Talk:Geometry

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Contents

[edit] Separation from History of geometry article

(Note: I couldn't find the message I composed to explain the move, so here it is to best of my ability from memory:)

"What the #%&%%^?! What happened to the geometry article?!"

I bet that's what you are thinking. Well, there wasn't one. What was here was the article "History of geometry" posted under the wrong name. This problem had been pointed out years ago by Larry Sanger, and has been a recurring theme on the talk page since, so I took the liberty to correct the problem. It's pretty amazing to find a stub on such a high-profile subject, isn't it? I'm sure Wikipedia's mathematics experts will have fun with this one.

Many students taking geometry at the middle- and high-school level will probably be visiting this article for help understanding the subject. Therefore, this article should be written with them in mind, as well as provide an overview to the overall subject leading to the various advanced subtopic articles on the subject.

I look forward to seeing what you guys/gals come up with.

Good luck, and have fun. --The Transhumanist 01:41, 3 October 2006 (UTC)

For previous discussions see Talk:History of geometry.

While I like the new article so far a lot better than the previous one as a general introduction to geometry, it all seems (except for the history summary) very much aimed at describing current research-level mathematics. Shouldn't there be something about high-school-level geometry, very early in the article? —David Eppstein 01:52, 3 October 2006 (UTC)
Definitely, this article is one of our most important summary articles so needs to cater for a wide readership. --Salix alba (talk) 08:18, 3 October 2006 (UTC)

The emphasis on research: Quine said there are people interested in philosophy, and people interested in the history of philosophy, and the implication was (partisan and) that these were different bunches of people. The history having moved out, it was interesting to me to tackle the question from the other end: what would be an acceptable survey of the 'state of the art'? Another analogy: a cosmology basic article could go back the Babylonians, and on the other hand school students might well expect to find the Big Bang, age of the Universe 14.5 billion years, dark matter discussed. It has been interesting so far ( a day or so): I hope some more can be added that does illuminate geometry. Charles Matthews 16:12, 3 October 2006 (UTC)

[edit] Suggestions on further expansion

Section(s) on analytic geometry, projective geometry, Non-Euclidean geometry, affine geometry should be added. (Igny 13:53, 3 October 2006 (UTC))

Also finite geometry. -- Cullinane 14:16, 3 October 2006 (UTC)

[edit] Re: references and examples

Examples don't seem appropriate. References should be restricted to general reading on contemporary geometry. References for individual topics make much more sense on a per article basis. Charles Matthews 17:53, 5 October 2006 (UTC)

[edit] Questionable

The article now includes a suspect sentence:

Frankly, this sounds like obscure ax grinding. (Surely Procrustean bed and Hilbert should be linked.) Nor am I convinced by the claim. For example Abhyankar's 1990 monograph, Algebraic Geometry for Scientists and Engineers, AMS (ISBN 978-0-8218-1535-9), devotes Lecture 19 (pp.145–158) to “Infinitely Near Singularities” including points in Nth neighborhoods (a.k.a. infinitely near points). How lost is that? --KSmrqT 00:39, 9 October 2006 (UTC)

Give me a little time, and I'll support this with a quote from Weil. Yes, infinitely near point is something that was recovered when birational geometry was put on a foundation. Cf. Manin talking about 'bubble space', when you blow up the projective plane everywhere, and again and again ... There is a genuine topic here. Charles Matthews 07:56, 9 October 2006 (UTC)

[edit] pictures

I noticed that the illustrations so far are historical. Perhaps some pictures, somehow illustrative of some aspects of "contemporary geometry" could be included. I think it would be cool for the lay-reader to see something like that, even if it appears quite mysterious. --C S (Talk) 01:32, 9 October 2006 (UTC)

[edit] Geometrist

I had a dream the other day, in which I was studying to become a "geometrist" and I wondered (when I woke up), whether this is a real term, one I made up (which could be real), or the by product of all the drugs. Any insight into this problem would be appreciated. In any case it's my new favorite word. —The preceding unsigned comment was added by Verbally (talk • contribs) 15:26, 23 October 2006 (UTC).

The usual word for someone who does geometry is "geometer". —David Eppstein 15:53, 23 October 2006 (UTC)
Isn't one of the earliest texts on geometry in practical application to earth measure also one of the earliest texts on al gebra.Rktect 23:39, 4 September 2007 (UTC)
  • 17. Somers Clarke and R. Englebach(1990). Ancient Egyptian Construction and Architecture. Dover. ISBN 0486264858.
A text found at Saquara dating to c 3000 BC or 5000 years BP has a picture of a curve and under the curve the dimensions given in fingers to the right of the circle as reconstructed to the right.

David, Is this geometry where the architect involved drew the figure and then took off the data for a specific problem, perhaps just finding an easy way of describing the construction template of an arch to the carpenter building it, or is it something else? Are the lists of fingers intended to describe a special type of coordinate series expressable in unit fractions?

It was presumed the horizontal spacing was based on a royal cubit but the results if based on an ordinary cubit are different. It appears the value for PI being used may be 3 '8 '64 '1024. which at 3.141601563 is slightly better than the Rhind value.

The circumference of the circle is 1200 fingers and the diameter of the circle is 191 x 2 = 382
3 '8 '64 '1024 x 382 ~= 1200.0
The side of the square is 12 royal cubits and its area is 434 square feet.
The area of the circle is ~ 191^2 x 3.141601563.
The algorithm suggests working with coordinates and numerical analysis to define a curve.
1
1 1
1 2 1
1 3 3 1
1 4 6 4 1
1 5 10 10 5 1
3
3 + 1/2y^3 is 3 '8, = 3.125
3 + 1/2y^3 + 1/2y^6 is 3 '8 '64,= 3.140625
3 + 1/2y^3 + 1/2y^6 + 1/2y^10 is 3 '8 '64 '1024 = 3.141601563
For purposes of comparison(3 '7 = 3.142857143) Rktect 23:53, 4 September 2007 (UTC)

[edit] Different forms of geometry

I think that there should be a section of this article explaining the different forms of geometry and the differences between their concepts. That is why I wrote that section about the different forms of geometry. But every time I put a section in about it it gets removed. But the readers of this article are not told anywhere in the article about what the difference between Euclidean and non-Euclidean geometry is. That is a very important aspect of geometry that should included in this article. There ought to be a section of the article explaining what Euclidean geometry is and what non-Euclidean geometry is and how they differ and also explaining what the other forms of geometry are. Prb4 16:15:12 February 14, 2007 (UTC)

Does anyone have a response to what I just said and would it be possible to reinsert a revised version of my explanation of the different forms of geometry into the article as a section titled different forms of geometry? Does anyone else here agree with me that the section should be reinserted into the article? The reason I feel this way is because the article does not deal with the issue of parallel lines which is the most important difference between Euclidean geometry and the two non-Euclidean geometries, hyperbolic geometry and elliptic geometry and because in general I think that this article does not give a sufficient overview of the differences between the concepts in different geometries and because in general I think this article is a second rate article which does provide enough information on geometry. Prb4 20:40:21 Wednesday February 14, 2007 (UTC)

Are degrees considered parallel lines in the spherical geometry of antiquity?Rktect 20:31, 27 July 2007 (UTC)

Content at User:Rktect/degrees —METS501 (talk) 21:00, 27 July 2007 (UTC)

[edit] History

Since this is a mathematics article does it really need to have a history section. I think the article should focus on current geometry and math concepts and the history of geometry should only be discussed in the history of geometry article. Prb4 19:20, 15 February 2007 (UTC)

Yes it does. It is hardly possible to explain what 'geometry' means in the 21st century without some history. The fact that you have been adding a late nineteenth century version of 'forms of geometry' rather tends to support that. Please don't remove historical context from mathematics article. It certainly helps the general reader, it may help mathematicians and physicists outside their specialisms; and we anyway have thousands of purely technical mathematical articles. Charles Matthews 19:40, 15 February 2007 (UTC)
I agree with Charles. Your insertions show a rather misguided idea of what "geometry" means today. Your proposed "different forms of geometry" is just a mess. Really, you should learn more mathematics before attempting to characterize the "different forms". --C S (Talk) 22:32, 15 February 2007 (UTC)

[edit] Variable shape geometry?

A new topic just popped up called Variable shape geometry. Should this wikipedia page be for advertising everything that has the word "geometry" in it, regardless of whether or not it is deemed relevant or useful? I don't really know much about this "variable shape geometry" but it seems strange that it is here while more mainstream forms of geometry such as non-commutative geometry, is not here. Rybu 19:08, 15 April 2007 (UTC)

Should this wikipedia page be for advertising everything that has the word "geometry" in it? Clearly, no. I removed that section. The article it pointed to is under AfD. —David Eppstein 02:59, 16 April 2007 (UTC)

[edit] Kant and Geometry

I believe the discussion under the Geometry beyond Euclid heading is wrong. Kant never denied the possibility of of non-euclidean geometry. He stating Geometry was A priori does not mean that non-euclidean geometry can not be developed or that he denied the possibility. That would mean that he stated it was Analyic which is precisely what he denied. He stated it was synthetic a priori truths, such that it was truths about the world directly understandable by the human mind by pure reason alone but but that also it could be denied without contradiction, which it could as non-euclidean geometry shows. This does not however mean it was not a priori, i.e. truths about the way the world works as it does, we do after all live in Euclidean 3 space, however much we may be able to conceive other geometries.

I will change the article to reflect this if no one has any objections. —Preceding unsigned comment added by 137.205.26.170 (talk • contribs) 18:52, 2007 June 10

I think the subject is contentious. I have restored the previous version, but with some modifications toning down the bald suggestion that Kant was wrong. I believe that he was, but others (including some experts) disagree. So the article no longer makes the assertion. Other points of view (including yours) are treated in a footnote. Silly rabbit 21:29, 10 June 2007 (UTC)

Thank you, I think this is a better compromise, give my edit was almost as POV as the original content, though the article does now read rather clumsily at that point. Maybe it would be best to remove the reference to Kant at that point, instead just sayng 'some philosophers' or something else, especially since Kant arguably allowed for the possibility of non-euclidean geometry by arguing geometry was synthetic rather than analytic truths as did say Hume or Hegel.

I wrote the offending sentence. I am far from an expert in philosophy, but all literature on history of mathematics that I am familiar with is critical of Kant's contribution, and some authors did call him 'wrong' on that, including, if I am not mistaken, Felix Klein, in Lectures on the development of mathematics in XIX century. It was precisely Kant, and not 'some philosophers', who claimed that euclidean geometry was one of the inherent truths, making this claim into one of the central arguments of his philosophical system. His position, its influence, and its subsequent assessment throughout 19th and 20th century are amply documented and deserve to be mentioned, but if presently another interpretation of Kant's view has become dominant, we should acknowledge it.
One has to be careful with hypothetical arguments concerning what someone did or did not allow for. Kant did not deny possibility of Quantum mechanics or Special relativity, but does it follow that he 'anticipated' them in any sense? Arcfrk 01:56, 11 June 2007 (UTC)

I fear that the problem here is indeed that of different understandings of philosophy, hence I suggested removing the reference at all. I think the problem is understanding the subtleties of the idea of synthetic a priori truths, which Kant did indeed make the centre of his philosophy. The point he was trying to make at great length though was the synthetic part instead of the a priori part which does not quite mean an inherent truth, though this is a somewhat reasonable approximation, it just means they can be appreciated by pure logic rather than through experience (a posteriori). The argument he made central to his system was that these were synthetic truths, against the nigh-universal contemparary view, as held by Hegel or Hume (hance some philosophers), that they were analytic. Analytic means they can't be denied without contradiction, whereas synthetic means they can be denied without a logical contradiction arising. If they were Analytic non-Euclidean geometry would be impossible, as it would generate a contradiction, if they were synthetic they would be possible, in this sense he was proved right in the end with the creation of non-euclidean geometry, the a priori part is another debate. I don't suppose the issue is too important though, I doubt someone's understanding of geometry or Kant will be destroyed either way by the compromise in place, though as I said it doesn't read too fluently.

It's quite a common error made among even among literature on the subject as the a priori, a posteriori, synthetic, analytic distinction is quite subtle, especially for those not well versed in the philosophy in question. As I said though, I don't suppose it's that important an issue.

[edit] Butterfly Theorem

Um,I've added a proof of Euclid's Butterfly Theorem as well. I'd love to furnish the Napoleonic triangle proof. Could someone tell me how to upload a picture? Rohan Ghatak 16:27, 31 July 2007 (UTC)

Yes, to upload a picture, you can click "File upload wizard" on the sidebar under the "interaction" heading, which takes you to Wikipedia:Upload, and then follow the directions from there. If you need more information, see Wikipedia:Uploading images. —METS501 (talk) 16:30, 31 July 2007 (UTC)
I'd like to add that it's preferable to add pictures to Wikimedia Commons instead of here. The procedure is almost the same and they show up automatically with the same names here, but are usable directly on other language editions of Wikipedia as well. —David Eppstein 17:28, 31 July 2007 (UTC)


[edit] geometry beginnings

i was wondering if geometry was ever based on other forms of mathematics 65.33.196.220 23:01, 18 September 2007 (UTC)deathdealer

This question is more suited to the reference desk. —METS501 (talk) 01:23, 19 September 2007 (UTC)

[edit] Picture selection

I think we should include basic geometry structures images into the article, such as basic shapes, solid figures, images related to trigonometry rather than exotic and less-known for wider audience structures. What do you think about that? Visor (talk) 20:05, 5 February 2008 (UTC)

[edit] Geometries vs. Spaces

Each article on a specific kind of space tends to have an associated article on the geometry, for example Euclidean space and Euclidean geometry. I have started making some changes to the article on Projective geometry, and I am wondering if the Wikipedia community have established any guidelines as to which aspects (e.g. axiomatic development, analytical treatment, visualisations, history, etc.) should primarily come under the space or the geometry article or both for readability? -- -- Cheers, Steelpillow 10:24, 12 May 2008 (UTC)

Well, you can read about probability spaces but not about probability geometry, and about topological spaces but not about topological geometry, and about vector spaces but not about vector geometry, and about Stone spaces but not about Stone geometry. I'm pretty sure I could list a bunch of others if I felt like looking around a bit. So the word "each" above may be a bit exaggerated. Michael Hardy (talk) 05:09, 17 May 2008 (UTC)
Sorry, by "space" I meant to imply "geometric space". Perhaps I should also specify "spatial geometries" as opposed to more abstract kinds. The point is, why have two articles on what is essentially one topic? -- Cheers, Steelpillow 13:22, 17 May 2008 (UTC)
It is a good question, and I'm not sure how to answer it. Obviously we can't have two articles with identical content, so duplicating the axiomatic treatment, etc, seems to be a bad idea. You might consider having a look at Hyperbolic space and Hyperbolic geometry. Hyperbolic space contains the analytical results, and a description of the models of the geometry, whereas Hyperbolic geometry emphasizes the axioms, history, and intuitive aspects of the theory. silly rabbit (talk) 13:31, 17 May 2008 (UTC)
It might be good to note why those articles are that way. Originally, hyperbolic geometry started as an elementary article on the concept of nonEuclidean geometry, and hyperbolic space was created as a more technical treatise on the hyperbolic manifold called hyperbolic space. There is another article on hyperbolic manifold but of course the case when the manifold is simply connected is the most important case and fully deserves its own article. This was kept up because some people (including me) thought that the layman would probably search for hyperbolic geometry not hyperbolic space, the latter term being less friendly. These reasons should hold also for projective geometry and projective space. --C S (talk) 07:07, 28 May 2008 (UTC)
To my mind, the word "space" usually describes a specific space, i.e., a concrete geometric object. This is usually presented as a model of the space in question—hyperbolic space has three common models that all represent the same space, for example. These are typically homogeneous and isotropic. All articles with "space" in the title could look roughly the same. (I haven't really checked to see if they do;in fact, they probably don't.)
The word "geometry" on the other hand, is a bit trickier. In full generality, it is the study of all spaces that have local properties that correspond to the space in question. So hyperbolic geometry is really the study of hyperbolic manifolds, spaces that are locally hyperbolic, but aren't necessarily isomorphic to the canonical hyperbolic space. But this isn't really how the terms are used commonly. Both Euclidean geometry and hyperbolic geometry more commonly refer to the study of objects living in these spaces that are invariant up to some kind of transformation. So in Euclidean geometry (the kind we all learned in junior high), we study congruence laws of triangles, and the like. Hyperbolic geometry, at this level, might study the fact that there are an infinite number of lines through a point that fail to intersect a given line.
Since the "space" articles seem to have a more clear purpose and definition, it might be best to start there, and then use the corresponding "geometry" article to treat the subjects that don't make it into the space article. Things like "history" would probably be better in the geometry article. VectorPosse (talk) 02:15, 18 May 2008 (UTC)
There seems to be some agreement here:
  • Projective geometry: Axioms, history, elliptic property, homogeneous coordinates.
  • Projective space: Models and their analytical treatments.
I think that intuitive aspects are best summarised on both pages, as they help the reader to understand the rest. But I am not sure where to discuss finite projective spaces/geometries, i.e. PG[m,n]. In a sense these are different spaces, but they are usually referred to as geometries and treated algebraically (i.e. abstractly) as a family. Or, should they have a page of their own (as they did until I unthinkingly tagged them onto the end of Axioms of projective geometry) - see the associated discussion page. -- Cheers, Steelpillow 20:41, 18 May 2008 (UTC)
Sometimes, the obvious takes a while to surface. The articles on "space" assume 3 dimensions. Those on "geometry" should be agnostic. -- Cheers, Steelpillow 11:45, 19 May 2008 (UTC)