Scaliform cells

Discussion of tapertopes, uniform polytopes, and other shapes with flat hypercells.

Scaliform cells

Postby ndl » Sun Dec 15, 2019 5:42 am

Has anyone figured out: what is the set of polyhedra that can be used for cells of scaliform polychora? I would have assumed it's all the uniforms and their axially symmetric facetings (as delineated by Dr. Klitzing in his paper), but the new S11 (formed by fusing 12-gadtaxady) uses this really cool pentagrammic orthobirotunda which doesn't seem to fit into that set. Any research done on this?
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Re: Scaliform cells

Postby Klitzing » Sun Dec 15, 2019 8:38 pm

The definition of scaliforms asks for vertex transitivity (of the whole polytope). Thus especially all vertices are on a single hypersphere. As facets OTOH are situated on an afine hyperplane, the intersection too would be some hypersphere, i.e. each facet too is bound to have a unique circumradius (on its own).

Thus for 4D convex scaliforms e.g. all the according Johnson solids, which already had been enlisted in my paper on convex segmentochora, would be allowed. But here similar multistratic figures become allowed too. And, in general, for non-convex cases, similar (still orbiform) stary figures.

BTW. you mentioned axially symmetric cells. Remember that the square pyramid within convex segmentochora not only occurs as "point atop square", but also as "edge atop triangle"!

Scaliformity, like orbiformity, restrict to both: existance of a unique circumradius, and existance of a unique edge length. Therefore cells of scaliform polytopes generally are orbiform polytopes. Orbiformity is more general than scaliformity so: there is nothing being said about the number of equivalence classes under polytopal symmetry.

Note that there is a nice further Observation known: if you use a non-scaliform (so orbiform) polytope for cell in a scaliform polytope, say with vi vertices of vertex type i and with v=sumi vi vertices in total, then there exists some natural number k, such that all ratios ri=k*vi/v become integral and moreover describe the exact values of occurences of incidences of this cell type at each polytopal vertex, when being considered as divided in its respective cell vertex types. E.g. when a square pyramid is being used in a scaliform polychoron, then its incidence count by its tip is just one fourth of the count of vertex incidence by the base vertex (of the pyramid).

--- rk
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Re: Scaliform cells

Postby Klitzing » Sat Mar 21, 2020 1:59 pm

Thought about new scaliform polytopes. Especially the new term of alterprisms looks great. An alterprism being defined as a lace prism where both bases are the same polytope essentially, simply within an alternate (symbol) orientaion. Cases with selfdual linear groups are known for sure. Cases with tridental graphs, where the 2 arms would swap, are known for long as well.

Currently it occured to me that oo3xo5/2ox3*a (= sidtidap) is uniform, its Stott expansion xx3xo5/2ox3*a (= siida (previously siidcup)) is purely scaliform.
Therefrom derived ones then are: oo3oo3xo5/2ox3*b (= sidtaxhiap) again uniform, while its Stott expansions xx3oo3xo5/2ox3*b (= stut phiddixa), oo3xx3xo5/2ox3*b (= wavhiddixa), and xx3xx3xo5/2ox3*b (= sphiddixa) all are purely scaliform.
Those I already have investigated, incidence matrices are set up. (Be prepared for my next website update. :) )

Next I'm just thinking about o5/2o3x5/2o3*b (= sitpodady) and o5/2x3x5/2o3*b (= swavathi) - the other ones with leading x5/2... would become Grünbaumian here -:
These 2 ought provide further examples here - ain't they?

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Re: Scaliform cells

Postby Klitzing » Sat Mar 21, 2020 3:40 pm

Haha, le voilà:

sitpodadia, the sitpodady alterprism, has a circumradius of sqrt([7+sqrt(5)]/8) = 1.074481, and here is its incidence matrix:
Code: Select all
oo5/2oo3xo5/2ox3*b &#x  -  height = sqrt[(sqrt(5)-1)/2] = 0.786151

o.5/2o.3o.5/2o.3*b     & | 240 |   60   12 |   60   30   90 |  12  20   80  30 | 1  13  20
-------------------------+-----+-----------+----------------+------------------+----------
..   .. x.   ..        & |   2 | 7200    * |    2    1    1 |   1   2    2   1 | 1   1   2
oo5/2oo3oo5/2oo3*b &#x   |   2 |    * 1440 |    0    0   10 |   0   0   10   5 | 0   2   5
-------------------------+-----+-----------+----------------+------------------+----------
..   o.3x.   ..        & |   3 |    3    0 | 4800    *    * |   1   1    1   0 | 1   1   1
..   .. x.5/2o.        & |   5 |    5    0 |    * 1440    * |   0   2    0   1 | 1   0   2
..   .. xo   ..    &#x & |   3 |    1    2 |    *    * 7200 |   0   0    2   1 | 0   1   2
-------------------------+-----+-----------+----------------+------------------+----------
o.5/2o.3x.   ..        & |  12 |   30    0 |   20    0    0 | 240   *    *   * | 1   1   0 gike
..   o.3x.5/2o.3*b     & |  20 |   60    0 |   20   12    0 |   * 240    *   * | 1   0   1 sidtid
..   oo3xo   ..    &#x & |   4 |    3    3 |    1    0    3 |   *   * 4800   * | 0   1   1 tet
..   .. xo5/2ox    &#x   |  10 |   10   10 |    0    2   10 |   *   *    * 720 | 0   0   2 stap
-------------------------+-----+-----------+----------------+------------------+----------
o.5/2o.3x.5/2o.3*b     & | 120 | 3600    0 | 2400  720    0 | 120 120    0   0 | 2   *   * sitpodady
oo5/2oo3xo   ..    &#x & |  13 |   30   12 |   20    0   30 |   1   0   20   0 | * 240   * gikepy
..   oo3xo5/2ox3*b &#x   |  40 |  120   60 |   40   24  120 |   0   2   40  12 | *   * 120 sidtidap

--- rk
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Re: Scaliform cells

Postby username5243 » Sat Mar 21, 2020 4:24 pm

So swavathi alterprism (swavathia?) should have as facets then: 2 swavathis, 120 siicups, and 240 gid || tiggy segmentochora ("gidatiggy"). And will exist, given it's just a suitable expansion of that last one.

And I think the other group will have as facets:

Sidtaxhiap: 2 sidtaxhis, 120 sidtidaps, 1200 pens
Wavhiddixa: 2 wavhiddixes, 120 siidcups, 1200 octatuts
Stut Phiddixa: 2 stut phiddixes, 120 sidtidaps, 1200 tetacoes, 720 stappips (5/2 antiprism prism)
Sphiddixa: 2 sphiddixes, 120 siidcups, 1200 tutatoes, 720 stappips

In addition, there are two regular polychora with hyic symmetry that are self-dual, gohi and gashi, that could form alterprisms (or maybe just antiprisms?) in 5d, gohi and gashi. I think Klitzing worked it out, and while the gohi case seems to be degenerate, the gashi case does work. Maybe there are more alterprisms found in those groups (o5o5/2o5o and o5/2o5o5/2o)?
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Re: Scaliform cells

Postby Klitzing » Sat Mar 21, 2020 8:09 pm

username5243 wrote:So swavathi alterprism (swavathia?) should have as facets then: 2 swavathis, 120 siicups, and 240 gid || tiggy segmentochora ("gidatiggy"). And will exist, given it's just a suitable expansion of that last one.

probably, not yet done...
And I think the other group will have as facets:

Sidtaxhiap: 2 sidtaxhis, 120 sidtidaps, 1200 pens
Wavhiddixa: 2 wavhiddixes, 120 siidcups, 1200 octatuts
Stut Phiddixa: 2 stut phiddixes, 120 sidtidaps, 1200 tetacoes, 720 stappips (5/2 antiprism prism)
Sphiddixa: 2 sphiddixes, 120 siidcups, 1200 tutatoes, 720 stappips

yep, those all are correct. And this shows moreover that sidtaxhiap is uniform, as ist facets all are uniform themselves, while the other 3 have multiform facets (octatuts, tetacoes, tutatoes) and so are purely scaliform.
In addition, there are two regular polychora with hyic symmetry that are self-dual, gohi and gashi, that could form alterprisms (or maybe just antiprisms?) in 5d, gohi and gashi. I think Klitzing worked it out, and while the gohi case seems to be degenerate, the gashi case does work. Maybe there are more alterprisms found in those groups (o5o5/2o5o and o5/2o5o5/2o)?

Gashia = xo5/2oo5oo5/2ox&#x has a facet total of 2 gashi + 240 sissidpy + 1440 stasc, so too is just scaliform. Might be that there are Stott expansions like xo5/2xx5oo5/2ox&#x, xo5/2xx5xx5/2ox&#x as well (same height each, only varying around radius), provided according facets won't get Grünbaumian. - Gohia = xo5oo5/2oo5ox&#x came out to have zero height only. Thus any Stott expansion here would have the same degenerate height too.

--- rk
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Re: Scaliform cells

Postby username5243 » Sat Mar 21, 2020 8:50 pm

Not what I was referring to, I was wondering about the possibility of taking other truncates from those families (such as righi or ragishi) and constructing alterprisms from them (for instance, righi alterprism = oo5xo5/2ox5oo&#x). Does anything of this sort work?
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