NOTICE: Citizendium is still being set up on its newer server, treat as a beta for now; please see here for more.
Citizendium - a community developing a quality comprehensive compendium of knowledge, online and free. Click here to join and contribute—free
Click here to participate in the November 2016 election.
CZ thanks our previous donors. Donate here. Treasurer's Financial Report -- Thanks to our content contributors. --

Talk:Plane (geometry)/Archive 1

From Citizendium, the Citizens' Compendium
Jump to: navigation, search
This article has a Citable Version.
Main Article
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]


Sorry, again. But this is even more a bad idea! This picture may -- perhaps! -- be used to illustrate the topological concept of a surface, certainly not that of a plane. --Peter Schmitt 00:42, 18 March 2010 (UTC)


"A plane is a surface on which a line perpendicular to a line which lies on that surface also falls entirely on the surface" — where is it taken from?? Hopelessly bad "definition".

"A plane is made up of an infinite number of straight lines" — it surely contains infinitely many straight lines, as well as infinitely many triangles, circles etc. But is it "made up" of them??

"Surfaces can be parallel" — really? what is the definition of this notion?

"Thus the surface has on it point A, point B, and point C is called surface ABC" — a plane is determined by three points (if not on a straight line), but a surface is not.

"If this crumpled picture of the Earth was spread flat on a perfectly flat table, and the picture had absolutely no thickness, then it would be a plane" — no, it would be a finite domain on a plane.

Boris Tsirelson 15:29, 28 March 2010 (UTC)


I was bold enough to rewrite it completely. Hope you do not object. Could someone please add an appropriate lead, and probably introduction with a completely informal idea of plane? It would be also nice to have pictures to the three geometric definitions. Boris Tsirelson 09:57, 29 March 2010 (UTC)

Analytic geometry

I added some high-school-level analytic geometry plus two drawings. --Paul Wormer 10:46, 1 April 2010 (UTC)

Getting better

Excellent article! This ain't a plain-old article by any flat stretch.--Thomas Wright Sulcer 16:07, 1 April 2010 (UTC)

Thank you for the compliment to Paul and me. However, a good article must have a lead, and probably introduction, right? Maybe you can try? Boris Tsirelson 17:51, 1 April 2010 (UTC)
Maybe in a bit. I'm off doing errands now. Thanx for your vote of confidence in me but I'm not scientific by any flat stretch!--Thomas Wright Sulcer 17:55, 1 April 2010 (UTC)
I looked it over. I think you're making excellent progress on it! I'm impressed. It soon gets into technical areas that are above my pay grade that I'm not going to understand like the equations (I was an anthropology major in college! -- arrgh). A suggestion I might offer at this point is to have the first paragraph focus more on the conventional (ie simple, that is -- three points in space define a plane etc) sense of plane -- a flat surface -- because I think this is what will be sufficient for most people. So maybe add a few more sentences perhaps to the first paragraph which explains the basic sense, perhaps, if you feel it's warranted. Then, I think it would be good to make a case for why one should consider exploring the more difficult mathematical questions about a plane -- that is, why a reader will benefit fro getting more involved in this subject. And then keep your great stuff you've got thereafter. Overall, highly impressed!!! My son is into math and he may want a look at it.--Thomas Wright Sulcer 20:30, 1 April 2010 (UTC)
By the way, I've linked this article from the corresponding Wikipedia article. It does not increase our Google rank (because of "nofollow" tag...) but could attract some readers. Boris Tsirelson 05:39, 2 April 2010 (UTC)
About lead/intro I feel I am too much mathematical for doing it. I am hardly understanding what do non-mathematicians think about planes (and other mathematical objects). Maybe an antropologist or chemist would do it better? Boris Tsirelson 05:43, 2 April 2010 (UTC)
About lead/intro: isn't it sufficient what is there now?--Paul Wormer 07:17, 2 April 2010 (UTC)
Oops... I am sorry, I did not note that it has appeared! Yes, probably it is OK. Boris Tsirelson 08:40, 2 April 2010 (UTC)
About the "flat surface in which a straight line joining any of its two points lies entirely on that surface" I wonder: does "flat" already mean "in which a straight line joining any of its two points lies entirely on that surface", or does it mean something different? Boris Tsirelson 08:50, 2 April 2010 (UTC)


The first sentence is from the Oxford Dictionary and hence "flat" is as the non-mathematician perceives it. (Undefined, intuitively clear, and perhaps somewhat redundant). I added to the lede a paragraph about Hilbert's work on classical Euclidean geometry. I found it illuminating that Hilbert assumes the plane to be an undefined object. Further the mentioning of Hilbert's book is (IMHO) a piece of information of encyclopedic nature. Feel free to change/delete/add anything. --Paul Wormer 09:37, 2 April 2010 (UTC)

"(Undefined, intuitively clear, and perhaps somewhat redundant)" — Ah, yes, this is why I am reluctant to write introductions; I know that I am too much mathematical for this business.
"Hilbert assumes the plane to be an undefined object" — Oh yes; this is what I mean by "In the axiomatic approach points, lines and planes are undefined primitives" in "Axiomatic approach".
"mentioning of Hilbert's book is (IMHO) a piece of information of encyclopedic nature" — surely it is. Boris Tsirelson 12:24, 2 April 2010 (UTC)
Overall highly impressed with this excellent article. Not that I understand it. It's just that stuff with equations in it looks cool. Just one comment about the first sentence -- maybe it's because I'm not quite mathematical enough to grasp it -- but I have trouble wrapping my mind around the idea of a plane with only one line in it. I keep thinking multiple lines, or 2-D; but it's the handyman in me to think of a flat board, not warped, since warped boards cause problems, particularly when trying to build stuff.--Thomas Wright Sulcer 14:26, 16 April 2010 (UTC)
Thomas the first sentence says "line between any two points". For a mathematician the adjective "any" implies an infinite number of lines, because there are no restrictions whatsoever on the choice of points. Grab (any) two points in the plane and you have a line (or a line segment) in the plane.--Paul Wormer 15:41, 16 April 2010 (UTC)

Related lemmas

I was about to write lemma articles for the following but wasn't sure if I'd get it correct. Is a nonagon meaning nothing there? To write a lemma, click on the space to the right where it says in fuzzy lettering "Add brief description or definition" (that is, don't click on the term itself in red, and don't click on the tiny "r"; rather, when you hover your cursor above the Add brief description... a blue line should appear underneath) , then write the definition under the subpages stuff, save it. Then, when the term is red, such as Hexagon, click on it, erase the stuff inside the <!---- this stuff erase it---> so only the "subpages" thingie at the top remains, and then save it as a "lemma". Sorry this explanation isn't very good.--Thomas Wright Sulcer 15:15, 16 April 2010 (UTC)

...said Thomas Wright Sulcer (talk) 15:10, 16 April 2010 (UTC) (Please sign your talk page posts by simply adding four tildes, ~~~~.) forgot, sorry.--Thomas Wright Sulcer 15:15, 16 April 2010 (UTC)

Also, one more thing; when writing lemmas or any article, try to include as many relevant (ie pertaining to the subject of the definition, wikilinks as possible, since they provide us with much needed Google Juice which helps boost our PageRank on Google searches and helps Citizendium appear earlier on a user's SERP page.--Thomas Wright Sulcer 15:15, 16 April 2010 (UTC)
I would modify that to try to include as many relevant wikilinks as possible. --Daniel Mietchen 15:52, 16 April 2010 (UTC)
Agreed. Exactly. Precisely. The key is relevance. --Thomas Wright Sulcer 16:02, 16 April 2010 (UTC)
Tom, I do not agree with your tactics regarding "Google Juice".
First of all, I do not think that you are right that heavy interwiki links will help at all. In the published PageRank method, only links from other sites (best: highly ranked ones) help to increase the overall PageRank of CZ pages. Interwiki links may only shift the weight from one page to another. Heavy cross-linking (a "thicket", as you call it) will likely give the corresponding pages the same value. (Of course, not all methods Google uses are known).
Secondly, even if you were right: Having a lot of stubs, definitions, lemmas will not be a reason to visit CZ again, even if a user accidently is lead to one by Google. The only sustainable method to advertise CZ are excellent articles (even if there are only a few). We should not waste time and energy to create content that is inferior to most other content that can be found! Write a definition if it comes naturally when working on an article, but do not write definitions just to have them.
Links on pages and Related Articles pages should be relevant and useful for the reader. For instance, it does not make sense to link mother in a definition of a mythological figure (like you did). They should guide the reader where to get relevant information, what to read next!
I am opposed to "lemmas" for titles that (probably) never will be extended to pages. In such cases, definitions serve the same purpose. Most polygons do not deserve a page, thus also not a lemma.
Many pages on Google rank give the advise NOT to try to boost it artificially. I agree for one more reason: It is not professional and does not fit expert-guidance. Moreover, it certainly should be a community decision whether to apply any such methods (and which ones).
--Peter Schmitt 17:37, 16 April 2010 (UTC)

(outdent)Peter, thanks for expressing your opinion. Let me say I am not certain about anything regarding PageRank, but I have been fussing with this issue as well as having done a mini-experiment here on CZ articles which you are free to peruse. I agree strength of incoming wikilinks from other websites is a big plus, and is to be encouraged. But let's say we agree about the following points:--Thomas Wright Sulcer 19:12, 16 April 2010 (UTC)

  • Wikipedia gets terrific web exposure. It's got a huge database of highly interlinked articles. It has 3 million+ articles. WP is often on the first or second SERP. It has several layers of links -- category links; "see also" links; "infobox" links, etc etc. WP doesn't have forums.--Thomas Wright Sulcer 19:12, 16 April 2010 (UTC)
  • Citizendium gets lousy web exposure. CZ's articles number perhaps less than ten thousand (am I right?). It is rarely on the first ten SERP pages. CZ lacks infoboxes, categories (but has the "related articles" stuff). CZ does have forums (not sure whether this helps us with interlinking or not).--Thomas Wright Sulcer 19:12, 16 April 2010 (UTC)
  • Overall, if memory serves correctly, CZ's main page has a PageRank of 5; Wikipedia's main page has an 8; Amazon's (the bookstore) has a 9. (Amazon also is highly inter-linked, allowing book reviewers to post link within reviews to other products).--Thomas Wright Sulcer 19:12, 16 April 2010 (UTC)

Do we agree about this? If so, let me say what I've learned recently. What I've noticed, after writing articles on SEO and PageRank, and doing the mini-experiment CZ:PageRank analysis of Citizendium articles, is that the CZ articles which perform best are not only well-written and good and packed with great information (which I'm striving for too, btw) but have the MOST numbers of inbound internal links. I'm talking Biology and Anthropology. They've been around a while (newer CZ articles don't do as well, generally). It's fully possible that sites out there on the web link to these articles, and boost traffic. But my sense is definitely that the internal inter-linking phenomenon is at work, just like Google suggests.--Thomas Wright Sulcer 19:12, 16 April 2010 (UTC)

About the ethical dimension. We're an online encyclopedia. We're not selling stuff. It's perfectly fair and good and acceptable to permit internal wikilinking on any kind of article. I don't think we're doing "black hat SEO" and I don't think Google will come to this conclusion -- rather, we're imitating WP, and Google clearly doesn't have a problem with WP and its methods. And I don't agree that ine interlinks wash each other out like you suggest. But frankly I'm not sure. I'm doing an experiment with the article Aeneid to see if creating an article thicket boosts its web presence; I'll let people know after a month or so what the outcome is. If this experiment fails, then it's back to square one. --Thomas Wright Sulcer 19:12, 16 April 2010 (UTC)

What I love about lemma articles is that they're the fastest way to create an article, since it bypasses the drudgery of creating the related articles subpages and such. Boom, it's done. And it's easy for subsequent contributors to expand it into a fuller article. I think it's perfectly fine to have an article about each type of polygon.--Thomas Wright Sulcer 19:12, 16 April 2010 (UTC)

Further, I think any effort here to bring more readers will have the effect of causing more contributors to want to join us. So, with this in mind, I think it's perfectly acceptable to build thickets. In this sense, my idea of what to put in a definition is inclusive -- if I can add material that's even marginally relevant, I'll try to do so, with the view that subsequent contributors, perhaps years later, can fix it or modify it, and I try to make things as relevant as possible. I agree perhaps the mother one was a stretch, but my thinking is that perhaps it might feed future articles about family or love or parenting or whatever comes along. I'm doing my best to try to make this encyclopedia a viable competitor to Wikipedia.--Thomas Wright Sulcer 19:12, 16 April 2010 (UTC)

But overall I won't know until perhaps a month, after the crawlers come by, and I can see whether the Aeneid experiment works.--Thomas Wright Sulcer 19:12, 16 April 2010 (UTC)

Tom, Google already knows about Aeneas, Iliad, and Odyssey (at least). They have been cached about April 1-7.
Yes but I still have trouble finding them via Google.--Thomas Wright Sulcer 23:21, 16 April 2010 (UTC)
Your bulleted statements are correct, of course. But from them it does not follow that interlinking is the decisive factor. WP has (almost) a monopoly, it is cited and linked from many (and many important) sites (and mirrored in various ways). Moreover, (at least) part of the ranking is determined by a "manually" determined factor. Certainly the rank of individual pages is determined to a large part by the sites rank.
The links between different domain names of the Wikimedia project may have a positive effect. (In our case: links from the forum may count.)
I'll try to keep open-minded about this. My sense is, still, that the forums are a distraction unless they have specific links to places within CZ.--Thomas Wright Sulcer 23:21, 16 April 2010 (UTC)
The examples you cite (Anthropology, Biology) may be linked from outside. But that CZ has many links to them (and less from them) fits my claim that these links determine the internal position ("important page on CZ"). If you link A to B, B to C, and C to A their value will be (about) the mean value of these three pages. (This is how it is explained.)
You may be right about this. I bet Anthropology and Biology have external links. But what's clear to me is that there's a clear correlation between the number of internal links and PageRank -- I bet if someone here was mathematically inclined, that they could do a scatterplot of CZ:PageRank analysis of Citizendium articles and have a more definitive measure. Overall, however, I admit that I don't know. I'm exploring. What you say makes sense mathematically in a way since it kind of assumes Google Juice is like some kind of liquid that can flow in and flow out. If this is the case, that GJ can dribble out, then I don't understand why Wikipedia would permit external links. There are tons of external links on WP (Amazon has fewer). I'm still trying to learn about this; I want to see what happens in a month or so with a few core articles like Aeneid highly interlinked, and supported by a thicket. Then I'll have a better handle on what you're saying.--Thomas Wright Sulcer 23:21, 16 April 2010 (UTC)
Definitions should be short -- they are meant for related pages. You can create a stub without creating a lemma. If you do not want to bother with the Metadata or subpages you need not -- just do not add the subpages template. But I am still convinced that a good and long article is much better for CZ than many stubs which make the user look for another source (WP) because they want to read more.
--Peter Schmitt 22:49, 16 April 2010 (UTC)
OK maybe I'll write more short articles without subpages if that's okay. But I'm writing rather long definitions since I assume that someday they'll become real articles. What I'm asking for is a grace period of perhaps a month or so, in which we can have a review of the experiment. In the meantime, please permit me to write some rather longish wikilink-heavy multi-sentence definitions; it's easy to convert them to real articles, with more detail, when the time comes.--Thomas Wright Sulcer 23:21, 16 April 2010 (UTC)
Also, here's how I see the essence of the difference between our general approaches.--Thomas Wright Sulcer 23:21, 16 April 2010 (UTC)
  • Quality articles --> more readers --> more contributors (your view?)--Thomas Wright Sulcer 23:21, 16 April 2010 (UTC)
  • Heavily wikilinked thickets around a few good core articles --> greater web exposure --> more readers --> more contributors --> greater quality articles (my current view which I may change)--Thomas Wright Sulcer 23:21, 16 April 2010 (UTC)
And what I'm saying is that I think CZ has been trying the first approach for a while now (several years). Are people happy that it's working? What I'm saying is let's try something different and see if anything changes, that's all. I think there's strong consensus among current CZers that there are not a lot of readers here; I'm trying to change that.--Thomas Wright Sulcer 23:21, 16 April 2010 (UTC)
I am not forbidding anything (I can't, by the way, and I do not want to). I only argue my opinion (hoping, of course, that some of my arguments are convincing). And I suggest that this should be discussed on a broader scope, a discussion that might lead to some policy guidelines (or might not) ... but this will not happen until there is an EC ...
I would be unhappy, however, if you'd extend your strategy to mathemtics. There are already too many poor stubs which should be improved.
Independent of your strategy there is a question nobody has answered me to my satisfaction. (I do not expect you to answer it):
What is the advantage of a lemma (that transcludes a definition to a page) over a definition copied to that page? (Or better: A short definition on the subpage and a slightly extended stub on the page?)
--Peter Schmitt 23:47, 16 April 2010 (UTC)

Continuation of discussion

(break for easier editing) OK, I won't lemma-ize mathematics. I only have a few thickets I'm working on and none of them are mathematical. I suggest you take a peek at the bottoms of your mathematics articles where it says "number of page views". I have no idea whether the thicket approach will do anything to page views of Aeneid; I'll try to keep everybody posted. About your lemma vs stub question -- I don't know. It's faster making a lemma. But I'll do either.--Thomas Wright Sulcer 02:55, 17 April 2010 (UTC)

Tom, I had never noticed the count on the bottom until you mentioned it. I put a question on the Forums at,3128.0.html, because I'm confused.
Homeopathy, arguably the most criticized article, had a count of 885. Certainly, that count could not include logged-in users; I'm sure I've visited that page more often than 885 times.
Yet, to my pleased surprise if it's accurate, cloud computing had a count of 972.
Cloud computing is an excellent and highly informative article; it's impressive how it keeps improving in terms of information and writing style as you keep working on it. Great job!!!--Thomas Wright Sulcer 16:00, 18 April 2010 (UTC)
Neither seems plausible. Howard C. Berkowitz 03:16, 18 April 2010 (UTC)
Given the amount of negative publicity the Homeopathy article has generated with CZ critics, I'm surprised at how low the edit count is too. Either they are just taking pot shots without even visiting the page or the page counter has a bug. Meg Ireland 03:47, 18 April 2010 (UTC)
Good question. Btw, Meg, your excellent suggestion about moving the "Draft Article Warning" tag at the top of most articles to the bottom of the article -- just a reminder that I agree that's an excellent idea.--Thomas Wright Sulcer 16:00, 18 April 2010 (UTC)
About the suggestion from HP about me participating in the forums. I'll consider it but I'm not enthusiastic, if it means I'll only find the kind of bickering and finger wagging and criticism there that I've been subjected to here. Plus I get lost in the forums and don't know which topics to follow, or what's new. What I enjoy is writing, creating, not arguing, not bickering. If people need to bug me or, as usual, criticize me, feel free to adorn my talk page with your wisdom.--Thomas Wright Sulcer 16:00, 18 April 2010 (UTC)
Finger pointing? There are one or two with whom finger pointing would be a deescalation. At times, I wonder if the code duello would be simpler. Seriously, if one wants to break the Charter deadlock, that's where pressure is needed.
I'd rather write than argue, with the caveat that I'm seriously interested in information structures and contextualizing, something I expect to see in a revived EC. Howard C. Berkowitz 16:10, 18 April 2010 (UTC)

On Google Juice

15% of the google juice received by a page is consumed by this page; 85% flows to other pages via (outbound) links. If the page contains no links, these 85% scatter over the whole net.

Thus, CZ loses some juice when a page (1) contains no links at all, or (2) a substantial portion of its links are external.

Note that the second (third, ...) internal link saves some juice only if external links exist.

In contrast, WP does not lose juice via external links, since these are declared "nofollow". Boris Tsirelson 16:53, 17 April 2010 (UTC)

And do not forget: "Google is known to penalize link farms and other schemes designed to artificially inflate PageRank" [1] Boris Tsirelson 16:57, 17 April 2010 (UTC)

My thoughts on all this...

This entire discussion has been moved to the Forums at:,3129.0.html

Tom, Chris, *everyone*, kindly conduct this sort of discussion in the Forums, NOT on a Talk page where it is totally irrelevant! Hayford Peirce 03:20, 18 April 2010 (UTC)