Talk:Spherical polyhedron

Table of images

I dropped in a table of spherical tiling images I had on Wythoff symbol. Tom Ruen 21:21, 31 March 2007 (UTC)

Thanks, Tom. Steelpillow 09:46, 1 April 2007 (UTC)

Rigorous Definition

Nice attractive article, lovely pictures, good intuitive approach. But us theoreticians need a more formal definition also. Also, the article should give non-examples - such as a toroidal polyhedron. I'm not expert enough on this subject (yet). SteveWoolf (talk) 13:06, 14 November 2008 (UTC)

Well, other than describing it as a tiling of a sphere I am not sure what else one can say. The terms "tiling" and "sphere" are rigorous enough (unless one invokes higher-dimensional spheres, which is not the case here). Certainly neither Poinsot nor Coxeter gave a less intuitive definition, and I am not aware of any other (that's not to say that someone reputable might have done somewhere). I do not think that non-examples would do anything other than confuse the casual reader, unless you have some rather subtle point to make. -- Cheers, Steelpillow (Talk) 18:00, 14 November 2008 (UTC)

I agree this is lacking in details. I think it's valuable (somewhere) to recognize the 3 classes of constant curvature spaces - spherical (K=1), Euclidean (K=0), and hyperbolic (K=-1), with geometric properties that a face can be translated without distortion, while scaling can preserve lengths, but not area. Something like that.

A wlink Spherical tiling goes to a tessellation section, which maybe shows (or ought to show) a connection to the wider potential of tiling other surfaces, but underdeveloped there. There's the general idea of tessellations and tilings, and then there's the more specific form topologically related to polyhedra with edge-to-edge connection of faces. The MAIN value I know for spherical polyhedra is the relation to symmetry, and degenerate forms like hosohedrons and dihedrons which exist as tilings on a sphere.

It should reference spherical geometry, also underdeveloped article. Perhaps it ought to be clarified here that edges are "straight", following great circle arcs. Also spherical trigonometry is important, if you want to compute information about polygonal faces on a spherical tiling.

Another relation I know is the Conway polyhedron notation, which creates topological operators to relate polyhedra, and it was implemented by George Hart for spherical tilings specifically.

Tom Ruen (talk) 18:43, 14 November 2008 (UTC)

The topics of curved spaces, isometric transformations and tilings generally, all have their own pages: all this page needs for these is a few links, either in the main text or a See also section. However an important aspect of spherical polyhedra does arise from all this - the spherical kaleidoscopes, from which we derive tilings (i.e. spherical polyhedra) of Möbius triangles and the more general Schwarz triangles, which in turn lead to the convex and star uniform polyhedra respectively. I guess this is probably the symmetry aspect that Tom mentions. -- Cheers, Steelpillow (Talk) 21:51, 14 November 2008 (UTC)

Hi again...my point is that as an abstract polytopist, I'd like to see a formal definition - i.e. what is a "sphere" in an abstract context? But I think this highlights a much wider issue here - namely that different polytope enthusiasts have very different aims and perspectives. Many, many are quite content to explore and enumerate all the pols that have some property X - such as Johnson solids. Others enjoy the graphics aspect of it all. These worthy people do not sully themselves with the finer points of theory. And to be sure, their work is useful and extremely attractive, respectively. (And a lot more attractive than 99% of what allegedly savvy art critics consider modern art). Some people like Euclidean spaces - me, I like abstract pols and a fully rigorous treatment. The problem is, abstract pols and their Euclidean relatives are precisely that - related, but not the same.

Which means: Some articles need to be SPLIT, with a new one for the abstract case. Others simply need an abstract section in the same article. For example, nearly all of the Polygon article has little to do with an abstract polygon. On the other hand, we hardly need 2 separate articles for Icosidodecahedron. This will be a great deal of work, given the never-ending proliferation of polytopic articles. (Actually it's really great - that so many have contributed so much). But if the articles were worth creating, as indeed they are, then our goal should be to make them RIGHT - and abstract pols, in my view, deserve a lot more recognition than they now have. See ya SteveWoolf (talk) 01:14, 15 November 2008 (UTC)

Hi Steve. Your criticisms of "realists" are well founded, and extend even to those who are concerned with the theory - Grünbaum has called this the "original sin" in the theory of polyhedra, and it stretches all the way back from Coxeter to Euclid. This is in fact a major (if not the) problem that abstract poltyopes were developed to address. However in the context of "spherical polyhedra", this term is usually used to distinguish a tiling of a real geometric 2-sphere from its flat-faced cousin; this is a purely (geo)metric distinction and has no abstract significance. So for the main body of this article the introduction of abstract polytopes is not appropriate. There may be an exception, in that abstractly, an orientable polyhedron of Euler characteristic 2 is topologically equivalent to a 2-sphere, and here the term "spherical polytope" might be appropriate. If it has appeared in this context in a reputable source then we can talk about this context in the article, but if it has not then an article titled "spherical polyhedron" is the wrong place to discuss it - the Abstract polytope article would be a better place. -- Cheers, Steelpillow (Talk) 09:41, 15 November 2008 (UTC)

I don't think you are right about this, but I'm not sure, this is an area I need to understand better. What seems clear to me is that the cube or square pyramid, e.g., are spherical. Now take 4 thin trapezoid "cubes", and glue them together on their bevilled edges to form a closed circuit with a hole. This is a toroidal polyhedron.

Note that the graph of this toroid (i.e. vertexes and edges only) is the same as for a 4-cube. To make it a 4-cube you need two more cells (3-faces) - the inner hole and the "outer" cube; then you remove the old maximal 3-face and replace it with a new maximal 4-face four more cells.SteveWoolf (talk) 22:08, 10 January 2009 (UTC)

Anyway, the point is that the "spherical" concept, in my view, is of central importance to abstract theory in classifying polytopes. I'll invite Mike and CG to comment, as they are likely more knowledegable about this. Meanwhile I'll try to improve my understanding. Either way, none of this topic is "new" ground, I am sure. Nice weekend SteveWoolf (talk) 11:46, 15 November 2008 (UTC)

I can't say I've worked with this myself, but here is what McMullen and Schulte's book has to say: "An abstract n-polytope P is called spherical if it is isomorphic to the face-lattice of a convex n-polytope Q." (p 153). Similarly, an abstract polytope is locally spherical if each proper section (equivalently all facets and vertex figures) are spherical. They say very little else about general spherical polytopes, though there is somewhat more about regular spherical polytopes. --CunningGabe (talk) 01:16, 17 November 2008 (UTC)

Thanks for that - at least you have confirmed what I intuitively feel to be spherical. However, I feel that we (a) Need to define this is a purely abstract manner, without recourse to classical ancestors, and then, (b) Prove that the abstract version is equivalent to the "cross-cultural" one. I don't think this will be especially difficult, and the exercise itself may produce new insights. I shall think on this.... SteveWoolf (talk) 07:16, 17 November 2008 (UTC)

Thanks for the example usage in an abstract context. One thing is certain, if by "cross-cultural" is meant the established usage when discussing spherical geometry, kaleidoscopes, etc. then it is quite distinct from the abstract usage - the one is geometric, the other abstract. It is a not even a necessary condition for any geometrically spherical polyhedron to be topologically (abstractly) spherical, as may be seen for example in the original exhaustive study of Uniform polyhedra:

H.S.M. Coxeter, M.S. Longuet-Higgins, J.C.P. Miller, Uniform polyhedra, Phil. Trans. 1954, 246 'A, pp 401-450.

where several topologically non-spherical polyhedra are derived from kaleidoscopic tilings of the sphere. However the idea of convexity is a slightly better match - for a polyhedron to be geometrically convex it must be a topological sphere (though the converse is not true).

Any remarks on spherical topology are probably best included in a general discussion of topological types. For some starting points, see for example topology and genus. -- Cheers, Steelpillow (Talk) 20:48, 17 November 2008 (UTC)

Okay, I now have a quite elegant abstract defn of spherical! Correct? We'll see. But I think it's better to centralise our abstract talk in one place, so you can see this in Talk:Abstract polytope#Have Abstract Defn of Spherical. SteveWoolf (talk) 04:26, 18 November 2008 (UTC)

Projective polyhedra

I have not seen the term "projective polyhedron" used in this context. Either this needs referencing (e.g. did Coxeter or Grünbaum use it?), or replacing with a more general description. -- Cheers, Steelpillow (Talk) 09:12, 14 April 2010 (UTC)

Hi Guy,

You raised this and I addressed this on my talk page, but for reference:

I’ve referenced with McMullen and Schulte, and the terminology is further discussed at projective polyhedron and Talk:Projective_polyhedron#Unimportant neologism? if anyone would like to read or discuss further on this.

—Nils von Barth (nbarth) (talk) 00:34, 23 April 2010 (UTC)

Limiting cases

I moved this deleted section here below, topologically valid tessellations, in case someone else can someday find some references that can support them. My only source is Coxeter:

Introduction to geometry, 21.3 Regular maps, p. 388:

Exercise: Describe the maps {2,1} and {1,2} on the sphere. (The former has one face, a digon {2}; the latter has two faces which are monogons {1}.)

Tom Ruen (talk) 20:46, 6 February 2015 (UTC)

---

The simplest (and trivial) regular tiling of the sphere is {1,1}, a monohedron, consisting of a single vertex, no edge, and a single face as the sphere outside the vertex. It is self-dual, i.e. the vertex and the face center can be swapped, recreating {1,1} like a central inversion.

Limiting cases

Tiling name	Digonal dihedron / Digonal hosohedron	Henagonal dihedron	Henagonal hosohedron	Henagonal henahedron
Tiling image
Duality	Self-dual	Dual tilings		Self-dual
Schläfli symbol	{2,2}	{1,2} = h{2,2}	{2,1}	{1,1} = h{2,1}
Coxeter diagram(s)			=	=
Faces	2	2	1	1
Edges	2	1	1	0
Vertices	2	1	2	1

Should this title be made a disambiguation page?

It strikes me that the name 'spherical polyhedron' is ambiguous, because there are two obvious meanings:

1. A tiling of the 2-sphere by spherical polygons.
2. A polyhedron in the 3-sphere S³ ("spherical space"), i.e. a 3-dimensional shape with spherical polygons as faces, and great circle arcs as edges.

It might be worth moving this article to spherical tiling, adding an article called something like Polyhedra in the 3-sphere or Polyhedra in spherical space, and making the title spherical polyhedron into a disambiguation page. –jacobolus (t) 07:41, 7 January 2026 (UTC)

@Radlrb33: any thoughts about this idea? –jacobolus (t) 17:34, 8 September 2026 (UTC)

I do not, as spherical tilings and spherical polyhedra are simply, synonymously and interchangeably used in the literature. You are misunderstanding, I believe, as S² represents a tiling on the surface of a three-dimensional sphere (a flat 2D surface on a 3D object). The 3-sphere is a 3D surface on a four dimensional-sphere, etc. Radlrb33 (talk) 21:16, 8 September 2026 (UTC)

You "do not" what?

The word "spherical polyhedron" (and related terms like "spherical tetrahedron", "spherical cube", "spherical prism", etc.) is used in two different senses in published literature. (1) to refer to a tiling of the 2-sphere, and (2) to refer to a shape analogous to a Euclidean polyhedron but occurring in spherical space rather than Euclidean space.

"spherical tilings and spherical polyhedra are [...] interchangeably used in the literature." Yes, I agree, but while the term "spherical tiling" is unambiguously used for just this specific thing, the term "spherical polyhedron" is ambiguous, because it is also a natural (and common) name for the analog of a polyhedron in spherical geometry, a different topic which also deserves a Wikipedia article. –jacobolus (t) 01:02, 13 September 2026 (UTC)

..."[have] thoughts about this idea." Radlrb33 (talk) 01:38, 13 September 2026 (UTC)

They do not deserve seperate articles, you'd be copying 95% of the information, if not more. Radlrb33 (talk) 02:00, 13 September 2026 (UTC)

The two topics are almost completely unrelated. It's like the difference between tesselation (of the plane) and polyhedron. Basically none of the material from here would be relevant in an article about polyhedra in the 3-sphere. –jacobolus (t) 04:13, 13 September 2026 (UTC)

Yet you agreed that the terms and notions are interchangeably used... Same symmetry groups at the core is more than 50 percent already, including non transitive types. Examples is another 15 to 25 percent. Can you give specific examples of these topics that would be so drastically unique as to require an entire seperate article, please? Oh, and aside from simple notation differences amongst related topics, and the such (that does not count). Radlrb33 (talk) 05:24, 13 September 2026 (UTC)

You are still not understanding my basic point. I've tried to be very explicit, but I'll try one more time.

The word "spherical polyhedron" has two completely different meanings, each of which is used by some mathematicians in the research literature.

1. The subject of this article, a tiling of the sphere by spherical polygons
2. Solid figures in 3-space of positive curvature such as the 3-sphere which consist of vertices connected by great-circle-arc edges and faces which are portions of great-2-spheres, and containing some volume

These are completely separate topics that just happen to share the same name.

An article about the second topic would include subtopics like the deconstruction of polyhedra on the 3-sphere into smaller polyhedral pieces, formulas for calculating the volumes or surface areas of polyhedra on the 3-sphere, various types of polyhedra on the 3-sphere and their properties, tilings of the 3-sphere by sets of polyhedra (the higher-dimensional analog of this article), and so on.

I think we should move this article to the title Spherical tiling, because that name is unambiguous and also a better description of this topic. Then we could potentially in the future change the Wikipedia title Spherical polyhedron to be a disambiguation page. –jacobolus (t) 06:38, 13 September 2026 (UTC)

I understood you, you should stop trying to repeat onward endlessly, like a broken record? I mean it's looking amusingsilly, I stated my point and you want to force an opinion it seems. I am opposed on a move without actual evidence (sandbox it) of points you will add to this "new spherical polyhedron" article, I am quite skeptical of its merits and uniqueness compared to what is already here, that is 90% unambiguously applicable to both spherical polhyedra and tilings, in so far as appearance, symmetry, and classifications.All of those things of secondary topics you listed can go here with ease, as examples will be applicable equally as tilings on spheres (areas between lunes or curves based on reflections taken, for example, or for surface areas, etc. are all equally applicable, I see no true beneficial reason for two articles, thus far). Radlrb33 (talk) 06:55, 06:59, 07:06, 07:17, 13 September 2026 (UTC)

The "point" you stated was not responsive to the question I asked, and it still seems like you don't understand what type of object I mean by a polyhedron on the 3-sphere. But I'll stop bothering you here, since I really don't know how to say it any more clearly. –jacobolus (t) 07:13, 13 September 2026 (UTC)

yeah tell me more on what I know or don't know please, it's so reassuring to hear your executive voice on my actual insights; bs Radlrb33 (talk) 07:19, 13 September 2026 (UTC)

"inadmissible vertex angles"

Hi @Radlrb33: the article currently says:

in the family of Johnson solids, 3 of 25 solids which are inscribable inside a sphere (specifically, the pentagonal pyramid ⁠$J_{2}$⁠, the square cupola ⁠$J_{4}$⁠ and the pentagonal cupola ⁠$J_{5}$⁠) produce inadmissible vertex angles between polygons, when projected onto ⁠$\mathbb {S} ^{2}$⁠.

Is this really true though? The source we cite here, an arXiv preprint, itself cites Wikipedia for this claim. Are there other sources discussing this topic?

It seems to me that this comes down to a choice of definition of "regular spherical polygon". As far as I can tell the issue that makes these "inadmissible" is that the polyhedra, when their vertices are placed on a sphere, take up less than a hemisphere, so that if we project through the sphere's center we end up with one "base" face projected onto the same regular polygonal portion of the sphere as all of the rest of the faces. But since, from the perspective of the center of the sphere, we are looking at the "outside" rather than "inside" of the base face, we could pretty reasonably choose to project the base onto the sphere in the other direction. We'd end up a face on the sphere which was larger than a hemisphere, with all of its internal angles ⁠$>180^{\circ }$⁠, but I don't see any reason why such a shape couldn't be considered a "regular spherical polygon", unless we restrict that name to only allow shapes smaller than a hemisphere. But that seems like a somewhat arbitrary choice.

Also ping @Steelpillow, @Tomruen. –jacobolus (t) 17:50, 8 September 2026 (UTC)

I fixed it up, and I will gather more sources, there are indeed some. This paper is remarkable though, and I'm not sure if it will get published, it seems to have been integrated into its latter published, referencing co-paper on the hyperbolic sphere that is also used here as a co-reference of sorts. Radlrb33 (talk) 19:28, 8 September 2026 (UTC)

Do you understand what I mean about the bottom face of e.g. the pentagonal pyramid? That is, we can make a tiling of the sphere by 5 regular spherical triangles and a "regular spherical pentagon", provided that the pentagon is larger than a hemisphere. Where is it that this is defined to be "inadmissible", and why? –jacobolus (t) 19:36, 8 September 2026 (UTC)

In their stricter sense, it is inadmissible in so far as reflecting the parameters seen in the flat analogues. Obviously, that is cosmetic, but an important insight nonetheless, in that it is evidence of an evolution in complexity inside positive curvature surfaces. I changed it to make it more universal to what is referenced in the literature; the paper on non edge-to-edge cases we have, mentions 43 total tilings on the sphere aside from the prisms and antiprisms (from the flat, regular), so we can for the time being use that reference if we wish. We will find better. Radlrb33 (talk) 20:46, 8 September 2026 (UTC)

I don't entirely know what "reflecting the parameters seen in the flat analogues" means. Can you elaborate a bit? –jacobolus (t) 21:44, 8 September 2026 (UTC)

Absense of angles greater than two pi, specifically. Radlrb33 (talk) 00:37, 00:39, 9 September 2026 (UTC)

Adams & al. (2024, doi:10.1007/s00454-024-00689-z) says:

Of these [Johnson solids], 25 are circumscribable by a sphere and can thus be projected out onto the surface of the sphere to obtain tilings by spherical regular polygons. That these 25 are circumscribable follows from their descriptions as modified Platonic and Archimedean tilings, allowing for gyration of collections of faces and diminishment

There's no mention that the ones which fit in less than a hemisphere are exceptions. They cite Zalgaller, but I'm not quite sure where (if anywhere) in that work this specific question is addressed. –jacobolus (t) 22:43, 8 September 2026 (UTC)

Good source! We may have to dig, we have at least one, for now. Radlrb33 (talk) 00:40, 9 September 2026 (UTC)

The base pentagon of the regular-faced spherical pentagonal pyramid is not convex. The interior angles > 180 deg and a line (segment of a great circle) cannot be drawn between any two interior points. Johnson's requirement for convexity entails convex faces. So there is a mismatch. You have to abandon either regularity or convexity. This is probably what someone had in mind when they felt that the spherical example was not an "admissible" Johnson image, and I assume is also what is meant by not reflecting the parameters. But nobody ever said it has to comply, to my knowledge (One can of course monster bar as much as one likes, but one still needs to acknowledge that one has done so). The current edit has cleaned the wording up and looks a lot better. — Cheers, Steelpillow (Talk) 19:41, 10 September 2026 (UTC)

I thought Johnson's requirement is about polyhedra in 3-space. Did he also write about spherical tilings? Anyway, I agree that the bottom face is not geodesically convex (perhaps we should include that keyword / wikilink). –jacobolus (t) 19:56, 10 September 2026 (UTC)

More specifically, he wrote about flat-faced polyhedra. Spherical polyhedra also exist in 3-space, but he did not write about them. And that is the point; he did not, and it is a fallacy to assume that the cases are equivalent. — Cheers, Steelpillow (Talk) 08:22, 11 September 2026 (UTC)

"Abandoning regularity" altogether is not necessary, of course, rather we extend the limits set, since there is a different behavior present that does not exist in Eucledian flat-land (i.e. no need to strictly comply). That's all, call it an extension to Johnson's or anyone else's more arcane definitions that were lacking greater perspective, or any other mathematical reason. From an editorial perspective, we acknowledge the views present in the literature with examples. It is true, that it is worth noticing that spherical cases admit five less classes of examples than their convex and concave (e.g., antiprisms) counterparts, if the restriction of regularity requires angles no larger than a hemisphere, and no hosohedra/dihedra that are without analogues in flat 3D. Radlrb33 (talk) 01:46, 01:47, 03:19, 07:10, 12 September 2026 (UTC)

If five spherical triangles share a vertex and are regular, they can only be faces of a regular icosahedron, and the complementary pentagon cannot be larger smaller than a hemisphere. —Antonissimo (talk) 01:15, 12 September 2026 (UTC)

The current statement, that "In total, 47 classes of spherical polyhedra can be projected as tilings from flat-faced forms in Euclidean space, using regular spherical polygons exclusively", is wrong on two counts:

1. These are not classes, there are no alternative morphs. They are just polyhedra.
2. Convexity is not stated as a requirement. The extra Johnson solids are therefore not excluded.

The decision remains as to whether the paragraph wishes to confine itself to convex polygons or not. Either way "classes of" should be deleted, and the choice should be made explicit. — Cheers, Steelpillow (Talk) 19:19, 12 September 2026 (UTC)

I don't understand how you are defining "class", or what a "morph" is, whose jargon are you using? I listed these solids by those with transitivity, and those without, which seems to be reflective of the literature that discusses spherical polyhedra with symmetry point groups and those without (i.e., the Johnson solids); those without well-defined symmetry groups are in their own class, therefore. As the paragraph clarifies in the first sentence, the choice is based on regular faces (inclusive of angles that are larger than the hemisoheric angle). Radlrb33 (talk) 01:26, 13 September 2026 (UTC)

"Classes" was used in the article, you just changed it to "families" which is equally undefined. I don't know what either is intended to mean; presumably you have some idea of what each means and why you think "families" is better? "Morph" is a common enough abbreviation of isomorph, meaning it has the same structural arrangement of faces but may have different geometrical characteristics. None of the Johnson solids is a class or a family, they are all unique polyhedra. Likewise their regular-faced spherical mappings. Counting them as classes or families is quite wrong. — Cheers, Steelpillow (Talk) 03:54, 13 September 2026 (UTC)

To follow-up, the Johnson solids are unique up to similarities, and these spherical tilings are unique up to rotation of the sphere. I agree with Steelpillow that calling each single unique example a "family" is confusing. –jacobolus (t) 03:57, 13 September 2026 (UTC)

I gave you those reasons you seek, and what else would you group them as then, if not the word "set"? I mean, you clearly see where I am trying to go with this as a reflection of the literature present, yet you are being a little rocky over something as simple as using the word "set" instead, or any synonymous use to say the rest of the solids that can be tiled by faces alone which don't showcase any form of transitivity (we can keep also that vocabulary used by Grünbaum and Shephard). How would you catalogue them, then, when literature both speaks explicitly on Johnson solids and the regular/semiregular sets viz. spherical tilings and regular-face restrictions - the central uniting theme, alongside transitivity and symmetry? Radlrb33 (talk) 04:07, 13 September 2026 (UTC)

I do not see where you are trying to go with this. Your remarks here appear to contradict the putative passage; your intent is thus obscured. Did you perhaps intend the passage to say that the five classes listed contain around 45 individual polyhedra and two infinite series? Then it should say so. Symmetries and transitivities have nothing to do with that (nor, I believe did G&S use the term "showcasing"). Worse, as pointed out repeatedly, the "central uniting theme" makes no stipulation as to the convexity of spherical faces; that is an unspoken assumption grafted on by somebody who has failed to make clear that they are monster-barring (the practice of adding arbitrary conditions to a definition, on order to confine the outcome to some preconceived but incompatible notion). — Cheers, Steelpillow (Talk) 10:39, 13 September 2026 (UTC)

You have yet to provide me a classification that fits better with what the literature says. Radlrb33 (talk) 10:49, 13 September 2026 (UTC)

Don't be childish. You make confused and contradictory statements. I am trying to help you out here. — Cheers, Steelpillow (Talk) 10:57, 13 September 2026 (UTC)

Having checked out the ArXiv paper cited, "Areas of spherical polyhedral surfaces with regular faces", it is of poor quality. It starts out, "For a finite planar graph, it associates with some metric spaces, called (regular) spherical polyhedral surfaces", then in the same para goes on, "We prove that for any graph ... which does not admit a spherical tiling" which appears to contradict the opener. Later it claims in "Remark 1.6" that, "...where F is not contained in a hemisphere. So F is not a regular spherical polygon," however nowhere does it define a regular spherical polyhedron in sufficient detail - it is just a monster-barring condition slipped in by the remark. So I cannot regard this unreviewed preprint as WP:RS in the present context, and I propose that we delete whatever tangles derive from it that we may find. — Cheers, Steelpillow (Talk) 17:28, 14 September 2026 (UTC)

monohedral degenerate case {1, 1}

@Radlrb33 can you rewrite the section § Limiting case using less jargon and a clearer explanation? And maybe add some additional sources? I find it almost completely incomprehensible, and I don't see how the content is supported by the only reference given, p. 384 of Coxeter's Introduction to Geometry, from which the relevant quotation as far as I can tell is "In fact, the universal covering surface of the projective plane is the sphere, and its fundamental group is of order 2, generated by the central inversion." –jacobolus (t) 21:11, 12 September 2026 (UTC)

It's very well-written, and clear, though run-offy as it stands. I find it excessive that you think it is "almost completely incomprehensible". I have sourcing for the {2,1} and {1,2} limiting casses mentioned above, and some potential good sourcing for {1,1} though under different terminology than the one introduced in this article. I'll update it tomorrow when I have the time. Radlrb33 (talk) 02:20, 13 September 2026 (UTC)

It's grammatically awkward, consists almost entirely of undefined jargon, some used in a non-standard way and some of which does not appear anywhere else in the article, and I find it very difficult to make sense of. The diagram of a yellow sphere with one dot on it is not helping.

Maybe you can explain the whole thing in like 5 times as many words, and then someone else can rewrite it in a more comprehensible way? –jacobolus (t) 03:46, 13 September 2026 (UTC)

Even after I just edited over it twice? It's significantly better already. Can you give me an example or two, please, rather than how it makes you feel only...? Jeez. Radlrb33 (talk) 03:56, 13 September 2026 (UTC)

I have spent years studying spherical geometry, and I find it extremely difficult to make sense of; I'm still not quite sure exactly what it's trying to say. I don't know what "very well-written, and clear" or "significantly better" means here. But I expect most readers will be completely mystified, and it would require long diligent effort to puzzle out the intended meaning. The entire thing is basically jargon soup.

Can you try to rephrase it here on the talk page, using limited jargon and as many words as required to convey the same message? Imagine you are writing for a reader who is a high school student. Then someone else can perhaps take a crack at translating that back into a version that can go in the article. –jacobolus (t) 04:09, 13 September 2026 (UTC)

You still do no provide a single example, of something that does not make sense and how it does not make sense (via contradiction, or a misnomer, or misrepresentation); or better any example that can be written differently than as is at the moment, while using the same vocabulary (or core notions at least), or even synonymous vocabulary. You want me to break it down for you sentence by sentence? Radlrb33 (talk) 04:30, 13 September 2026 (UTC)

Yes, can you break it down sentence by sentence? Or maybe take each sentence and rewrite as a paragraph or two. Each idea needs to be given enough space for readers to digest it. Each jargon word that isn't understood by anticipated typical readers needs to be glossed. Jargon needs to be used in a standard way, and consistently. Grammatical constructions need to be simplified.

To take the first sentence as a concrete example:

There exists a monohedral degenerate case {1, 1} where the surface area covering a sphere by one hosohedral lune of form {2, 1} (or equivalently, two monogons in {1, 2} dihedral arrangement) is represented through a vertex antipode that is symmetrical about its polar opposite face center

• The "there exists" construction is inherently awkward and cumbersome, we should if at all possible rewrite to be more straightforward
• Schläfli symbols are not defined/explained (this should be done near the top of the article)
• Terms "monogon", "monohedron", "monohedral", "henagonal" are not defined anywhere in the article. The picture has a "henagonal hosohedron", but the wikilink on it redirects to "monogon", so it's not clear what the difference is between "henagonal", "monogon", "monohedral".
• "surface area covering a sphere by one hosohedral lune of form {2, 1}" is not obvious to anyone who hasn't puzzled over it. This needs to be unpacked as a separate sentence or two
• It is not explained what a "hosohedral lune of form {2, 1}" is. "Hosohedral lune" seems to just mean "lune whose angle is some unit fraction of a full turn", and a "hosohedral lune of form {2, 1}" seems to just mean the entire surface of the sphere with one semicircular seam; that should be explained
• Why is a "hosohedral lune of form {2, 1}" equivalent to "two monogons in {1, 2} dihedral arrangement"?
• "surface area ... is represented through a vertex antipode" – I have no idea what it means for a surface area to be "represented through a vertex antipode"
• "vertex antipode" is non-standard and undefined terminology, which seems like a misuse of "antipodes"/"antipodal" or at best confusing. I think we should avoid the use of "antipode" as a noun, in favor of "antipodal point" or "point antipodes to ...". But it seems like this is supposed to just mean "vertex", and I'm really sure what you're getting at with it.
• I'd recommend avoiding the term "polar opposite", since it may be confused for the polar dual (which for a point is an oriented great circle). Instead, I'd recommend the terms "antipodal" or "diametrically opposite"
• What does it mean for one point to be "symmetrical about" another point?
• What does "its face center" mean when "it" is a point?
• Both hosohedra and dihedra are introduced in a section after this one. The article's structure should be changed so that concepts are introduced before they are discussed

Overall, I still don't understand what the main idea of this sentence is supposed to be / what the sentence is trying to say. I have similar problems with every other sentence in this section. –jacobolus (t) 06:18, 13 September 2026 (UTC)

I haven't read your entire thing, but I kno' already you're analizing an old version of this page. Radlrb33 (talk) 06:41, 13 September 2026 (UTC)

Can you still try to answer? You made some changes while I was in the middle of writing this, but they don't actually fix any of the problems I have, and the newer version is no more comprehensible than the previous one. If this section cannot be made clearer, it should be removed. The current version is a total disaster. –jacobolus (t) 06:44, 13 September 2026 (UTC)

Tomorrow yea right now I don't want to, you can do some work and help clean it if you want to, rather than dictate and be demeaning from a reflection of dissapointment and judgement. I don't think I've seen you say one nice thing on the improvements added thus far, ahah. : ) Radlrb33 (talk) 06:48, 06:51, 13 September 2026 (UTC)

I'm not trying to be mean. I just cannot make complete sense of your writing, and I find it hard to make any sense of it at all without extraordinary amounts of effort. If you can unpack your ideas to the extent that I can understand them, then I can try to help translate that for a wider audience. I think this section needs to be more or less totally rewritten. Just making incremental changes is not going to help. –jacobolus (t) 07:07, 13 September 2026 (UTC)

I know you understand them, you are being deceiving. : ) Anyways, I will add a couple of things and change that on limiting cases, and I will leave here since there is no point in communicating with you. Radlrb33 (talk) 07:21, 13 September 2026 (UTC)

Why would I lie and pretend to not understand something if I did understand it? What possible motivation would I have to do that?

If you can't clearly explain yourself in the article, and you can't explain any further in the talk page, then who are you expecting to read this section, and what are you expecting them to get from it? Wikipedia must focus on helping to convey information to readers. If no information is being conveyed by a particular part of a Wikipedia article, then it has failed and should be removed or rewritten.

But how about this: can you provide a reliable source which supports the claims of this section? Then I can go read that source which I might have a better chance with. If not, the section should be removed anyway, on grounds of WP:V, WP:RS, and WP:NOR. –jacobolus (t) 07:28, 13 September 2026 (UTC)

It's been at times constructive editing here alongside you recently, but at the moment I do not want to interact further. Thank you for understanding and respecting editorial boundaries; I find you being aggressive, assuming, and unacknowledging, as well as evasive. Tic toc Radlrb33 (talk) 08:10, 13 September 2026 (UTC)

That's fine. We should just remove this section then, and you can workshop it and come up with some sources, and maybe it can be put back in at a later point. –jacobolus (t) 08:15, 13 September 2026 (UTC)

I have now reverted the whole section, endorsing jacobolus' earlier reversion. @Radlrb33:, may I recommend that you create a user sub-page for it, and work on it there until it is of adequate quality to include in this article? — Cheers, Steelpillow (Talk) 10:43, 13 September 2026 (UTC)

I reverted your revert; please discuss my new sourcing and points added first. Radlrb33 (talk) 10:52, 13 September 2026 (UTC)

@Radlrb33:. WARNING! You are now edit warring and must stop. Not wanting to discuss with other editors, as you just stated, is not acceptable in our collaborative environment. I agree with jacobolus who removed your section once and now says it should be removed again. I agree, removed it myself a second time, and am about to remove it again. If you directly restore it for a third time, you will fall foul of the Three-Reverts rule and I will report you to the Administrators Notice Board for account sanction. Again, my strongest advice is to work it up in your user space and gain some community approval, before attempting to repost it here. — Cheers, Steelpillow (Talk) 11:09, 13 September 2026 (UTC)

@Radlrb33: You are a seasoned editor of some 20 years, with a long history of editorial abuse; alternative account, topic ban, breaches of good faith, etc. etc. You know full well that the outright personal abuse which I have just redacted is totally unacceptable. I have a mind now to take you to a full WP:ANI. — Cheers, Steelpillow (Talk) 12:59, 13 September 2026 (UTC)

Radlrb33's new version now says fewer things that don't make sense to me, so that's an improvement. But I'm not sure that Coxeter's exercise asking readers to prove that "There is no map of type {1, 1}." or the proposed solution of the exercise, "The positive integers p and q are not quite arbitrary. If one of them is 1, the other can only be 2. [...]" is adequate support for the text "On the other hand, the monohedron {1, 1}, with only a single vertex and face, is such an extreme degenerate case in spherical tilings of the form {p, q} that it does not form a map." –jacobolus (t) 11:54, 13 September 2026 (UTC)

The real difficulty is that sources differ as to the status of such degenerate constructions. How do we even define {1, 1} as a relevant figure, when it is in general forbidden by graph theory due to its triviality (a dot on the page), and by most polyhedronists due to its similar lack of the minimal structure required? I am well inclined to leave a dedicated section out, unless and until someone can write a sensible, relevant and verifiable account of how it brings any value to the subject. — Cheers, Steelpillow (Talk) 17:40, 14 September 2026 (UTC)

I think {1, 1} should be left out unless there's a clear source. Having no edge and a face "boundary" that is only one point makes it extremely different from other examples.

I agree that we probably don't need a dedicated section. The features of the {1, 2} and {2, 1} cases could be more explicitly described in a couple sentences in the section about the dihedron/hosohedron. I don't think we need to say anything connecting these to the projective plane, which seems to me like a possibly confusing tangent that would be better placed in some other article. –jacobolus (t) 18:31, 14 September 2026 (UTC)

To be honest, I think the inclusion of {2,1} and {1,2} is questionable on the same basis. To qualify as a polyhedron, every edge must have just two endpoints, and be the meet of just two faces. On this basis, the simplest spherical polyhedron is {2,2}, the digonal hosohedron or digonal dihedron (it is self-dual). Graph connectivity expresses these same conditions, though in different language. Without robust citation from genuinely reliable sources, mention of the degenerate aberrations should be confined to the reasons that such RS explicitly reject them. I'd suggest they be deleted from the tables in the general discussion. The projective plane is of course not a sphere and its tilings are not relevant to this article. I would only note that if an {n,m} is degenerate on the sphere, then it is equally so on the projective plane. — Cheers, Steelpillow (Talk) 23:01, 15 September 2026 (UTC)

We can find sources at least mentioning these. e.g. Coxeter in Regular Complex Polytopes includes as an exercise "Describe the simplest dihedron {1, 2} and the simplest hosohedron {2, 1}." Another example source, doi:10.1090/S0002-9939-96-03492-2: "A geodesic monogon on the round metric is a great circle, which has an interior angle of π".

If we consider a dihedron to be a division of the sphere into two hemispheres with any number of points along the great circle between them, each taken to be a vertex of a degenerate polygon with straight angles at each vertex, then allowing the case where we just have 1 point doesn't seem inherently different from allowing the case with 2 or more points. Similarly, if we consider a tiling by lunes to allow any unit fraction of a whole turn as the internal angle at the vertices, then allowing the case where the denominator is 1 also seems plausible enough. –jacobolus (t) 07:37, 18 September 2026 (UTC)

The key point here is that these structures are not polyhedra in any formal definition of the term: they do not conform to any standard definitions, such as in topology, graph theory and abstract theory. This article is about Spherical Polyhedra, and call these degenerates what you will they are not valid polyhedra in any usual sense. In Regular Complex Polytopes, Coxeter was teasing the student, inviting them to consider the figures' degeneracy. Note that {2,1} comprises two monogons, which get no mention in his Regular Polytopes (i.e. real regular polytopes), and you can't build a polyhedron out of things that are not polygons. However it is possible that some mathematical disciplines may allow them, as definitions can vary enormously. As I said earlier, if any significant source discusses them more seriously than just posing them as a riddle, then perhaps they can find mention in the relevant section. But a section of their own in this article is not justifiable. — Cheers, Steelpillow (Talk) 10:08, 22 September 2026 (UTC)

@Steelpillow I actually think the title Spherical polyhedron is a problem for this topic. In my opinion we should rename/move it to Spherical tiling, which is more descriptive, a bit broader, and seems to me like a more appropriate scope for an encyclopedia article. Then (a) we don't need to worry quite so much about what someone's precise definition is of "polyhedron", which is inherently confusing and prone to inconsistency because it has been generalized from term meaning a solid figure in Euclidean 3-space to a collection of flat figures in spherical 2-space, a context where many basic geometric features are different, and various authors can quite reasonably disagree on how to reconcile the differences. And also (b) we don't need to worry about the potential confusion that the term "spherical polyhedron" is also applied to the substantially unrelated topic of solid figures in spherical 3-space.

After a rename we could also add a section discussing tilings by other kinds of shapes. –jacobolus (t) 17:21, 22 September 2026 (UTC)