Talk:Geometry/Archive 1

Archive 1

Archive 2

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?!"

Geometry.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. Founded by Greeks.

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)

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)

Questionable

The article now includes a suspect sentence:

• The history of 'lost' geometric methods, for example infinitely near points, which were dropped since they did not well fit the Procrustean bed of Bourbakiste axiomatization trying to complete the work of Hilbert and Principia Mathematica, is yet unwritten.

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 N^th neighborhoods (a.k.a. infinitely near points). How lost is that? --KSmrq^T 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)

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, 19 October 2006 (UTC)

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)

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)

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.

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)

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)

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)

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)

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)

Persons interested in the subject matter of geometry should read and take note of Isaac Newton's discussion of Geometry in his preface to his Principia. In this he explains the need for geometrical (conceptually accurate) concepts in order for people to carry out what he calls the "mechanics" of drawing out lines and configurations. :Also worth reading in Motte's translation of the principia is Cotes's preface to the second edition, whereby Newton's procedural method of analysis is explained and favorably compared to alternative methods. WFPM (talk) 21:00, 10 November 2008 (UTC)

redirect from "geometric"

For some reason geometric redirects to the geometry page. I was attempting to access geometric distribution. You might want to consider removing that redirect.169.235.16.14 (talk) 15:50, 6 November 2008 (UTC)

I redirected it to Geometry (disambiguation). Tom Ruen (talk) 22:05, 22 December 2008 (UTC)

Picture not representative of Geology

Geology is a broad field of study. So there should be a picture that gives a general idea or example of geology. Something that should be recognisable to the layman. The current picture would only be recognised by experts on the field of Geology. 77.250.25.165 (talk) 18:08, 6 July 2009 (UTC)

Are you sure you're talking about the right article? This one is Geometry, not Geology. And the first two pictures span a range from the very ancient to the very modern. I don't think it's possible to encapsulate the whole field of geometry in a single picture. —David Eppstein (talk) 20:21, 6 July 2009 (UTC)

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)

• definitely! what is doing a calabi-yau manifold as leading image? I will work on that.  franklin  02:46, 18 January 2010 (UTC)

Infobox

An infobox for this article has been created. It is located at template:general geometry. Attention is needed to populate its entries with the most relevant and general information. The image can also be changed, but please choose one that is iconic for the topic of geometry, one that anyone can feel identified. A historic image can be a good choice but also a nice modern diagram with many of the most well known geometry theorem/problems would be appropriate.  franklin  03:26, 18 January 2010 (UTC)

Remark - equiform geometry

What about equiform geometry? It doesn't mention in the article. —Preceding unsigned comment added by 84.109.169.207 (talk) 05:16, 30 January 2010 (UTC)

pls help me with my assignment..

how many lines can pass through A?

                                           .A

                                .C

                                                  .B

How many lines can pass through A & B? B & C? What is the least numberof lines that can contain the three points?

give your conclusion about your findings in #s1,2 & 3 by answering the following questions.

How many points are needed to determine a line? How many points are needed to form a plane? How many lines may pass through a point? —Preceding unsigned comment added by 120.28.148.181 (talk) 13:45, 3 March 2010 (UTC)

Moved two sections from the article

The first of them duplicates other content; the second is highly polemical. Neither adds much of quality to the article. Arcfrk (talk) 03:45, 25 March 2010 (UTC)

Contemporary geometers

The E8 Lie Group Petrie Projection

Some of the representative leading figures in modern geometry are Michael Atiyah, Mikhail Gromov and William Thurston. The common feature in their work is the use of smooth manifolds as the basic idea of space; they otherwise have rather different directions and interests. Geometry now is, in large part, the study of structures on manifolds that have a geometric meaning, in the sense of the principle of covariance that lies at the root of general relativity theory in theoretical physics. (See Category:Structures on manifolds for a survey.)

Much of this theory relates to the theory of continuous symmetry, or in other words Lie groups. From the foundational point of view, on manifolds and their geometrical structures, important is the concept of pseudogroup, defined formally by Shiing-shen Chern in pursuing ideas introduced by Élie Cartan. A pseudo-group can play the role of a Lie group of 'infinite' dimension.

Axiomatic and open development

The model of Euclid's Elements, a connected development of geometry as an axiomatic system, is in a tension with René Descartes's reduction of geometry to algebra by means of a coordinate system. There were many champions of synthetic geometry, Euclid-style development of projective geometry, in the 19th century, Jakob Steiner being a particularly brilliant figure. In contrast to such approaches to geometry as a closed system, culminating in Hilbert's axioms and regarded as of important pedagogic value, most contemporary geometry is a matter of style. Computational synthetic geometry is now a branch of computer algebra.

The Cartesian approach currently predominates, with geometric questions being tackled by tools from other parts of mathematics, and geometric theories being quite open and integrated. This is to be seen in the context of the axiomatization of the whole of pure mathematics, which went on in the period c.1900–c.1950: in principle all methods are on a common axiomatic footing. This reductive approach has had several effects. There is a taxonomic trend, which following Klein and his Erlangen program (a taxonomy based on the subgroup concept) arranges theories according to generalization and specialization. For example affine geometry is more general than Euclidean geometry, and more special than projective geometry. The whole theory of classical groups thereby becomes an aspect of geometry. Their invariant theory, at one point in the 19th century taken to be the prospective master geometric theory, is just one aspect of the general representation theory of algebraic groups and Lie groups. Using finite fields, the classical groups give rise to finite groups, intensively studied in relation to the finite simple groups; and associated finite geometry, which has both combinatorial (synthetic) and algebro-geometric (Cartesian) sides.

An example from recent decades is the twistor theory of Roger Penrose, initially an intuitive and synthetic theory, then subsequently shown to be an aspect of sheaf theory on complex manifolds. In contrast, the non-commutative geometry of Alain Connes is a conscious use of geometric language to express phenomena of the theory of von Neumann algebras, and to extend geometry into the domain of ring theory where the commutative law of multiplication is not assumed.

Another consequence of the contemporary approach, attributable in large measure to the Procrustean bed represented by Bourbakiste axiomatization trying to complete the work of David Hilbert, is to create winners and losers. The Ausdehnungslehre (calculus of extension) of Hermann Grassmann was for many years a mathematical backwater, competing in three dimensions against other popular theories in the area of mathematical physics such as those derived from quaternions. In the shape of general exterior algebra, it became a beneficiary of the Bourbaki presentation of multilinear algebra, and from 1950 onwards has been ubiquitous. In much the same way, Clifford algebra became popular, helped by a 1957 book Geometric Algebra by Emil Artin. The history of 'lost' geometric methods, for example infinitely near points, which were dropped since they did not well fit into the pure mathematical world post-Principia Mathematica, is yet unwritten. The situation is analogous to the expulsion of infinitesimals from differential calculus. As in that case, the concepts may be recovered by fresh approaches and definitions. Those may not be unique: synthetic differential geometry is an approach to infinitesimals from the side of categorical logic, as non-standard analysis is by means of model theory.

Axiomatic vs. synthetic

Hi, I posted a question here about the definition of synthetic geometry. Can anybody help? -- Cheers, Steelpillow (Talk) 19:35, 13 April 2010 (UTC)

I think the phrase "The approach to geometric problems with geometric or mechanical means is known as synthetic geometry" is just plain wrong, so I've deleted it. Jowa fan (talk) 10:01, 9 May 2010 (UTC)

Misuse of sources

This article has been edited by a user who is known to have misused sources to unduly promote certain views (see WP:Jagged 85 cleanup). Examination of the sources used by this editor often reveals that the sources have been selectively interpreted or blatantly misrepresented, going beyond any reasonable interpretation of the authors' intent.

Please help by viewing the entry for this article shown at the cleanup page, and check the edits to ensure that any claims are valid, and that any references do in fact verify what is claimed. I searched the page history, and found 10 edits by Jagged 85 (for example, see this edits). Tobby72 (talk) 22:47, 24 September 2010 (UTC)

The reverted edits [[1]]

Hi, how are these edits irrelevant in the history section? [[2]].

I am puzzled how the history of Geometry makes no mention of any Indian and Geometry works! Is this a joke? The references are provided for too. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 17:36, 10 May 2011 (UTC)

The photograph you added seems to be of a scholar who is not currently mentioned in the article. Please add a reliable source instead of a photograph. The book you linked does not seem to be in English, so it is hard for me to gauge how reliable the publisher is. Tkuvho (talk) 17:44, 10 May 2011 (UTC)

The page number 52(pdf page) of the book (35 by printed page number) clearly mentions the line in English mentioned as it is.

The picture is Āryabhaṭa's, who wrote many treatises, its caption text mentioned. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 19:12, 10 May 2011 (UTC)

If the book is self-published it is not an acceptable source. For Aryabhata if he wrote many treatises it should not be difficult to source one of them, and add a sentence more informative than the photo. Tkuvho (talk) 19:21, 10 May 2011 (UTC)

One of the sources you recently added is by Ian Pearce. This is not a reliable source. Please personally check the reliability of whatever you add. Incremental change is the key to success. Tkuvho (talk) 04:10, 11 May 2011 (UTC)

The sources mention references too. For example, about Sulbasutras and so on. There is reference section in each page. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 06:12, 11 May 2011 (UTC)

It seems to me that most of the deleted text really belongs at History of geometry. The history section of Geometry is supposed to be short (people can go to the longer article if they want more detail), and is written in chronological order. I think it would be reasonable to add one paragraph about classical Indian geometry following the paragraph about Greek geometry. Hopefully someone can write something that's not too controversial. Jowa fan (talk) 05:05, 11 May 2011 (UTC)

I will try to present the same in few lines soon. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 06:12, 11 May 2011 (UTC)

Please keep in mind keeping a neutral point of view and using reliable sources. Pearce was a very bad choice from both of these points of view: it's a student essay, so it has very little reliability, and it's explicitly an advocacy piece. The Sulba Sutras are already mentioned in our article, by the way, in the first paragraph on history. If you think that should be expanded, bear in mind how briefly the same paragraph treats the mathematics of Babylon, Egypt, and China. —David Eppstein (talk) 06:24, 11 May 2011 (UTC)

I saw the comment after I saved edits. I had mentioned references, exactly as mentioned in the website [| diff]. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 07:10, 11 May 2011 (UTC)

You might find it helpful to read about secondary and tertiary sources. The web pages at http://www-history.mcs.st-andrews.ac.uk are a tertiary source, and one whose reliability is in doubt where this topic is concerned. I think it would be better to name the secondary sources directly in this case (e.g. look at the references on that web site and choose carefully which ones support your text). Jowa fan (talk) 04:28, 12 May 2011 (UTC)

Thanks. ..असक्तः सततं कार्य कर्म समाचर | असक्तः हि आचरन् कर्म.. Humour Thisthat2011 06:00, 12 May 2011 (UTC)

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)

If you're going to do a History of Geometry article, why no mention of India and Pre-7th century BCE Vedic alter construction? Are you all just daft or does History only begin when Greece learned to civilize itself and come out of caves? Cheers- JBalz — Preceding unsigned comment added by 142.59.203.143 (talk) 22:17, 13 July 2012 (UTC)

Einstein relativity theory as a start to differential geometry?

Quite unlikely. I wish to remind the author of that section, that before Einstein there were many scientists interested in the field of Classical Mechanics or theory of elasticity, plasticity and other related (to many to give names, not to omit anyone important). I do not understand this fascination in Einstein's works. He was neither most brilliant nor of most merit in the field. He did have several ideas, undoubtedly. However, as it seems to be common to attach this expression to him, the famous E=mc2 was first derived by Poincare! This is only one of the examples, perhaps most trivial, where mr Einstein is being brought up to a rank of super-scientist, which he most certainly was not... On top of that, there is no such thing as Einstein summation convention. Yes, he seems to have made it popular, but he most certainly did not invent it (If I remember correctly it was Gibbs who first applied it). Any thoughts? — Preceding unsigned comment added by 159.245.16.100 (talk) 08:25, 20 November 2012 (UTC)

While I tend to agree that Einstein's reputation is hyped-up, your reading of what is in this article is off base. The statement was that differential geometry has become important for mathematical physics, not the other way around as you claim in the title here. Your tirade against Einstein is a bit misplaced here ... try an article that actually talks about him. Bill Cherowitzo (talk) 18:28, 20 November 2012 (UTC)

Geometry

Geometry is little cool because its all over you — Preceding unsigned comment added by 71.72.218.124 (talk) 22:57, 3 January 2013 (UTC)

Etymology of "geometry"

On 07:25, 1 June 2013, ‎117.193.150.234 provided the following etymology for the word "geometry":

"The word geometry comes from the sanskrit word jyamithi meaning the mathematics of constructions."

I have found online sources that state that ज्यामिति (jyamiti) means "geometry", but they are not sources that could be considered reliable. Most online sources trace "geometry" to Greek roots without going any further.

We need a more complete history of the word.

Thank you, ‎117.193.150.234, for bringing up a very interesting bit of information.

Stigmatella aurantiaca (talk) 09:12, 1 June 2013 (UTC)

Thanks for that. I have completed a brief survey of the subject boundary of geometry, particularly in the west, and this is consistent with my findings, i think. Please checkout and comment on http://web.archive.org/web/20140220182415/http://my.opera.com/jehovajah/blog/2013/06/01/from-plato-to-legendre-geometry-in-aristotelian-crisis I started at where i thought the problem arose, not because i was unaware of the Harappan and Akkadian/Sumerian influences, in addition to the Egyptian. Jehovajah (talk) 08:33, 11 June 2013 (UTC)

The Grassmanns Justus, Hermann and Robert

Hi David,

I am suggesting that you add a link or a note referring to the work and influence of the Grassmanns in he section in which you discuss Riemann's Contribution.

I have written extensively on the Grassmanns in my blogs which you will find if you google "jehovajah Grassmann".

It is not appropriate to do more than a brief entry in this page , but it is less appropriate to leave them out! — Preceding unsigned comment added by Jehovajah (talk • contribs) 00:00, 27 May 2013 (UTC)

Which article are you talking about? —David Eppstein (talk) 00:02, 27 May 2013 (UTC)

This section in Geometry.

Geometry beyond Euclid

Differential geometry uses tools from calculus to study problems in geometry.

For nearly two thousand years since Euclid, while the range of geometrical questions asked and answered inevitably expanded, basic understanding of space remained essentially the same. Immanuel Kant argued that there is only one, absolute, geometry, which is known to be true a priori by an inner faculty of mind: Euclidean geometry was synthetic a priori. This dominant view was overturned by the revolutionary discovery of non-Euclidean geometry in the works of Gauss (who never published his theory), Bolyai, and Lobachevsky, who demonstrated that ordinary Euclidean space is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by Riemann in his 1867 inauguration lecture Über die Hypothesen, welche der Geometrie zu Grunde liegen (On the hypotheses on which geometry is based), published only after his death.There was also an influential contemporary Alternative view of Geometry expounded in the works of the Grassmanns Justus Grassmann ,Robert Grassmann ,Hermann Grassmann. Riemann's and the Grassmanns' new idea of space proved crucial in Einstein's general relativity theory and Riemannian geometry, and Hermann Grassmann's linear space which consider very general spaces in which the notion of length is distinguished and defined, is a mainstay of modern geometry.″

If yoou did not write it or contribute to it , i am sorry. However could you direct me to who did in that case?Jehovajah (talk) 00:17, 27 May 2013 (UTC)

It wasn't me, but it doesn't matter who wrote it; see WP:OWN. What matters is the consensus of the editors who remain interested in this article. So probably you would be best off taking this to Talk:Geometry where they could see it and discuss it. —David Eppstein (talk) 04:22, 27 May 2013 (UTC)

The above inquiry has brought me here, where i petition my case.Jehovajah (talk) 08:57, 27 May 2013 (UTC)

You write a great deal about this on your blogs. Why don't you write about it yourself? The Grassmanns are certainly very notable. Stigmatella aurantiaca (talk) 10:43, 27 May 2013 (UTC)

Etiquette! i thought it best to gain a consensus through discussion before changing anything. My knowledge grow but is not extensive enough to be definitive. In any case, my thought was just to put in hyperlinks to the articles already in Wikipedia.Jehovajah (talk) 08:12, 31 May 2013 (UTC)

Well, I've read some of your blogs and appreciate your level of understanding as well as your modesty and concern for consensus. I would say, go ahead and make your changes, refer to this talk discussion, and we'll all take it from there. You have been more than polite. I trust that you will do a good job and that any misunderstandings among us will be quickly resolved. Good luck! Stigmatella aurantiaca (talk) 11:50, 31 May 2013 (UTC)

Just a sandbox attempt to make the references. Jehovajah (talk) 07:01, 14 June 2013 (UTC)

Elementary geometry

The page on Elementary geometry redirects to this article, but the subject is not defined here. Can somebody remedy this? -- Cheers, Steelpillow (Talk) 18:27, 29 April 2010 (UTC)

That's because when there's no article for things like "elementary geometry," editors on wikipedia who don't want to wait or write the article tend to oversimplify the meaning of things by linking them to an article that they think is close enough to the meaning. What happens next is elementary terms like "geometric solid" get oversimplified to mean polyhedron too, but is inaccurate because it doesn't include geometric shapes like spheres, cones, or cylinders. Also elementary terms like "plane figure" get redirected to geometric shape and then has no mention of the term "plane figure" on that page either. Bottom line, wikipedia is not elementary school-friendly. Oicumayberight (talk) 06:19, 1 August 2013 (UTC)

Importance in fields outside of mathematics

It would be nice for the article to say something about the importance of knowledges of geometry in physics, chemistry, engineering, geology, astronomy, biology, navigation, cartography, etc... There are plenty of uses! Ariel C.M.K. (talk) 13:15, 3 June 2014 (UTC)

Geometry scavenger hunt

How many types of Geometry are there? Name and give a definition of each one. — Preceding unsigned comment added by 73.10.92.2 (talk) 00:18, 23 January 2015 (UTC)

External links modified

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Nomination of Portal:Geometry for deletion

A discussion is taking place as to whether Portal:Geometry is suitable for inclusion in Wikipedia according to Wikipedia's policies and guidelines or whether it should be deleted.

The page will be discussed at Wikipedia:Miscellany for deletion/Portal:Geometry until a consensus is reached, and anyone is welcome to contribute to the discussion. The nomination will explain the policies and guidelines which are of concern. The discussion focuses on high-quality evidence and our policies and guidelines.

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Sentence needs deletion

The Dimension section ends with this sentence:"Currently, the existence of the theoretical dimensions is purely defined by technical reasons; it is likely that further research may result in a geometric reason for the significance of 10 or 11 dimensions in the theory, lending credibility or possibly disproving string theory." The sentence suffers from being unsourced, poorly written, and wildly speculative. I see no value in it, especially since many of its phrases are gibberish: (currently), (theoretical dimensions),(purely defined),(technical reasons)(geometric reason), etc. It should be removed.40.142.185.108 (talk) 14:24, 5 September 2019 (UTC)

 Done D.Lazard (talk) 15:35, 5 September 2019 (UTC)

Addition of section on complex geometry to Contemporary geometry

I suggest the addition of a small subsection about complex geometry in the section on contemporary geometry, perhaps right after algebraic geometry (and definitely after both algebraic geometry and differential geometry). Complex geometry is a field of study that is partially encapsulated by both algebraic and differential geometry, but can also sit outside them (there are singular complex analytic varieties that are not algebraic varieties). In any case, the techniques of complex geometry are a combination of tools from both differential and algebraic geometry, along with many novelties, and the field is broad enough and active enough in contemporary mathematical research to be at least comparable in importance to convex geometry or discrete geometry, both of which have sections here. For example, in his preface Huybrechts writes

Complex geometry is a highly attractive branch of modern mathematics that has witnessed many years of active and successful research and that has recently obtained new impetus from physicists' interest in questions related to mirror symmetry.

Here is what I would suggest:

Main article: Complex geometry

Complex geometry studies the nature of geometric structures modelled on, or arising out of, the complex plane. Complex geometry lies at the intersection of differential geometry, algebraic geometry, and analysis of several complex variables, and has found applications to string theory and mirror symmetry.

Complex geometry first appeared as a distinct area of study in the work of Bernhard Riemann in his study of Riemann surfaces. Work in the spirit of Riemann was carried out by the Italian school of algebraic geometry in the early 1900s. Contemporary treatment of complex geometry began with the work of Jean-Pierre Serre, who introduced the concept of sheaves to the subject, and illuminated the relations between complex geometry and algebraic geometry. The primary objects of study in complex geometry are complex manifolds, complex algebraic varieties, and complex analytic varieties, and holomorphic vector bundles and coherent sheaves over these spaces. Special examples of spaces studied in complex geometry include Riemann surfaces, and Calabi-Yau manifolds, and these spaces find uses in string theory. In particular, worldsheets of strings are modelled by Riemann surfaces, and superstring theory predicts that the extra 6 dimensions of 10 dimensional spacetime may be modelled by Calabi-Yau manifolds.

Tazerenix (talk) 18:05, 8 January 2020 (UTC)

I support the addition of this section. However, sheaves links to a dab page and must be replaced by [[sheaf (mathematics)|sheaves]]. D.Lazard (talk) 18:32, 8 January 2020 (UTC)

References

Origin of the word Geometry

The latin word is derived from the Sanskrit word jya (place)-mithi (measurement) which dates further back than the greek usage. It would be good to research this topic further and clearly mention the full origin. An 8th century text (about at least 2 centuries before pythagoras) talks about the pythagoras theorem with its original name. It would be good to fully research there and make the text accurate. — Preceding unsigned comment added by 192.55.54.38 (talk • contribs)

Well, no. Latin, in general, does not derive from Sanskrit; both derive from a common ancestor, proto-Indo-European. And in the specific case of "geometry", reliable sources trace it to Greek. The dating of the Sulba Sutras as a document to the 8th century BC is dubious; it was an oral tradition, with the oldest written texts only being much later. Also, your capitalization is odd; why capitalize Sanskrit but not Latin, Greek, and Pythagoras? —David Eppstein (talk) 17:00, 16 April 2020 (UTC)

Semi-protected edit request on 30 June 2020

In section 2.9, change "Length, area, and volume describe the size or extant of an object" to "Length, area, and volume describe the size or extent of an object" Martibttbtt (talk) 20:39, 30 June 2020 (UTC)

 Done - good catch. --Bill Cherowitzo (talk) 20:56, 30 June 2020 (UTC)

Geometry is more general than that

Geometry includes the study of manifolds in which the concepts of line and plane are meaningless, spaces in which the concept of angles is meaningless and spaces in which the concept of distance is meaningless. Some examples:

• Projective Geometry has lines and planes but no angles or distances. The closest analog is the cross ratio
• A manifold without a connection has no lines or even geodesics. Shmuel (Seymour J.) Metz Username:Chatul (talk) 22:10, 4 September 2020 (UTC)

@D.Lazard:user:D.Lazard reverted my changes with the summary Reverted 1 edit by Chatul (talk): Non-neutral point of view. If you disagree with the current formulation of the lead, discuss first in the talk page; I assume that he did not first check whether I had, in fact, discussed it. How should I proceed? Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:48, 7 September 2020 (UTC)

This is the general article on geometry, and your edit concern the general introduction of this article. This means that only the most important aspects of the subject can appear in the introduction. As the introduction must be accessible to the most general audience (including college students, physicists, engineers and computer scientists who are interested in image processing), technical details and advanced subtopics have not their place here (see MOS:LEAD).

Moreover, our articles must present a neutral point of view (see WP:NPOV), that is the main-stream point of view. Your edit does respect this fundamental pronciple, as it suggests that "point, line, plane, distance, angle, surface, and curve" are not fundamental concepts in modern geometry. This is definitively wrong, even if the properties that remain true without some of these concepts have been widely studied. Also, neutrality implies that emphasis must not be given to relatively minor subtopics, such as connections, when fundamental subareas such as algebraic geometry, fractals, or computational geometry are not even mentioned.

The lead and the article can certainly be improved, but this is not an easy task. So, any significant change of the lead must reach a consensus here before being implemented. D.Lazard (talk) 14:48, 7 September 2020 (UTC)

The existing lead violates WP:NPOV, and conflicts with much of Geometry#Contemporary geometry. Would you accept deleting While geometry has evolved significantly throughout the years, there are some general concepts that are fundamental to geometry. These include the concepts of point, line, plane, distance, angle, surface, and curve, as well as the more advanced notions of topology and manifold entirely?

How can the concepts be fundamental in modern geometry when they are missing from broad fields of Geometry? Even in the 19th Century length and angle were no longer fundamental, and Projective Geometry is far from an unimportant byway in modern geometry, a fact that you implicitly accepted when you mentioned Algebraic Geometry,

Also, connections are not minor, either in Mathematics per se or in applications of Mathematics. In particular, thye play an important role in Gauge Theory, which is central to modern Quantum Field Theory. — Preceding unsigned comment added by Chatul (talk • contribs) 17:58, 7 September 2020 (UTC)

The resolution to this problem is very simple. Just find some reliable sources that support your opinion. If there's a contest, we look to most mainstream point of view. What sources are you quoting to establish that point, line, etc. are no longer fundamental?

On the other hand, if we can't find reliable sources for the quote as it stands, I think deleting it would be just fine. It has one source, but I'm not sure how 'mainstream' it is. An easy fix would be to say that these concepts are fundamental to 'much of geometry', which is certainly true.

Although, I'd be hard pressed to find anything called 'geometry' that doesn't include at least one of those topics. Do you have any examples of geometry without points, lines, planes, distances, angles, surfaces, curves, topologies or manifolds? (Edit: I just looked over your proposed edits and saw that you included examples!) I think it's an 'at least one of these apply', not 'every one of these apply'. Perhaps that could be made more clear.Brirush (talk) 22:54, 8 September 2020 (UTC)

Edit: Just wanted to point out that I'm the one that wrote that line years ago, and I was just a silly young graduate student instead of the silly adult I am now. I just searched google books for definitions of geometry and chose one that I saw, and put the reference in.Brirush (talk) 22:56, 8 September 2020 (UTC)

user:D.Lazard mentioned Algebraic Geometry; while the classical approach has points, lines and planes, even points were abstracted away in the 20th Century. There's also Noncommutative Geometry, though I might classify that as Algebra or Analysis. Shmuel (Seymour J.) Metz Username:Chatul (talk) 04:36, 9 September 2020 (UTC)

Note that you also deleted the sourced changes with regard to lapses of rigor in Euclid's Elements, without giving your objections in Talk:Geometry#Flaws in The Elements. Shmuel (Seymour J.) Metz Username:Chatul (talk) 04:36, 9 September 2020 (UTC)

Before answering a previous post of Chatul, I was waiting that it was signed by Chatul himself or a bot. Unfortunately, this has not been done, and I had to sign it myself. Also, I have fixed the indentation of the last Chatul's posts as well as two section links, because they were confusing. D.Lazard (talk) 08:36, 9 September 2020 (UTC)

I agree with Brirush's comments on the disputed sentence. I agree also with Chatul's assertion that the lead violates the rules of MOS:LEAD, but this is not for the reasons given (implicitly) by Chatul. This is because it gives a too large place to the history of Euclidean geometry (we have a section "History", and also the articles Euclidean geometry and History of geometry for that), and the place of modern geometry is reduced to half a sentence. This cannot be fixed by changing some sentences as Chatul tried. I have not yet a clear idea for a convenient lead, so I'll come back here later for suggesting a new lead. D.Lazard (talk) 08:36, 9 September 2020 (UTC)

@D.Lazard:I've called for 3O

Flaws in The Elements

One of the discoveries since the ancient Greeks is that there are incorrect proofs in Euclid's Elements and that, in fact, some of the stated theorems do not follow from the stated axioms and postulates. Should this not be discussed when mentioning the Elements?. In that context it would be appropriate to mention, e.g., Hilbert's Grundlagen der Geometrie, as not merely reformulating Geometry but as adding the missing axioms Shmuel (Seymour J.) Metz Username:Chatul (talk) 02:07, 8 July 2020 (UTC)

The assertion that "there are incorrect proofs in Euclid's Elements" is an anachronism, as you assert yourself that the problem is that he did not follow the modern rules of rigor that have been established more than 2,000 years later. On the contrary, the sentence "geometry was put into an axiomatic form" is together another anachronism and a understatement, and sould be written something like: "Euclid introduced what is now called the axiomatic method, for a unified presentation of geometry as a succession of rigorously proved theorems".

In any case, this is already discussed in Euclid's elements#Criticism. D.Lazard (talk) 09:07, 9 September 2020 (UTC)

The words rigorous and rigorously are themselves anachronisms.

The issue isn't how much detail to go into, but rather whether a false claim should be corrected. Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:58, 9 September 2020 (UTC)

Here there is no claim that is mathematically false. There is only an epistmologic assertion that is controversial, which you have modified in a way that is even more controversial. D.Lazard (talk) 15:37, 9 September 2020 (UTC)

The claim that the proofs in The Elements are rigorous is false, and I gave references.

• Bill Casselman (February 2011). "Why Only Five?". Feature column. American Mathematical Society.
• Meder, A. E. (1958). "What is wrong with Euclid?". The Mathematics Teacher. 51 (8): 578–584. JSTOR 27955773.
• Kleiner, Israel (December 1991). "Rigor and Proof in Mathematics: A Historical Perspective". Mathematics Magazine. 64 (5). Taylor & Francis, Ltd.: 291–314. doi:10.2307/2690647. JSTOR 2690647.
• Timothy Peil. "Survey of Geometry". web.mnstate.edu.
• Beeson, Michael; Narboux; Wiedijk, Freek (2017). "Proof-checking Euclid". arXiv:1710.00787. {{cite journal}}: Cite journal requires |journal= (help)

Shmuel (Seymour J.) Metz Username:Chatul (talk) 23:55, 9 September 2020 (UTC)