Johnson Conjugates

[ubersketch]

Does anyone have a list of conjugates of johnson solids?

[student91]

What is the conjugate of a polytope?

[ubersketch]

What is the conjugate of a polytope?

Kind of hard to explain, but a conjugate of a polytope is one that has the exact same incidences as another. Apparently they only come in pairs or conjugacy involves more than just identical incidences. For example, a gad and sissid are conjugates because they both have 5 sides per face and each vertex has 5 of these faces and each face has the same number of faces next to it. Same with doe and gissid. Something you should note is that conjugates are in the same army.

[Klitzing]

What is the conjugate of a polytope?

Kind of hard to explain, but a conjugate of a polytope is one that has the exact same incidences as another. Apparently they only come in pairs or conjugacy involves more than just identical incidences. For example, a gad and sissid are conjugates because they both have 5 sides per face and each vertex has 5 of these faces and each face has the same number of faces next to it. Same with doe and gissid. Something you should note is that conjugates are in the same army.

Conjugate polytopes is a term derived from algebraic conjugacy, i.e. the other solution of mostly a quadratic minimal polynom of some involved irrationality. Wrt. polytopes this most generally comes out to replace within the Dynkin diagrams the link marks 5/1 <-> 5/3, 5/2 <-> 5/4, 4/1 <-> 4/3 (independently). This is because incidence matrices would just consider the numerators and not the denominators of these link marks for their calculations.

No, the remark of Übersketch about belonging to the same regiment generally is wrong. That happens to be correct only for uniform polytopes. E.g. the conjugate of bilbiro = J91 is gibil biro. And those definitely are not within the same regiment!

I don't have a full listing of those as ready to access. But some are mentioned within the individual incmats files on my incmats website (cf. above) below the introductary stuff of the specific polytope, stating "As abstract polytope ... is isomorphic to ..., thereby replacing ...".

Or you might want to have a look at Jim McNeill's polyhedra page http://www.orchidpalms.com/polyhedra/, where those "great" versions often also are mentioned.

--- rk

[ubersketch]

What is the conjugate of a polytope?

Kind of hard to explain, but a conjugate of a polytope is one that has the exact same incidences as another. Apparently they only come in pairs or conjugacy involves more than just identical incidences. For example, a gad and sissid are conjugates because they both have 5 sides per face and each vertex has 5 of these faces and each face has the same number of faces next to it. Same with doe and gissid. Something you should note is that conjugates are in the same army.

Conjugate polytopes is a term derived from algebraic conjugacy, i.e. the other solution of mostly a quadratic minimal polynom of some involved irrationality. Wrt. polytopes this most generally comes out to replace within the Dynkin diagrams the link marks 5/1 <-> 5/3, 5/2 <-> 5/4, 4/1 <-> 4/3 (independently). This is because incidence matrices would just consider the numerators and not the denominators of these link marks for their calculations.

No, the remark of Übersketch about belonging to the same regiment generally is wrong. That happens to be correct only for uniform polytopes. E.g. the conjugate of bilbiro = J91 is gibil biro. And those definitely are not within the same regiment!

I don't have a full listing of those as ready to access. But some are mentioned within the individual incmats files on my incmats website (cf. above) below the introductary stuff of the specific polytope, stating "As abstract polytope ... is isomorphic to ..., thereby replacing ...".

Or you might want to have a look at Jim McNeill's polyhedra page http://www.orchidpalms.com/polyhedra/, where those "great" versions often also are mentioned.

--- rk

I mean army, a group of polytopes that share vertices. But yeah, usually a conjugate of a polytope is in a seperate regiment in the same army.

[ubersketch]

I found a list!
http://www.orchidpalms.com/polyhedra/jo ... hnson.html
However, I'm not exactly sure why diminished is used instead of replenished.
Too bad there isn't a VRML of the great pentagonal rotunda.

[Klitzing]

I mean army, a group of polytopes that share vertices. But yeah, usually a conjugate of a polytope is in a seperate regiment in the same army.

Still wrong, Übersketch, not even within the same army!
This is because longuish ellipsoidal circumsurfaces become forshortened ellipsoidal ones. Just reconsider the provided counterexample:

No, the remark of Übersketch about belonging to the same regiment generally is wrong. That happens to be correct only for uniform polytopes. E.g. the conjugate of bilbiro = J91 is gibil biro. And those definitely are not within the same regiment!

--- rk

[Klitzing]

I found a list!
http://www.orchidpalms.com/polyhedra/jo ... hnson.html
However, I'm not exactly sure why diminished is used instead of replenished.
Too bad there isn't a VRML of the great pentagonal rotunda.

Well, there has been an old project of mine, enlisting the higher symmetrical ones of the edge facetings of the uniform polyhedra, the published paper on that project is

Axial-Symmetrical Edge-Facetings of Uniform Polyhedra, Dr. R. Klitzing
Symmetry: Culture and Science Vol. 13, No.3-4, 241-258, 2002

and can be found here. Its members, even when not yet on my incmats website, are being on electronic display as a 3 views pic as well as a VRML here.

There too you can find the conjugate of pero = J6, i.e. GJ6 = gid-6-10-1 as well. It is the leftmost one on this page.

And again, from this figure you'd see as well, that the vertex set of pero and of its conjugate would be heavily different!


"Diminished" is a term which can be applied to convex figures, refering to chopping off some bit. "Replenished" in opposition is the term of Jim McNeill (or Jonathan Bowers?) to describe the same effect (in terms of alike incidence numbers) as diminishing was for convex figures, when being applied to the conjugate figures thereof. That is, those no longer can be described by some chopping off process. Rather these new facets, being inserted underneath the chopped off bit in case of the convex starting figures, mostly would be partially outside in those conjugates.

--- rk

[ubersketch]

Some two things I would like to talk about
I find that the field of star regular faceds is unexplored. Most of the polytopes of interests are usually CRFs.
Another thing is that we should come up with Johnson naming equivalents for to Johnson Conjugates rather than slap "great" somewhere and call it a day. More descriptive terms would be nice.

[quickfur]

[...]
I find that the field of star regular faceds is unexplored.
[...]

So, explore them, and tell us your findings!

[ubersketch]

[...]
I find that the field of star regular faceds is unexplored.
[...]

So, explore them, and tell us your findings!

Hm, good idea, I'll have to find a way to discover some, most likely through conjugacy.