Talk:Monogon

message: 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)
replies: text: 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)
replies: text: 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)

text: 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)

text: 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)
replies: text:  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)
replies: text:  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)
replies: text:  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)

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

text: 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)

text: 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)

text: It has one vertex. Its edge has been blended away with itself. Double sharp (talk) 04:39, 27 October 2013 (UTC)
replies: text: 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)
replies: text:  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)
thread_title: Henagonal henahedron doesn't exist

message: 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)
replies: text: I renamed to monogon. Tom Ruen (talk) 06:48, 6 January 2015 (UTC)
thread_title:  Henagon or monogon? 

message: 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)
thread_title:  Removal of the monogonal cases 

message: 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 − . 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 Cp (order p): this is mentioned by Coxeter on p.15 as well. Of course (p 2 2) gives the order-2p dihedral group Dp, 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)
replies: text: {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)
replies: text: 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)

text: How did star polygons get involved in digons and monogons?! Tom Ruen (talk) 22:42, 21 February 2015 (UTC)
replies: text:  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)
thread_title:  Coxeter on the monogon and 0-gon