Talk:Geometry

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

@D.Lazard: Revision https://en.wikipedia.org/w/index.php?title=Geometry&oldid=1144056819 added the text This enlargement of the scope of geometry led to a change of meaning of the word "space", which originally referred to the three-dimensional space of the physical world and its model provided by Euclidean geometry; presently a geometric space, or simply a space is a mathematical structure on which some geometry is defined. However, the word space can refer to mathematical structures that are not geometric, e.g., vector spaces over arbitrary fields. I'm not sure how it should be worded, since the term Geometry is itself murky. Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:04, 12 March 2023 (UTC)[reply]

This depends of your definition of “geometric”. Currently, nobody pretends that algebraic geometry and finite geometry are not geometry, and vector spaces over a finite field belong to both areas. There is nothing murky in geometry. Simply, this is a scientific area and not a mathematical term, and, as such, it is not subject to a mathematical definition. D.Lazard (talk) 19:51, 12 March 2023 (UTC)[reply]

How is Geometry not a mathematical discipline? --Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:47, 13 March 2023 (UTC)[reply]

We hear geometry-related words all the time: ‘what’s your angle?’ and ‘everyone should eat three square meals a day!’ and ‘she ran circles around me!’, often with little thought to how fundamental those shapes are to the discipline called geometry. Barrista hex (talk) 10:36, 20 December 2023 (UTC)[reply]

wanna learn from you... Barrista hex (talk) 10:32, 20 December 2023 (UTC)[reply]

Geometry just refers (except in very limited cases in NCG) to any set whose elements we can describe as "points" because in addition the set has some information about how its elements have a "position" relative to each other. "Space" is just a catch all term used to describe such structures, so I think its sort of tautological to say Geometry is the study of Spaces.

There's a more limited definition of geometry in the context of topology which refers to spaces with some particular kind of rigidifying geometric structure on them such as a metric, Riemannian metric, volume form, algebraic structure, etc. But I don't think that really applies to "Geometry" in the large. Tazerenix (talk) 23:09, 12 March 2023 (UTC)[reply]

I've never seen an Algebra text refer to the elements of, e.g., a vector space, a Fréchet space , as a point. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:47, 13 March 2023 (UTC)[reply]

The requirement is not that a textbook refers to them as "points" but that there is a relation between elements which provides information about their relative position. In the case of a vector space, the relation is linear (you can specify when two elements lie along the same line). In particular there is an affine structure (and more, as there is a distinguished point at the "center", another positional relationship). Of course an algebra book will not think of vector spaces as spaces if its goal is to do algebra, but they certainly don't refer to them as "vector sets". Tazerenix (talk) 23:07, 13 March 2023 (UTC)[reply]

In Topology there is no concept of relative position. Does that mean that a topological space is not a space.? --Shmuel (Seymour J.) Metz Username:Chatul (talk) 08:58, 14 March 2023 (UTC)[reply]

Closeness is the basis of topology, and is a sort of relative position. However, although although Tazerenix's definition of points and spaces is ingenious, I am not sure that I completely agree with it, and it is

WP:OR. So, it is better to say that space, point, geometry, geometric method, geometric space, etc. are what is so called by the community of mathematicians. These terms do not require to be formally defined as they are only used to provide an intuitive support to reasonnings, which otherwise would be more difficult to understand. For example, learning the axioms of vector spaces is easy, but understanding the richness of the concept cannot be done without considering the geometrical aspects of the concept. D.Lazard (talk) 10:31, 14 March 2023 (UTC)[reply

]

See for example Kuratowski closure axioms in which topology is defined entirely using the concept of a point being "close" to a set. This is an example of information about the relative positions of points: If a point x is close to a set A and a point y is not, then x is closer to A than y! Tazerenix (talk) 22:58, 14 March 2023 (UTC)[reply]

Not so. None of the axioms refer to closeness. There is a derived concept of a point being close to a set, but none of the axioms use it. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 00:23, 15 March 2023 (UTC)[reply]

If you define the relation "${\displaystyle x}$ is close to ${\displaystyle A}$" as "${\displaystyle x}$ is contained in ${\displaystyle \mathrm {cl} (A)}$" then the axioms of a topology can be specified as

1. No point is close to the empty set
2. Every point of ${\displaystyle A}$ is close to ${\displaystyle A}$
3. The points of ${\displaystyle X}$ which are close to ${\displaystyle A\cup B}$ are the points close to ${\displaystyle A}$ or to ${\displaystyle B}$
4. If a point ${\displaystyle x}$ is closeto the set of points close to ${\displaystyle A}$, then ${\displaystyle x}$ is close to ${\displaystyle A}$

A set ${\displaystyle X}$ with a relation between points and sets of "closeness" is equivalent to specifying a topology (precisely, define the closure operator ${\displaystyle \mathrm {cl} :{\mathcal {P}}(X)\to {\mathcal {P}}(X)}$ by ${\displaystyle \mathrm {cl} (A):=\{x\in X\mid x{\text{ is close to }}A\}}$). Tazerenix (talk) 02:23, 15 March 2023 (UTC)[reply]

Speaking as a topologist, I don't believe that every topological space ought to be described as geometric, however one might reasonably define the term. While there is, of course, a close connection between topology and geometry, I don't think topology is best described as a subset of geometry. Paul August ☎ 16:50, 13 March 2023 (UTC)[reply]

I would probably classify Topology as part of Geometry, although topologies not satisfying the separation axioms might be counter-intuitive. I could probably make an argument for considering it to be a part of Analysis, albeit a weak one. --Shmuel (Seymour J.) Metz Username:Chatul (talk) 17:19, 13 March 2023 (UTC)[reply]

The Contemporary Geometry section describes ten different fields of Geometry. Shouldn't Geometric Algebra be in this list? — Preceding unsigned comment added by 50.206.176.154 (talk) 05:17, 26 April 2023 (UTC)[reply]

From the geometry point of view, geometric algebra is only a tool used in Euclidean geometry. So, it must not be listed among the main parts of modern geometry.

Nevertheless, section

coordinates and linear algebra (analytic geometry). The equivalence of the two approaches has been proved by Emil Artin in his book Geometric Algebra. The algebraic approach to Euclidean geometry led to the introduction of various algebraic concepts such as vectors, quaternions, dual spaces, and over all, Geometric algebra. D.Lazard (talk) 11:11, 26 April 2023 (UTC)[reply

]

The previous commenter is talking about something different from (though not entirely unrelated to) Artin's book. It is also not accurate to say that geometric algebra is "only a tool used in Euclidean geometry". For more context, you may perhaps be interested in Hestenes (2002) "Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics". For use beyond Euclidean geometry, see e.g. Hestenes (1991) "Projective Geometry with Clifford Algebra". –jacobolus (t) 14:13, 26 April 2023 (UTC)[reply]

My suggestion for the content of the article shows clearly that I am aware of the two meanings of "Geometric algebra". Also my sketch for the content of § Euclidean geometry does not imply that geometric algebra is not applicable outside Euclidean geometry; simply that it has been developed for the need of Euclidean geometry. Similarly, vectors and dual spaces are widely used outside Euclidean geometry. My opinion is that, for geometry, geometric algebras are not more important than, say, tensors. Both seem to be too technical and too specialized for having more than a single mention in this general article. D.Lazard (talk) 15:30, 26 April 2023 (UTC)[reply]

You said “From the geometry point of view, geometric algebra is only a tool used in Euclidean geometry.” I am just pointing out that that is not right. Of the sections listed in this article (which to be honest seem like a kind of arbitrary assortment), geometric algebra is a tool relevant to at least Euclidean geometry, Differential geometry, Non-Euclidean geometry, Algebraic geometry, Complex geometry, Discrete geometry, Computational geometry, Convex geometry.

developed for the need of Euclidean geometry – this doesn’t seem right either. Grassmann's work was pretty general and later mathematicians applied his products to all sorts of contexts. Clifford was very interested in modeling non-Euclidean geometry (though he died young and never got the chance to fully develop his ideas). Hestenes started out explicitly trying to model (both flat and curved) spacetime.

I don't think focusing on Artin's book as in your sketch here is the right approach to a section about Euclidean geometry (per

WP:DUE

), though IMO the current section ("geometry in its classical sense" etc.) is pretty useless.

Inre (Grassmann/Clifford/Hestenes style) geometric algebra I think it would be better to instead add 'vectors' and 'multivectors' to

geometric transformations are shoved into the "symmetry" subsection also seems like a poor choice. These should probably both be elevated to (separate) sections. –jacobolus (t) 16:15, 26 April 2023 (UTC)[reply

]

Removal of pleonasm "periodic periods" in Algebraic geometry section

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The second sentence in the Algebraic geometry section reads "It underwent periodic periods of growth...". The use of the adjective "periodic" to describe the noun "periods" seems pleonastic and therefore hampers readability. I suggest changing the sentence to open with either:

1. It underwent periods of growth---implying that "Algebraic geometry" underwent "a length of time" of growth (see period).
2. It underwent periodic growth---implying that the growth of "Algebraic geometry" was "happening repeatedly over a period of time" (see periodic).

Given the context and subsequent text in the sentence, the first option seems more appropriate than the second. Kyle F. Hartzenberg (talk) 01:23, 6 September 2023 (UTC)[reply]

I agree that the sentence is confusing. It seems that the intended meaning is that, since its origin, algebraic geometry had several distinct period of growth, implicitly separated by periods of relative stability; this is a controversial assertion. Moreover, the provided list (projective geometry, birational geometry, algebraic varieties, and commutative algebra) does not correspond to the beginning of the sentence, and is essentially non sensical as a list, as the first item is not specific to algebraic geometry, the second and the third are subjects of study in algebraic geometry, and the third is the fundamental tool for linking geometry and algebra into algebraic geometry. Moreover, this sentence is too vague for having any encyclopedic value.

So, the whole paragraph must be rewritten. Clarifying only "It underwent periodic periods of growth" cannot be done without introducing a controversial assertion. So, before the needed rewrite, it seems better to remain ambiguous. D.Lazard (talk) 08:45, 6 September 2023 (UTC)[reply]

Rewrite done. D.Lazard (talk) 17:51, 6 September 2023 (UTC)[reply]

How do I say this word? Please add IPA. 1.127.110.251 (talk) 11:03, 25 November 2023 (UTC)[reply]

View the proposal here. Writehydra (talk) 04:59, 2 February 2024 (UTC)[reply]