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about the constraints placed on the topic by the author, one can be caught out. Of course, geometers can be notoriously sloppy about getting their boundary conditions clear before they start, so that gives us encyclopedists a double-challenge in presenting verifiable knowledge that happens to be inconsistent - but seldom verifiably so. One must be cautious. One can interpolate where the bounding assumptions are clear, but one cannot extrapolate beyond any clear boundary. This is especially so when close to a supposedly strict boundary where the authorities' unconscious assumptions, for example that a digon always has equal sides, might be invalid.
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bevelling and truncation have been well described, so your illustration here should have no trouble in finding sources. That is to say, one can source a comment on the degeneracy of squares into digons and then present illustrative examples. But if one were to source a comment on digons as valid constructions and then present them as degenerate in
Euclidean space, that would not be acceptable: One needs a reference to say that they are degenerate in Euclidean space. We can interpolate, but we cannot extrapolate.
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As a general principle, I would suggest that we can take a general comment about a degenerate situation and give illustrative examples. What we can not do is to take a general comment about a non-degenerate situation and give a degenerate example. For example the progressive transformations involving
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be said, then we are faced with how to say it. For example, should we say it once and scatter around links to that article? Or, should we repeat it over and over on every darn article and in every darn image that it is relevant to? Or, maybe throw everything it is relevant to into one giant article?
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For example the question, "Why can't you have a digon constructed with two edge lengths, one long and short path?" is actually wrong-headed. You can, for example as a step in the topological analysis of a smooth manifold. But it's a digon constructed to slightly different rules. Without being clear
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at the
Illinois Mathematics and Science Academy defines digons (and monogons) as "two special curved polygons that have no analogs among polygons with straight edges." Clearly a straight-edged digon would be physically impossible -- but do polygons necessarily have to have straight edges? Could two
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So is it original research to take generalization and give explicit examples? If it is, if every fact expressed on
Knowledge has to be defended by explicit references, then we're not allowed to explain anything that sources don't explain well. Then there's no use in knowing anything, and we should
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represent both active mirrors, which can make a "rectangle" in general but by convention is assumed to be the equilateral solution, thus becomes a "circular square". So I drew the square with 4 colors of edges, and you can see the digons represent degenerate "rectangles" where either set of edges
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So some where we have to explain each node represents a mirror, and these two mirrors are orthogonal, usually with an implicit branch order 2 by no connection. The ring represents an active mirror where there's a virtual image of the generating point across the mirror, while a ringless node is
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has a website discussing, "the two-sided polygon called a lune," including illustrations on how to calculate the area of such two-sided polygons. I'm not a mathematician, but after just five minutes of google searches I've found at least three sources that seem to be at-odds with the wikipedia
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And without detailed OR examples derived as simplest examples from a general theory, we can't ask "Why can't you have a digon constructed with two edge lengths, one long and short path?" We have to say "We don't know, because
Coxeter didn't tell us and he's dead."
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can be hard for some folks to swallow, but it cannot be helped. If one's own thinking struggles to verify some aspect of what one is writing, then it is most likely that one has not fully appreciated what the sources are - and are not -
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If a circle is not degenerate then neither is a digon. If a digon was degenerate then why can you project a 3d digon on a 2d screen? Other shapes with two sides can be found in
Euclidean geometry, a crescent for example two sides.
559:. A Coxeter diagram without circles or holes is assumed to be a Coxeter group because its imposible for all mirrors to be inactive (except if the generator point was at the center of the circle, which for tilings is not allowed.)
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I don't think it makes sense to distinguish a regular digon. That is to say all digons on the sphere are regular since the edges must be great circles and two nonparallel great circles must intersect on two opposite points.
385:, there can be two or more different minimum length paths, and so the digon can be defined with two different edge paths. But even on a sphere, two arbitrary points won't be antipodal, and so will also be degenerate.
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to define "straight lines". In
Euclidean space, there's only a single minimum length path (a straight line) between two points, so a digon can be defined only if it is degenerate. However on curved surfaces, like a
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Poverty may arise where the sources fail to give a consistent picture, or where we ourselves muddle up zeal for verifiability with zeal for presentation. But poverty is better than misinformation. — Cheers,
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Excessive brevity and link-clicking get unreadable and tiresome, excessive clutter and page length get unreadable and tiresome, there is a balance to be struck, but it is a different argument from what
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Small correction perhaps. A digon can exist in a degenerate form on a sphere, just like in a degenerate form on the plane. So the only NONDEGENERATE digon on the sphere exists with polar vertices.
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has an illustration of a digon, along with a very different definition: "The digon is the degenerate polygon (corresponding to a line segment) with Schläfli symbol { 2 } ." Also,
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I think it means all straight lines (but that is highly redundant, that should mean circles are degenerate shapes they are not) We should link to
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just type verbatim what is written in books, or summarizing it, leaving examples out, unless such examples are given explicitly by a source.
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No, the point is that a digon with straight lines is degenerate. With curves it is not really a polygon (and hence not really a digon).
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When you have finished reviewing my changes, you may follow the instructions on the template below to fix any issues with the URLs.
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So ALL of this is expressed in
Coxeter's work, but he never bothers to explain these elements in the most basic cases like this.
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Someone left this in the main article. I don't really know what to do with it other than shunt it here, perhaps...
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to delete these "External links modified" talk page sections if they want to de-clutter talk pages, but see the
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is the horizontal mirror (cyan). The nodes can also be colored to match the lines of reflection in the diagram,
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https://web.archive.org/web/20150714082609/http://www.math.iastate.edu/thesisarchive/MSM/EekhoffMSMSS07.pdf
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definition. Could someone more knowledgeable about geometry expand (and correct, if need be) this article?
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If you have discovered URLs which were erroneously considered dead by the bot, you can report them with
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That's a rather poverty strickened system of teaching new understanding to readers.
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Here's an example graphic (right) I made to demonstrate the nature of digons in a
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Finally we can add two final cases on the right most edge of the graphic: h{2},
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represents an active horizontal mirror, so the 2 vertices are on the y-axis.
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This sort of degeneracy exists in all of the
Wythoff constructions, so a
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represents an active vertical mirror, with 2 vertices on the x-axis. And
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Okay, I got creative, expanded an explantion of digons in polyhedra for
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http://www.math.iastate.edu/thesisarchive/MSM/EekhoffMSMSS07.pdf
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for additional information. I made the following changes:
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inactive, so the generating point must be on the mirror.
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points connected by two curved lines qualify as a digon?
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One way of looking at the "teaching" issue is to recall
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vertices from {2} which becomes a monogon, and s{2},
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30:on 10 December 2015. The result of
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