Talk:Geometry

Edit to Notes

Note 3 full citation is Greek and Vedic Geometry Frits Staal Journal of Indian Philosophy 27 (1/2):105-127 (1999)

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)

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)

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#. 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.

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)