subsymmetry. The 600-cell itself could be represented within that subsymmetry as the tegum sum  xfooxo-5-xofxoo-2-oxofox-5-ooxofx-&#zx. In fact this is nothing different than the lace city information of the hexacosachoron:

Code: Select all

                 o5o           o5o                 
                        o5x                       
                                                   
            x5o                     x5o           
     o5o                                   o5o     
                        f5o                       
                 o5f           o5f                 
                                                   
     o5x                                   o5x     
            f5o                     f5o           
                                                   
o5o                     x5x                     o5o
                                                   
            o5f                     o5f           
     x5o                                   x5o     
                                                   
                 f5o           f5o                 
                        o5f                       
     o5o                                   o5o     
            o5x                     o5x           
                                                   
                        x5o                       
                 o5o           o5o                 


The EKF of consideration then applies the edge reversal to the first node in the first and in the 5th level. The first level reads x.....5x.....2o.....5o..... . Edge reversal while keeping the vertex set results in (-x).....5F.....2o.....5o..... . The 5th level reads ....x.5....o.2....o.5....f. and the edge reversal here results in ....(-x).5....f.2....o.5....f. . (F, as usual, is ff or equivalently f+x, and f is the golden ratio.) This is the kaleido faceting part. The expansion part then is to be taken to that first node position (each), in order to get rid of these retrograde edges. Thus, the final EKF of consideration then is oFxxox-5-Fofxfo-2-oxofox-5-ooxofx-&#zx. As already displayed on my website, that tegum sum representation is equivalent to the following lace city display:

Code: Select all

                 x5o           x5o                 
                        x5x                       
                                                   
            o5f                     o5f           
     x5o                                   x5o     
                        F5o                       
                 x5f           x5f                 
                                                   
     x5x                                   x5x     
            F5o                     F5o           
                                                   
x5o                     o5F                     x5o
                                                   
            x5f                     x5f           
     o5f                                   o5f     
                                                   
                 F5o           F5o                 
                        x5f                       
     x5o                                   x5o     
            x5x                     x5x           
                                                   
                        o5f                       
                 x5o           x5o                 


Its cell total is
10 gyepips (J11)
25 ikes
10 pips
150 squippies (J1)
75 tets
50 trips

This EKF now allows for 2 independend types of diminishing. These diminishings exactly match to those tegum sum levels, which got involved by these kaleido facetings, i.e. the first resp. the 5th level. That is, those diminishings give rise to 3 further CRFs. Those then are:

A) diminishing just the first level results in .Fxxox-5-.ofxfo-2-.xofox-5.oxofx-&#zx - or as lace city:

Code: Select all

                 x5o           x5o                 
                        x5x                       
                                                   
            o5f                     o5f           
     x5o                                   x5o     
                        F5o                       
                 x5f           x5f                 
                                                   
     x5x                                   x5x     
            F5o                     F5o           
                                                   
x5o                                             x5o
                                                   
            x5f                     x5f           
     o5f                                   o5f     
                                                   
                 F5o           F5o                 
                        x5f                       
     x5o                                   x5o     
            x5x                     x5x           
                                                   
                        o5f                       
                 x5o           x5o                 


Here the cell totals are:
25 ikes
10 paps
15 pips
125 squippies (J1)
75 tets
50 trips

B) diminishing just the 5th level results in oFxx.x-5-Fofx.o-2-oxof.x-5-ooxo.x-&#zx - or as lace city:

Code: Select all

                 x5o           x5o                 
                        x5x                       
                                                   
                                                   
     x5o                                   x5o     
                        F5o                       
                 x5f           x5f                 
                                                   
     x5x                                   x5x     
            F5o                     F5o           
                                                   
x5o                     o5F                     x5o
                                                   
            x5f                     x5f           
                                                   
                                                   
                 F5o           F5o                 
                        x5f                       
     x5o                                   x5o     
            x5x                     x5x           
                                                   
                                                   
                 x5o           x5o                 


Here the cell totals are:
10 gyepips (J11)
25 mibdis (J62)
35 pips
25 squippies (J1)
75 tets
50 trips

C) applying both types of diminishing at the same time results in .Fxx.x-5-F.fx.o-2-.xof.x-5-.oxo.x-&#zx - or as lace city:

Code: Select all

                 x5o           x5o                 
                        x5x                       
                                                   
                                                   
     x5o                                   x5o     
                        F5o                       
                 x5f           x5f                 
                                                   
     x5x                                   x5x     
            F5o                     F5o           
                                                   
x5o                                             x5o
                                                   
            x5f                     x5f           
                                                   
                                                   
                 F5o           F5o                 
                        x5f                       
     x5o                                   x5o     
            x5x                     x5x           
                                                   
                                                   
                 x5o           x5o                 


Here the cell totals are:
25 mibdis (J62)
10 paps
40 pips
75 tets
50 trips

But then note that the same level-wise diminishings could have taken place on the original 600-cell itself (when represented in this subsymmetry)! This likewise gives rise to 3 further CRFs:

D) diminishing just the first level results in .fooxo-5-.ofxoo-2-.xofox-5-.oxofx-&#zx - or as lace city:

Code: Select all

                 o5o           o5o                 
                        o5x                       
                                                   
            x5o                     x5o           
     o5o                                   o5o     
                        f5o                       
                 o5f           o5f                 
                                                   
     o5x                                   o5x     
            f5o                     f5o           
                                                   
o5o                                             o5o
                                                   
            o5f                     o5f           
     x5o                                   x5o     
                                                   
                 f5o           f5o                 
                        o5f                       
     o5o                                   o5o     
            o5x                     o5x           
                                                   
                        x5o                       
                 o5o           o5o                 


Here the cell totals are:
10 paps
450 tets
This one kind is a mixture of ex and gap. In fact just a single circuit of paps is being used, whereas in gap two completely orthogonal such rings are in use.

E) diminishing just the 5th level results in xfoo.o-5-xofx.o-2-oxof.x-5-ooxo.x-&#zx - or as lace city:

Code: Select all

                 o5o           o5o                 
                        o5x                       
                                                   
                                                   
     o5o                                   o5o     
                        f5o                       
                 o5f           o5f                 
                                                   
     o5x                                   o5x     
            f5o                     f5o           
                                                   
o5o                     x5x                     o5o
                                                   
            o5f                     o5f           
                                                   
                                                   
                 f5o           f5o                 
                        o5f                       
     o5o                                   o5o     
            o5x                     o5x           
                                                   
                                                   
                 o5o           o5o                 


Here the cell totals are:
25 mibdis (J62)
225 tets
Here the chopped off Vertices are those of an inscribed (x,f)-pedip; accordingly those mibdies align as rings of 5 around their 5-5-edges, and these patches then align on a larger cycle, like pearls on a necklace.

F) applying both types of diminishing at the same time results in .foo.o-5-.ofx.o-2-.xof.x-5-.oxo.x-&#zx - or as lace city:

Code: Select all

                 o5o           o5o                 
                        o5x                       
                                                   
                                                   
     o5o                                   o5o     
                        f5o                       
                 o5f           o5f                 
                                                   
     o5x                                   o5x     
            f5o                     f5o           
                                                   
o5o                                             o5o
                                                   
            o5f                     o5f           
                                                   
                                                   
                 f5o           f5o                 
                        o5f                       
     o5o                                   o5o     
            o5x                     o5x           
                                                   
                                                   
                 o5o           o5o                 


Here the cell totals are:
25 mibdies (J62)
10 paps
75 tets

What would be interesting here, is that (C) is nothing but the Stott expansion of (F) wrt. the first node. Moreover (A) is obtained from (D) as its EKF wrt. the edge reversal at the first node of the (then remaining) 4th level and the afterwards expansion wrt. the first node positions each. And conversely (B) is obtained from (E) as its EKF wrt. the edge reversal at the first node of the first level and the afterwards expansion wrt. the first node positions each. - In fact, those considered diminishings do commutate with the respective EKF operations!

--- rk

[Klitzing]

One more new CRF within this run on diminished EKF's of ex or EKF's of diminished ex.

Today I'm looking on the axial o3o3o subsymmetry.
I'm reconsidering the already known EKF xoofxxFfooFxofo 3 xxFxoFxxxFoxFxx 3 ofoxFoofFxxfoox &#xt with cells list
  16 octs
  24 ikes
  72 squippies (J1)
  198 tets
  24 tricues (J3)
  48 trips
  2 tuts
Even so this figure is axially top-down symmetric, it does not belong to o2o3o3o symmetry. In fact it uses tetrahedral inversion here. (The same, btw., applies to the 600-cell itself.)

This EKF then was derived as follows. One starts with ex within this subsymmetry. That one can be given as xoofoxFfoofxofo 3 oofoxfooofxofoo 3 ofoxfoofFxofoox &#xt . Next one applies kaleido faceting to the fifth and - by top-down symmetry - also to the 11th layer, using o3x3f → x3(-x)3F resp. f3x3o → F3(-x)3x . Thus we have the kaleido faceted figure xoofxxFfooFxofo 3 oofo(-x)fooof(-x)ofoo 3 ofoxFoofFxxfoox &#xt . Finally one applies a Stott expansion wrt. the central node simultanuously at each layer. This is, how the above EKF was derived.

 

Today now I would like to point out the third and - by symmetry - the 13th layer. These layers could be omitted without loss, resulting in an according 12-diminishing of the EKF. (In fact, thoe diminishing in the vertex sets of that F-scaled oct in each of these layers would be Independent. Thus we would result in 2 different CRFs here.) The more symmetrical one then results in xo.fxxFfooFx.fo 3 xx.xoFxxxFox.xx 3 of.xFoofFxxf.ox &#xt , having then a total cell count of
  24 gyepips (J11)
  12 mibdies (J62)
  16 octs
  72 squippies (J1)
  24 tricues (J3)
  48 trips
  78 tets
  2 tuts

Again it happens here that the applications of diminishing and of EKF do commutate. In fact, one also could apply this very diminishing to the 600-cell itself in the first run. This then results in xo.foxFfoofx.fo 3 oo.oxfooofxo.oo 3 of.xfoofFxof.ox &#xt (a figure with 12 icosahedra and 600-12*20=480 tetrahedra). Only in the second run one then applies the EKF construction wrt. the (formerly) 5th and 11th layer resp. to the middle nodes each. - Either construction would result in this new CRF with the above provided 276 cells and with 192 vertices.

In this construction (as an afterwards applied diminishing) it happens that all the former icosahedra get diminished into gyepips, that the octahedra, truncated tetrahedra, triangular cupola, and square pyramids all remain unchanged, but that patches of 10 tetrahedra each combine into mibdies. Thus the former count of 198 tets get reduced into 198-12*10=78 remaining tetrahedra. Here the gyepips will adjoin to the pentagonal faces of the mibdies exclusively.

--- rk

[Klitzing]

That "new lace term", asked for be student91 way back in 2014, meanwhile setled to become the "tegum sum".
In fact, when A and B are both convex Wythoffians, based on the same (undecorated) symmetry group symbol,
then that tegum sum happens to be = hull( compound( A, B )).

They are denoted similar to other lace figures (lace prisms, lace towers, lace simplices, ...) in a stacked way,
laying out the symmetry symbol once, covering each node position as a sequence of the respective decorations from either of the layers.
In order to become the tegum sum notion, the trailer was setled to be "&#zx" for tegum sums.

In fact, they are closely related to degenerate lace prisms (&#x), i.e. having zero height (z = zero),
however, as can be seen from the above usage of the hull function, they omit the substructure and rather use the periform instead.

Despite of emphasizing above the hull and periform terms, tegum sums are valide for non-convex structures too.
Then the respective elements still are obtained either from the elements of the components (if they are exposed),
or are lace prisms (or somtimes tegum sums) in turn.

This construction device has proven very usefull over the last 8 years.
Quite many finds only became accessable to evaluate incidence matrices thereof, when dealing with that device.

Like in other constructions too, many of these tegum sum derived polytopes could well be described differently as well.
Thence so far no unique general naming scheme was required, rather names so far where given from scratch,
mostly from these different representations, eg. as a xyz-diminishing of ABC.

However, since this construction proved so usefull, and as the number of such derived polytopes increases quite fast,
it would be great to have an (at least alternate) naming scheme, which allows tegum sum derived polytopes to call that way directly.

Sure, one always could call them "A,B-tegum sum".
But what I have in mind here, is that such a thing also should get an according OBSA-device as well.

Any suggestions @Jonathan Bowers?

--- rk

[wendy]

I use 'thatch' for this. I used to use tegumate or something like that, but eventually the intent was not to equalivantly cunfuse 'prism-cut' for 'truncate' or 'intersection'. That would be prolly grand if we could do a word for that, so ID = fI xD intersection,

[Klitzing]

e.g.
xo3ox4xx xo4xx&#zx

it is the tegum sum of
x.3o.4x. x.4x. = osirco
.o3.x4.x .o4.x = squatic

-> OBSA = "osirco + squatic + [...]" ???

--- rk

[Klitzing]

At least that previous example has been solved in the meantime:

Consider o3x3o3o4x = span within its B2xB3 subsymmetry.
Then it becomes wxx4oxo oxo3oox4xxx&#zx.
Thus you can spot that osirco as its 2nd and that squatic as its 3rd layer.
The first, then to be omitted layer consists of 4 cyclically arranged (w4o . . .) cubes (. . o3o4x).
Therefore it could be described as a "cydispan" (or "cubcy dispan" for "cubical ...").

However, this solution just emphasizes once again the specificallity of a singular case.
No generally applicable rule for any tegum sum whatsoever.

--- rk