Talk:Noble polyhedron

Noble polytopes/honeycombs

Interesting category. Jonathan Bowers uses it for higher polytopes as well, but no hard references. Meaning: facet-transitive and vertex-transitive. Examples - bitruncation of self-dual regular polychora/honeycombs: bitruncated 5-cell and bitruncated 24-cell, bitruncated cubic honeycomb. Tom Ruen 03:23, 7 March 2007 (UTC)

I wouldn't be surprised if nothing has been published on this. A great deal of recent polychoron work has gone on but has never made it past Jonathan's and George Olshevsky's websites. Steelpillow 20:20, 7 March 2007 (UTC)

p.s. perfect polytope seems to be a more general category of vertex/facet-transitives, including the cases above and their duals. Like [1] Tom Ruen (talk) 20:45, 3 January 2013 (UTC)

In that reference, "perfect polytope" is used in quite a different sense, more akin to the term "pristine" for a spacefiller. For example Table 2 lists several polyhedra having faces in different symmetry orbits - they are perfect but not noble. Equally many noble figures, such as the stephanoids, are not perfect. — Cheers, Steelpillow (Talk) 20:55, 3 January 2013 (UTC)

Noble tilings?

Noble Euclidean tilings might be useful to add as topological infinite polyhedra: apparently regular tilings: square tiling, triangular tiling, and hexagonal tiling, PLUS a geometrically stretched triangular tiling with isosceles triangles, and stretched square tiling as rectangles. Wait, there's also at least two distorted hexagonal tilings with rectangles - like a brick wall, and Herringbone pattern.

Surely Grünbaum in Tilings and Patterns must talk about this, maybe just doesn't use the term noble? yes, like a subset of 107 isohedral types in 9.1.1. I think they could be enumerated fully from the table with all α elements in vertex transitivity column!

(10) P₃: 2-7,11-14

(16) P₄: 17-22, 43, 46-47, 50-56

(5) P₆: 7-9, 12-13

I think that's good, although topologically, they could be counted differently, so mid-edge vertices counted as two edges. Then they should all be topologically identical to the regular tilings. Anyway, this seems an interesting subset to consider! SockPuppetForTomruen (talk) 22:54, 31 December 2012 (UTC)

By topology:

16 Square tilings: P₃-2, P₃-3, P₃-4, P₃-5, P₃-6, P₃-7, P₄-43, P₄-46, P₄-47, P₄-50, P₄-51, P₄-52, P₄-53, P₄-54, P₄-55, P₄-56

4 Triangular tilings: P₃-11, P₃-12, P₃-13, P₃-14

11 Hexagonal tilings: P₄-17, P₄-18, P₄-19, P₄-20, P₄-21, P₄-22, P₆-7, P₆-8, P₆-9, P₆-12, P₆-13

SockPuppetForTomruen (talk) 23:18, 31 December 2012 (UTC)

It looks to me as if any uniform stretching of one of the regular tilings will be noble, as will any tiling of congruent quadrilaterals. I don't know if there are any more. But unless you can find a reference to someone isolating them from the full list on the basis of their being noble, then sadly they have no place in this article. — Cheers, Steelpillow (Talk) 12:20, 1 January 2013 (UTC)

I'm not sure what uniform stretching would means, but it isn't fully free, and "any tiling of congruent quadrilaterals" definitely won't be, as the Tilings and Patterns full P₄ listing shows. But if a full article is added on that full list of 107, with the table listing vertex transitivity, then I'd imagine at least a cross link would be worthy from here, to express the relation to the tilings. The main objection I can see is the interpretation of edge-to-edge vs midedge tilings. If a polygon isn't allowed to have colinear adjacent edges (not strictly convex), and non-edge-to-edge connections are excluded (how could these be included anyway?), then most won't qualify as noble. SockPuppetForTomruen (talk) 00:20, 2 January 2013 (UTC)

Yes, I should have qualified that any quadrilateral can be tiled nobly. By "uniform stretching" I mean a simple linear stretching or squashing in a constant direction - everywhere by the same amount, i.e. uniformly. I don't think you should tabulate vertex transitivity unless that has already been done, that would be OR: I don't have the book to hand, so I can't comment on their list.

Nobility assumes no mid-edge connections: "wallpaper" rectangles must be taken as hexagons having 180 deg internal angles, or nobility is not applicable. Whether you allow tilings with such hexagons, even coptic polygons, is arbitrary in that it is down to the author's self-imposed rules. For example even a cross-quadrilateral can be tiled nobly, though different cross-quads require different rules/assumptions about the interior (Note that this sentence is OR on my part). — Cheers, Steelpillow (Talk) 10:08, 2 January 2013 (UTC)

Thanks for your thoughts. Sorry I can't share the book contents easily, not much online to link. I admit I'm most interested in non-intersecting ones now. I did find a cheaper soft cover edition (Tilings and Patterns) is coming out in April 2013! [2] It's got about 100,000 man-hours of content I think! The tables explicitly list vertex transitivity, at least an encoded format, so vertex uniform are listed with all α terms. When I get a chance I'll add these visually as topological variations to the triangle/square/hexagon tiling articles. What's not given explicitly is topological equivalences for each, although one of the table columns may imply these classes with some care. It looks like all 107 are topologically identified as one of the 3 regular tilings, or 7 semiregular dual (Catalan) tilings. I should read the text a bit more carefully too, might actually say that! I know the 107 isn't a fixed number, like herringbone pattern (as a topological hexagonal tiling) is included once, even if you might distinguish between rectangle versus parallelogram faced tilings. Tom Ruen (talk) 22:56, 2 January 2013 (UTC)

Here's the cases, scanned and colored (to help compare, although should be single color for isohedral), with the indices from Tilings and Patterns. So it would take a bit of work to describe how and why are they are all different, or what degrees of freedom exist in each example. But if we wanted to identify noble tilings, this would apparently be the list! Tom Ruen (talk) 02:23, 3 January 2013 (UTC)

Triangle	Quadrilateral
Hexagonal

Now that's an interesting one. We know that Nobility is defined as both face- and vertex-transitivity. We have a list of face-transitive tilings in which some are identified as (also) vertex-transitive. Certainly, in an article titled "face- and vertex-transitive tilings" we could extract those from the list. But in an article or section titled "Noble tilings", I am not so sure. Does bringing these separate ideas together constitute OR? — Cheers, Steelpillow (Talk) 14:33, 3 January 2013 (UTC)

I'll sit back and admire them for a bit, but eventually face-transitive and vertex-transitive tiling or something like that sounds good. Noble is a nice word, but it doesn't bother me to exclude it, and a new article can be linked here under See also. Tom Ruen (talk) 20:30, 3 January 2013 (UTC)

Noble scaliform polychora and polytera

From Klitzing's website: http://bendwavy.org/klitzing/explain/noble.htm. Added constraint is that vertices must be congruent and edges equal in length. The noble scaliform polyhedra are the regular polyhedra. (One known one is missing, the polychoron affic with 48 cotcoes. There may be more unknown ones not listed. Certainly there are some known noble scaliform polypeta not listed here.) Double sharp (talk) 14:38, 8 April 2014 (UTC)

This site is self-published by a German-speaker and is neither peer reviewed nor edited in translation. We cannot use it as a reliable source. — Cheers, Steelpillow (Talk) 16:31, 8 April 2014 (UTC)

Yes, this was for my own reference instead of for the article. (Besides the article is called "noble polyhedron", and therefore noble polychora and above wouldn't fit here, even if they had RSes. Which incidentally makes me think that Net (polyhedron) is certainly not the right title for that article, since it covers nets of polychora as well, but I digress!)

Irritatingly the nonconvex uniform polychora and above are not really covered anywhere in reliable published sources, except for the regular ones: it's honestly about time this changed. (Bowers states that 3 more uniform star polychora were known to Gosset. Those three are semiregular by his definition, having only Platonic and Kepler-Poinsot solids as cells and being vertex-uniform: did Gosset ever publish anything on semiregular star polytopes?) The only excuse I can come up with for the polychora is that the star ones arise naturally from considering facetings of vertex figure polyhedra (which almost always seem to be able to be derived by Coxeter-Dynkin diagrams from the Platonic or Schläfli-Hess polychora for some reason), but that is a rather far-fetched excuse IMO. Double sharp (talk) 15:22, 9 April 2014 (UTC)

A good few years ago some enthusiasts discovered that there were a large number of uniform polychora, and some of them started what they called the Uniform Polychora Project to discover and list them all. Prof. Norman Johnson (of Johnson solid fame) was writing a book on uniform polytopes at the time but, perhaps unwisely, delayed it at the last minute while the spate of new figures kept pouring out. As far as I know this book has, sadly, never been published (the draft contained a lot of other important stuff besides). This has not stopped this Mathworld page from listing it among its references - it was certainly not published in 2000! This other Mathworld page claims there are over 8,000 known uniform polychora, while this site claims it has since been whittled back down to 1,849 by discounting certain forms deemed unacceptable. Last I heard (a few years ago still), the complete discovery and proof of same were looking like they were beyond reach. Perhaps unsurprisingly, nobody seems keen on a peer-reviewed incomplete story, so RS is pretty much impossible. Not much help, really. — Cheers, Steelpillow (Talk) 16:36, 9 April 2014 (UTC)

For a polyhedron to be a uniform polychoron vertex figure, it needs to have a circumsphere that all its vertices touch, and all its faces need to be among the vertex figures of the 75 uniform polyhedra. (The first ensures scaliformity: the second uniformity.) If we take a polyhedron that satisfies these rules, and we facet it to form another polyhedron that satisfies these rules, we may get another uniform polychoron. That's not a lot to go on for a proof that the set is complete. Double sharp (talk) 11:26, 10 April 2014 (UTC)

p.s. The ONLY "published" place for ALL of the 1849 uniform (non-scaliform) polychora is Stella_(software) 4D, all under short Bower names, and pretty confusing outside of the convex ones that I can understand via Coxeter-Dynkin diagrams. I don't know what will be in Norman's book from the longer 8000 or shorter 2000 list. Tom Ruen (talk) 19:49, 9 April 2014 (UTC)

They're not really that confusing after a while: they were until I realized the correspondence they had to facetings of the verfs of the convex polyhedral vertex figures. George Olshevsky had a good case study (pentagonal prism as vertex figure for rectified 600-cell) on his Multidimensional Glossary... But yes, after that I now kinda see that now pretty much the only way (barring overlooked omissions in the existing list) to find new uniform polychora is to locate new regiments (which last happened in 2006), i.e. a set of polytopes sharing the same vertices and edges.

The longer 8000 list contains a lot of objects that would not be traditionally called polytopes, with degenerate cells (e.g. the small complex icosidodecahedron), completely coinciding faces, etc. I'm inclined to allow the addition of fissaries back into the 2000 list. Compound vertex figure, but is not a compound: an admittedly not too good example is the small hexagonal hexecontahedron if you consider the coinciding vertices one vertex. Double sharp (talk) 11:18, 10 April 2014 (UTC)

Re. Gosset, according to Coxeter's "Regular polytopes", Gosset only published a brief abstract of his otherwise unpublished essay. Most or all of his findings were later rediscovered by Stott and others. Coxeter's book references a reasonable historical bibliography, including Gossett's abstract and Stott's work. — Cheers, Steelpillow (Talk) 20:46, 9 April 2014 (UTC)

Face configuration

The nobel polyhedra, being face-transitive, would seem to be represented by a single face configuration (as a platonic solid, or catalan solid), although there's geometric ambiguity when vertex figures are not regular. Tom Ruen (talk) 17:58, 18 May 2015 (UTC)

p.s. My guess is new examples may come from Regular skew polyhedron#Finite regular skew polyhedra of 4-space, like the face configuration the crown polyhedra is regular V4.4.4.4, with 4 crossed quadrilaterals around every vertex. Tom Ruen (talk) 01:03, 19 May 2015 (UTC)