A few questions I guess.

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

A few questions I guess.

Postby ubersketch » Thu Oct 18, 2018 11:28 pm

Long time no see.
I haven't done much with polytopy but I have done a few large number stuff. I'm slowly returning to this subject.
Assume that vertex-transitive takes on this definition.
Is tutcup weakly uniform?
Are all the scaliforms weakly uniform?
Is there a definition of uniform polyhedra based on incidences?
And that's it.
I'm just looking for some discussion and to answer some really burning questions in my head.
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Re: A few questions I guess.

Postby Klitzing » Fri Oct 19, 2018 7:12 am

Dear Übersketch,

you would be fully free to pile up any new set of defining settings and investigate the set of polytopes, which would follow those rules. Moreover it is strongly recommended to do so, as such like revisitions possibly bring up new insights or interrelations.

But you neither should aim to redefine settled notions nor cite those wrongly, as this only encreases all that Babylonean confusion.


As to vertex transitivity: that one is defined by the action of some symmetry group which thereby allows any vertex of a polytope to get mapped onto any other of that polytope, but with the additional requirement that the polytope as a whole will then be undistinguishable from the former orientation. This sometimes also is represented with the term flags, i.e. the incidence structure of all boundary elements incident to a specific vertex (not just the faces only, and esp. including the figure itself). So e.g. the Archimedean small rhombicuboctahedron and Miller's solid (one of the Johnson solids) both have just a single, in fact the same vertex figure [3,4,4,4] throughout. But only the former is vertex transitive, while for the latter the symmetry group action of the whole figure is being devided into 2 orbits of vertices!

For further definitions of the other terms, you are dealing with, cf. https://bendwavy.org/klitzing/explain/polytope-tree.htm#uniform, https://bendwavy.org/klitzing/explain/polytope-tree.htm#scaliform, https://bendwavy.org/klitzing/explain/polytope-tree.htm#orbiform. In those days of the dawn of this millenium the investigation of the segmentotopes lead to a find, which got the name tut || inv tut = tutcup, which then was the first observed figure deviating only little from full uniformity. It then was called to be "weakly uniform" (just to have a working title). That working title later became an own name and since is known as "scaliform". There even is an own page dealing with this and much more of those: https://bendwavy.org/klitzing/explain/scaliform.htm.

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Re: A few questions I guess.

Postby ubersketch » Mon Oct 22, 2018 10:11 pm

Yeah, but in most contexts, it shouldn’t matter whether you are saying pseudo-vertex transitive or vertex transitive.
Also you haven’t quite answered my questions, but thanksgiving for reminding me how I got the name in the first place.
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Re: A few questions I guess.

Postby Klitzing » Mon Oct 29, 2018 7:52 pm

Dear Übersketch,
I'd think you're after a longer and more detailed reply.
ubersketch wrote:Yeah, but in most contexts, it shouldn’t matter whether you are saying pseudo-vertex transitive or vertex transitive.

In the contrary, there is a huge difference between those just local and the full global concept! - Or would you agree in calling pi a rational number? Simply because of
    3 = 3/1,
    3.1 = 31/10,
    3.14 = 314/100 = 157/50
    etc.
all are rational numbers?

Or consider the quasiperiodic tilings. Those implement a higher degree module (symmetry of spectrum of difraction) than the number of basis vectors of the tiled space. E.g. the well-known Penrose tiling implements just 2 types of rhombs. But that very building rule of quasiperiodicity does not allow any random combination of those rhombs, rather there are some few allowed vertex surroundings only. But note, the number of those local patches depends from the radius of those patches! For any given radius (edge size assumed to be unity throughout) there will be a finite set of local "allowed" configurations. But in order to get the resulting tiling to be identified uniquely (not just whole equivalence class of such Penrose tilings) the size of the patches would have to be infinite. - This is what I tried to mention wrt. Miller's solid resp. the Archimedean rhombicuboctahedron.

Different authors address that difference between "pseudo (= seemingly) vertex transitivity", which considers just local patches only, and (true) vertex transitivity, which considers the global object, as "equigonal" vs. "isogonal". Just to mention.

ubersketch wrote:Also you haven’t quite answered my questions, but thanksgiving for reminding me how I got the name in the first place.


Well, that highly depends on the definition of "weakly uniform". I for one used that very term in the dawn of the 20th century for the relaxation of the 3rd rule of uniformity, i.e. that one, which asks for dimensional hierarchicallity (= all its elements have to be uniform in turn). In fact, I'd found a figure, that very tutcup, which does follow other rules of uniformity, except of that 3rd one. In the sequel by a joined research of different authors brought up a great number of convex and even non-convex polychora of that type (e.g. cf. Jonathan Bowers' (aka PolyhedronDude's) according listing). So that working title just took about 4 years to settle down into an own specific term: the term "scaliform" got born. - In that sense, for sure, tutcup will be "weakly uniform" right by concept.

If you'd like to relax uniformity in a different sense, and call that again "weakly uniform", then you will have to investigate for new, whether it does follow your rule set or not.

But be warned. There are infinitely many choices for rule sets. So you cannot aim to address those all by an own label (term). It rather is whether the result set of Elements (polytopes) could be achieved to enumerate all contained elements (polytopes) explicitely, or at least whether you can include/exclude specific well-known groups of elements (polytopes), or the set of rules in any other way opens an "interesting" direction of research (e.g. like the "CRF"s) - only then that rule set deserves a specific own name. ("Interesting", btw. is not being attributed by the inventor, rather by his audience.)

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Re: A few questions I guess.

Postby Marek14 » Tue Nov 13, 2018 9:01 am

Klitzing wrote:I for one used that very term in the dawn of the 20th century...


How old ARE you? :o
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Re: A few questions I guess.

Postby wendy » Tue Nov 13, 2018 9:09 am

He is younger than me, so i imagine he means the "dawn of this century". Some of us folk who were born near the middle of last century tend to think the 'last century' refers to the 1800s, and people born then would served in the wars might be seen in veteran marches from that era. There are kids alive today, born since 2000. Hard to imagine, but it is surely true.
The dream you dream alone is only a dream
the dream we dream together is reality.

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Re: A few questions I guess.

Postby Klitzing » Tue Nov 13, 2018 10:36 am

Marek14 wrote:
Klitzing wrote:I for one used that very term in the dawn of the 20th century...

How old ARE you? :o

You trapped me. ;)
Ment to write "already somewhere before Y2K", so I'd better had to write "in the dawn of 21st century".
Haha. :D

Btw., I'd suppose that you both, Marek14 and Wendy, still remember that evolution of scaliformity as being outlined in the polyhedron mail archive…

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Re: A few questions I guess.

Postby Marek14 » Fri Nov 16, 2018 10:41 pm

Well, not really -- I have been on the list for a long time now and as with most of my interests, geometry comes and goes in waves, so I don't remember that particular discussion.
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