Talk:Monogon

Henagonal henahedron doesn't exist

Number of edges of a polyhedron = Sum of number of edges of faces / 2.

But according this formula, since face of henagonal henahedron is single henagon,
an henagonal henahedron has a henagonal face, a vertex, and half an edge.
Thus, henagonal henahedron don't exist and its image is a misconception. --Nikayama NDos (talk) 03:09, 15 February 2013 (UTC)

The formula you are using assumes that two different faces meet at every edge, explaining the factor of two. In the henagonal henahedron, there is only one face, so the factor is unnecessary. Double sharp (talk) 15:13, 15 February 2013 (UTC)

But this formula also fits in henagonal hosohedron, {2,1}, single digon tessellates the sphere, two edges meets each other. --Nikayama NDos (talk) 00:05, 16 February 2013 (UTC)

Also, we can't expect what shape {1,3}, {1,4}, {3,1}, etc, will be. They're impossible.--Nikayama NDos (talk) 00:07, 16 February 2013 (UTC)

Finally, henagonal henahedron, {1,1}, single henagon tessellates the sphere. but there is no edge for the henagon. This is a discrepancy. --Nikayama NDos (talk) 00:17, 16 February 2013 (UTC)

The henagonal henahedron is certainly special case, and can be seen as a henagon rotated around an axis of symmetry, so the single edge is upgraded to a face. I agree {n,1} and {1,n} do not exist for n>2. Tom Ruen (talk) 00:54, 16 February 2013 (UTC)

Why can it be seen as that? If that explanation is correct, henagonal hosohedron can also be explained like that - similarly one edge is missing. --Nikayama NDos (talk) 01:07, 16 February 2013 (UTC)

To help, I added pictures for all 3 cases. Topologically they are all consistent. All have the spherical Euler characteristic of 2. It appears in general, {1ⁿ}, i.e. {1}, {1,1}, {1,1,1},... represent an n-sphere with one vertex, one facet representing the entire sphere. Tom Ruen (talk) 01:36, 16 February 2013 (UTC)

P.s. Here's George Olshevsky's definition: (Calling it Mono- instead of Hen-) Tom Ruen (talk) 01:41, 16 February 2013 (UTC)

Monogon: The simplest tiling or tessellation of a circle, consisting of a single vertex and a single edge, which is the periphery of the circle starting and ending at the vertex. The Schläfli symbol of a monogon is {1}; it is trivially a regular tessellation. The body of a monogon is the interior of its circle. Although a monogon cannot be used as an element of a Euclidean polytope, it can be used as a cellet in rudimentary n-dimensional hyperspherical tessellations. For example, the monogonal dihedron, whose Schläfli symbol is {1,2}, is the regular tiling of the sphere by two hemispheres that share a common equatorial monogon. Its dual {2,1} is the digonal monohedron, a regular tiling of the sphere that has two vertices (at the poles; hence the adjective digonal) joined by a single edge. Its single face (hence the name monohedron) is the entire surface of the sphere. Having but a single face, the digonal monohedron is the simplest hosohedron. There is also an even simpler monogonal monohedron {1,1}, which is the simplest regular tessellation of a sphere. It consists of a single vertex (so it is monogonal), no edges, and a single face (so it is monohedral), the entire surface of the sphere except for the vertex. It is self-dual.

I see one problem in the text. Hosohedron doesn't talk about {2,1} and {2,2} but these should be called henagonal and digonal hosohedrons. This article wrongly says {2,1} is a digonal henahedron. So there are two types of henahedrons: {2,1} and {1,1}. The first has one edge-self-contacted digonal face, and the second one point-self-contacted henagonal face. Tom Ruen (talk) 21:06, 15 February 2013 (UTC)

I had corrected the text, as "The henagonal dihedron has two faces, one edge, and one vertex (F:2,E:1,V:1). The henagonal hosohedron has one face, one edge, and two vertices (F:1,E:1,V:2). The henagonal henahedron consists of a single vertex, no edges and a single face (F:1,E:0,V:1)." and reverted Nikayama NDos override until discussion complete. Tom Ruen (talk) 00:41, 16 February 2013 (UTC)

But why is the face of {1,1} a henagon? It has no edge. --Nikayama NDos (talk) 14:24, 26 October 2013 (UTC)

It has one vertex. Its edge has been blended away with itself. Double sharp (talk) 04:39, 27 October 2013 (UTC)

Correction: what I wrote above seems to be nonsense. Indeed, {1, 1} doesn't seem to make much sense. Double sharp (talk) 09:59, 8 August 2021 (UTC)

This is somewhat off-topic, but you might be interested in knowing that there's a generalization of polytopes that allows for things like {1, 1} to make sense. I wrote about it on the Polytope Wiki. – OfficialURL (talk) 10:10, 1 December 2021 (UTC)

Sources needed for {1}

This whole article is an embarrassment to Wikipedia. Neither of the sources cited is reliable, while definitions for polygons and polyhedra in reliable sources invariably exclude such simple figures as these. Even if somebody can find something verifiable to say about them, there will not be enough to justify a standalone article and they can be fitted in to the context of the subject matter of the source. Either find some reliable sources (WP:RS) with enough to say to fill a whole article, or get rid of it. — Cheers, Steelpillow (Talk) 17:28, 3 January 2015 (UTC)

I can't find any really helpful references so far, but its clear it exists as limiting cases in spherical geometry as a trivial tessellation. Coxeter talks about regular dihedra {p,2}, and hosohedra {2,p}, so a 1-dihedra, {1,2} contains a monogon {1} bounding ridge, but otherwise no mention. Similarly {1,1} is categorically outside of dihedra or hosohedra, but exists in a 1-ditope {1,1,2}, containing {1,1} as a bounding ridge, but no examples or explicit statements for these cases. Tom Ruen (talk) 00:36, 5 January 2015 (UTC)

Coxeter usually requires p ≥ 2 or even p ≥ 3. He is no justification for making p = 1. That whole game is WP:OR and moreover mathematically broken. — Cheers, Steelpillow (Talk) 16:51, 5 January 2015 (UTC)

John Conway appears to assert here that "monogon" can "be used in rather special circumstances", along with "digon", so perhaps implying it exists. Unfortunately he gives nothing about its properties, but his statement suggests that it might be covered somewhere. Double sharp (talk) 04:22, 6 January 2015 (UTC)

Sounds good, from 1994. And I've not seen any use for henagon compared to monogon. So at least a move would be in order. Tom Ruen (talk) 05:03, 6 January 2015 (UTC)

In Coxeter's Introduction to geometry, 21.3 Regular maps, p. 386:

If χ=1 or 2, the possible values for p and q [given by (p-2)(q-2)<4)] without the restrictions p>2, q>2. Thus the regular maps on a sphere (χ=2) are just the spherical tessellations: {p,2}, {2,p}, {3,3}, {4,3}, {3,4}, {5,3}, {3,5}, namely the dihedron whoses p vertices are evenly spaced along the equator, the hosohedron whoses edges and faces are p meridians and p lunes, and "blow up" variants of the five platonic solids. In the centrally symmetric cases we can identify antipodes to obtain the regular tessellations of the elliptic plane (χ=1): {p,2}/2, {2,p}/2, p even, {4,3}/2, {3,4}/2, {5,3}/2, {3,5}/2.

So without restrictions to me means p=1 or 2 are allowed by the description, like "1 vertex [trivially] evenly spaced along the equator" is a valid construction. So Coxeter doesn't talk directly about tessellations of the circle or saying monogon, but if a single-vertex equator exists on a spherical polygon, then monogon is categorically what we have. Tom Ruen (talk) 04:58, 6 January 2015 (UTC)

GOT IT, in an exercise (since its trivial!), Coxeter's 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) 05:47, 6 January 2015 (UTC)

Given Steelpillow didn't like a short article based on minimal sourcing, I moved the trival definition into digon#monogon as both are "improper" polygons. Tom Ruen (talk) 10:36, 6 January 2015 (UTC)

But monogons are not digons. Why not an article called degenerate regular polygon? (Because monogons must be regular by definition, and if I haven't missed anything the constructions of digons given forces them to be regular.) Double sharp (talk) 05:39, 7 January 2015 (UTC)

That's why I put under Related polygons. I'm not against a rename to degenerate regular polygon or something else. Another option might be sections in Spherical polyhedron, given that's the only application I know for them. Tom Ruen (talk) 05:59, 7 January 2015 (UTC)

Moved to improper regular polygon. Double sharp (talk) 06:56, 13 January 2015 (UTC)

Henagon or monogon?

Isn't hena- more commonly used with another prefix for ones in the units place, e.g. hendecagon, icosikaihenagon, etc.? Double sharp (talk) 02:13, 18 April 2014 (UTC)

I renamed to monogon. Tom Ruen (talk) 06:48, 6 January 2015 (UTC)

More questionable factoids

The boxout for the monogon includes:

• Schläfli symbol ... h{2}
• Coxeter diagram or
• Symmetry group [ ], C_s
• Dual polygon Self-dual

Can any of this be sourced or is it all Original Research? — Cheers, Steelpillow (Talk) 11:54, 6 January 2015 (UTC)

Come to think of it, where is the source that any of these circles with dots on are justifiably described as "tessellations"? — Cheers, Steelpillow (Talk) 12:11, 6 January 2015 (UTC)

Coxeter's papers on Regular and semiregular polytopes I, II, III covers usage Schläfli symbols, Coxeter diagrams, and bracket notation Symmetry groups for polytopes and honeycombs/tessellations. And regular polytopes covers alternations, half, h for regular figures, h{p}={p/2}, (p even). is equivalent to {1}, and is equivalent to h{2}. But if a monogon or digon can't be said as self-dual because Coxeter never explicitly said so, life gets complicated. Tom Ruen (talk) 20:25, 6 January 2015 (UTC)

p.s. Silly me, all regular polygons and 1D honeycombs/tessellations are self-dual! :) Tom Ruen (talk) 20:43, 6 January 2015 (UTC)

How is equivalent to {1}, BTW? I can understand , though. Doesn't this lead to ambiguity: for example, list of regular polytopes and compounds gives {2} and {2,1} the same symbol, ? Double sharp (talk) 12:10, 6 February 2015 (UTC)

Yes, for degenerate cases life gets complicated. One cannot make the same simple-minded inferences that one can with standard cases. If Coxeter did not expressly give a symbol or fact in some degenerate case, then to derive that symbol or fact here is original research. Can you please be more specific - which of these symbols/facts did he specifically give for these degenerate cases and which have you inferred? The need to avoid OR with degenerate cases also applies to tessellations. Also, if a fact is trivially true, e.g. because all polygons are self-dual, then it should in general be omitted from lists of properties for any particular polygon.

P.S. Not all regular tilings are self-dual. Consider {3,6} and {6,3}. — Cheers, Steelpillow (Talk) 21:59, 6 January 2015 (UTC)

Coxeter explicitly said, [h{p}={p/2}, (p even)], Regular polytopes, top of page 155. I agree some care is needed, but an alternation removes half of the vertices, half of two is one. Also I should have said regular polygons and regular 1-tilings of the circle "blown up" versions of regular polygons. And I dont agree trivially true facts should be omitted, since even trivial facts requires minimal thought, as I demonstrate. Tom Ruen (talk) 22:17, 6 January 2015 (UTC)

p.s. Guy, rather than labeling me a fanboy, a more rational discussion might challege readability, and whether Coxeter-Dynkin diagrams, are helpful for readers on trivial objects like digons and monogons or wider polygons. And I'll confess my agenda, that I hold appreciation for Coxeter's work, and his symmetry notation, and include notations on simple objects like polygons in hopes of clarifying symbols that become entirely "abstract" in higher dimensions with a steep learning curve for understanding things you can't "see". But I accept Schläfli symbols may look less mysterious to readers who want the simplest symbols. Tom Ruen (talk) 23:31, 6 January 2015 (UTC)

So that's one of the symbols sourced. What about the other two? And if trivially true facts are to be included, why have you left out the dimensionality, Euler characteristic and orientability? Should we not also note on every biographical boxout on Wikipedia that the person concerned is of the species homo sapiens and inhabits planet Earth? No, I really don't buy your argument for repeatedly listing the self-duality of every darn polygon we come across.

P.S. I used the term "fanboy" generically in a wider discussion on another Project page and in reply to someone else about a more general issue, please do not take it out of context. Rationally, Coxeter-Dynkin diagrams should not be used in degenerate cases where there is no supporting source. Let's stick to Coxeter's work, cite it properly, and not go beyond it with our own OR. He really does not need that. There are surely many cases where original research would help make a Wikipedia article more readable and informative, but that is not what we do. I do seem to be reminding you of this point rather often in these discussions. — Cheers, Steelpillow (Talk) 11:35, 7 January 2015 (UTC)

Your arguments are unfair, and standards no less arbitary than mine. Feel free to add any other properties you consider valuable, and I promise I won't challenge your facts because you can't find an explicit reference that says that specific fact. Tom Ruen (talk) 04:50, 8 January 2015 (UTC)

I seek to base my arguments and standards on WP:POLICY. If they are unfair or arbitrary, please show me the policies and guidelines which expose this. Is your "Feel free to add any other properties..." a genuine gross misunderstanding of my ridicule or an obscure riposte? — Cheers, Steelpillow (Talk) 10:41, 8 January 2015 (UTC)

A related issue shown here is that I can't decide what point group symmetry notation is most important, and so I've compromised by using 3 notations in many places like stat tables, Schoenflies, [Coxeter], and (orbifold notation), like D2, [2], (*2•). By history Schoenflies should win, although actually I'm skipping Internation notation used more in crystallography. So at least in further compromise I have a summary here List_of_spherical_symmetry_groups including all notations I've seen used in books I've referenced. So I learned Orbifold notation 20 years ago taking a class under John Conway, and really liked it's clarity, but its limited to 2D surface symmeties. Since then I learned from Coxeter the bracket notation, which is very similar to Orbifolds in appearance, and from Coxeter's papers and works which are more obscure, but directly connect to the Coxeter diagram, and completely general for reflective groups in any dimensions, and markups for subgroups, so when I add this bracket notation symemtry to Wikipedia, I'm risking "undue" attention to Coxeter's works, which previously has been kept out of traditional geometry and math books simply because people haven't been exposed to Coxeter's work.

So my advocacy by showing 3 symmetry notations, and Coxeter diagrams has a purpose. So people who are used to one notation will read it and be able to see how the same symmetry is represented ine each. However it does get unwieldy, so like for tetrahedral symmetry, I include a subgroup tree which has Bracket notation primary, and File:Tetrahedral_subgroup_tree.png, and other symbols tagged below, which isn't pretty, although it could be done in 3 graphics. Tom Ruen (talk) 04:50, 8 January 2015 (UTC)

All the more reason to provide the citations I asked for when I opened this discussion. If you can cite this stuff, then I can have no argument against it. If you don't provide proper citations, you can't be surprised if people question it. — Cheers, Steelpillow (Talk) 10:41, 8 January 2015 (UTC)

Anyway, one difference between you and me is that I believe in inclusiveness, and I believe Wikipedia can serve to help bridge divergent notations, terminology, and hopefully help readers see their overlapping uses, and contexts, and translate what they need, and decide for themselves what systems work best for their purposes. Still, even if I achieved that goal in any small corner, I accept I error on inclusion and I can't be sure what level of details illuminates and what level obscures by overwhelming or distracting from central ideas. So I need help in my blind-spots to see what helps me, but harms new readers in understanding or interest in reading more. Tom Ruen (talk) 04:50, 8 January 2015 (UTC)

"one difference between you and me..." Oh, yawn! Let's keep imagined personalities out of this, shall we? But I am puzzled - you talk of "So I need help in my blind-spots to see what helps me, but harms new readers" yet when I suggest that at least some light may be shed on your darkness by WP:OR and WP:CITATION, you spurn it. Why the disparity? — Cheers, Steelpillow (Talk) 10:41, 8 January 2015 (UTC)

So you can call OR anything that dares read two independent sources and notes that they are not independent but saying similar things in slightly different languages. And if you do that with vigilence, I'm sure Wikipedia could be systematically reduced to about 0.0001% of its current size and usefulness. Tom Ruen (talk) 04:50, 8 January 2015 (UTC)

You are mistaken in saying that "So you can call OR anything that dares read two independent sources and notes that they are not independent but saying similar things in slightly different languages." Have you actually read WP:OR? What you can do is to point out that, in geometry, degenerate cases are degenerate precisely because they do not obey all the rules and therefore the usual close associations cannot be taken for granted and need to be sourced. If you find that "unfair" or "arbitrary" then we will need to take this issue back to the Mathematics WikiProject. — Cheers, Steelpillow (Talk) 10:41, 8 January 2015 (UTC)

I can search harder to see if or where Coxeter actually uses the word degenerate. What he does use is improper. To me degenerate means overlapping elements (like skilling's figure has coinciding edges), and circle tessellations are used precisely because it avoids degeneracy, although only in the singular case of antipodal points. So in this regard I'm assuming {2}, {1}, {2,1}, {1,2} cases are improper rather than degenerate on a circle or sphere. Tom Ruen (talk) 10:50, 8 January 2015 (UTC)

Regular polytopes p. 61	Any tessellation can be regarded as a "degenerate" polyhedron whose center has receded to infinity.
Regular polytopes p. 123	We regard honeycombs as "degenerate" polytopes.

So both cases are not even talking about my meaning, although a qualifier that we could add to all honeycombs as what I'd call "unbounded polytopes". And its true, you have to consider unbounded figures differently, although in the case of a tessellation on an n-sphere, this doesn't apply. Tom Ruen (talk) 10:58, 8 January 2015 (UTC)

In regards to digons/dihedra:

Regular polytopes p. 4	It is sometimes desireable to extend our definition of a p-gon by allowing the sides to be curved; e.g. we shall have occassion to consider spherical polygons, whose sides are arcs of great circles on a sphere. This extension makes it possible to have p=2: a digon has two vertices, joined by two distinct (curved) sides.
Regular polytopes p. 12	As maps we have also the dihedron {p,2}, and the hosohedron {2,p}. The latter is formed by p digons or "lunes".
Regular polytopes p. 66-67	We have not mentioned the "improper" tessellations, where p or q=2, because much of the following discussion would break down to them.

So maybe we can agree "improper" cases have special consideration (which I never denied), and that "degenerate" is a questionable term for spherical tilings of any sort. Tom Ruen (talk) 11:13, 8 January 2015 (UTC)

Coxeter seems to be allowing 2-gons, but not 1-gons: by moving to spherical polytopes, he explicitly mentions that it means digons are allowed, but makes no mention of monogons. So I really think we shouldn't put the monogon and digon on the same level: the digon is far more standard and well-behaved. Double sharp (talk) 12:06, 6 February 2015 (UTC)

Removal of the monogonal cases

OK, so I looked through the pictures of {1,1}, {1,2} and {2,1} and deleted them from the articles where they were being used, because they illustrated these monogonal cases as being on par with more standard cases like the dihedra, hosohedra, and Platonics, which IMHO is more than they should get (they are not valid abstract polytopes and Coxeter seems to not support the inclusion of {1,2} and {2,1} in his collection of regular dihedra and hosohedra). So, for example, when {1,2} and {2,1} were illustrated as the first dihedron and hosohedron, I cut them (and made the table nice-looking again by making p in {p,2} and {2,p} range from 2 to 6, for example, instead of 1 to 5). I tried to get rid of everything on {1,1}, as its construction seems rather confused (we got it from Olshevsky, if I am not mistaken) and I do not think it is attested in any RS.

The next step would be to make monogon and digon once again separate, and maybe merge the discussion of the monogon somewhere if there's not enough info: but whatever info is stated about the monogon and polytopes using it should IMHO make it very clear that it is absolutely not a traditional well-behaved polytope. Double sharp (talk) 12:35, 6 February 2015 (UTC)

Coxeter on the monogon and 0-gon

From Regular Complex Polytopes, p.9: "When regarded as a spherical polygon, the digon {2} no longer needs to have coincident edges. There is even a monogon {1} consisting of a single point and a great circle through it."

But then just before that he writes: "Here p [in the Schläfli symbol {p}] is not necessarily an integer; for any rational number n/d ≥ 2, there is a spherical {n/d}, of density d.", completely ignoring the monogon! And while he lists the dihedra and hosohedra {n/d, 2} and {2, n/d} in his enumeration of the regular spherical polyhedra on p.10, he seems to be ignoring the case n/d = 1 entirely.

And then they appear in an exercise on p.14:

"If p and q are integers and {p,q} has N₀ vertices, N₁ edges, N₂ faces, then "pN₂ = 2N₁ = qN₀ and N₁⁻¹ = p⁻¹ + q⁻¹ − 1⁄2. What happens if p = q = 1 or 0?"

Well, I guess I can respond to this particular exercise. If either p or q is 0, the polyhedron is going to be markedly uninteresting! You'd need to have no edges, and unless both are 0 there would be necessarily no faces or vertices either by simple arithmetic. And in the case of {0,0}, there obviously cannot be any edges, and the whole thing is just an unmarked sphere with no tessellation on it. The second equation also doesn't work, as 0⁻¹ is undefined.

If p or q is 1, there's no way the second formula is going to make any sense unless the other number is 2. (Otherwise N₁ isn't an integer.) So the only cases not explicitly ruled out are {1,2} and {2,1}, which both give N₁ = 1. {2,1} would give N₀ = 1, N₂ = 2; obviously {1,2} is just the dual case.

But this appears to be just a special case of what he mentions on p.15:

"...every finite reflection group (in Euclidean 3-space, or on a sphere) has a fundamental region (p q r), where p, q, r are positive integers satisfying p⁻¹ + q⁻¹ + r⁻¹ > 1, with the restriction that, if one of these integers is 1, the other two must be equal."

The area of this Schwarz triangle is π(p⁻¹ + q⁻¹ + r⁻¹ − 1), and it is noteworthy to see that he says that this "clearly holds for (p p 1) as well as for a genuine triangle", therefore implying that (p p 1) is not a genuine triangle. And indeed it is not! (1 1 1) is just a hemisphere with three vertices marked on the great circle bounding it (all its angles are π), and (p p 1) is just a digon with a third vertex dumped on it somewhere (the antipodal vertices have equal angles π/p, and a third vertex can be inserted anywhere you like on the digon with angle π). These are at best degenerate triangles.

(p p 1) generates the cyclic group C_p (order p): this is mentioned by Coxeter on p.15 as well. Of course (p 2 2) gives the order-2p dihedral group D_p, and (3 3 2), (4 3 2), (5 3 2) give the polyhedral symmetry groups.

It appears that only {1,2} and {2,1} appear as regular spherical polyhedra from these cyclic groups. I admit that I have no idea what in the world groups like (3 3 1) and (1 1 1) would produce. A naïve extrapolation of the Wythoff symbol–vertex configuration correspondence gives vertex configurations like 1² (= {1,2}?) and 3.1.3.1. But what in the world the latter means, I do not know. (User:Steelpillow? Help? This is confusing, which is weird because these are surely the most simple and degenerate cases.) Double sharp (talk) 16:51, 21 February 2015 (UTC)

{2,1} is the monodigon, the hosohedron with one edge. {1,2} is the dimonogon, the dihedron with one vertex. Maproom (talk) 19:14, 21 February 2015 (UTC)

There are many different ways of looking at these issues, for example one might be considering the regular decomposition of a sphere, the kaleidoscopic constructions on a sphere, the spherical realization of some abstract alegbraic symmetry group - to name but three. Or, what about kaleidoscopes on arbitrary manifolds such as the projective plane or toroids? One may define a polygon in different ways, some more compatible with some of these approaches than others. Is it for example a geometric construction in the Euclidean plane, or some arbitrary surface, or even in higher dimensions, is it a combinatorial structure, an incidence complex, a topological decomposition, or a ranked partially-ordered set? Some of these will allow monogons, others not. What does a Schläfli symbol represent - one of these aforementioned constructions, or perhaps an algorithm for joining polygons (whatever they may be) around vertices? And when you get to stars, are your definitions going to catch up with you? The hoary chestnut is {6/2}: is it a star of David or a double-wound triangle. Did you lose touch of your initial axioms along the way? Oh no, don't tell me you were as lazy as Coxeter and never set them down in the first place! At this level, Coxeter's treatment is simply not rigorous. He would probably have been the first to admit it and he wouldn't have cared because he was not concerned with these borderline cases, only with the beautiful mathematics he was unfolding. When people pointed out mistakes he had made, he would often laugh in delighted approval. How on Earth is a poor Wikipedian to make head or tail out of this mess? Simple: follow WP:POLICY. This basically boils down to, turn to other reliable sources and only document what they directly corroborate. Do not even second-guess the time of day. But if you wish to educate yourself for your own satisfaction, then I would suggest a good few years hunting down the papers of Branko Grünbaum (many are available on his web site). After that, you will be ready to appreciate the dominance that abstract polytope theory is fast achieving. — Cheers, Steelpillow (Talk) 20:44, 21 February 2015 (UTC)

How did star polygons get involved in digons and monogons?! Tom Ruen (talk) 22:42, 21 February 2015 (UTC)

Because Coxeter here specifically allowed for star dihedra and hosohedra, such as {5/2, 2} and {2, 5/2}? Double sharp (talk) 04:08, 22 February 2015 (UTC)