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extra structure on them. Just what it means to "rigidify" is not clear (and precisely the discussion at hand) but, for example, just a topology is probably too weak to be called geometry, and a metric structure is definitely geometry, and things like smooth structures, projective structures, affine structures, algebraic/complex structures and so on all sit somewhere inbetween. Here I use "geometry" in the sense of "geometry and not topology" rather than the broader use of "geometry as in geometry and topology". This is neither here nor there as we can just broaden what we mean by rigid to include, for example, a topology. I think the page is aiming somewhere inbetween these two interpretations of the word "geometry"; for example whilst topology is mentioned, it is only given a small section of the page (despite being a relatively large part of "geometry and topology" as a combined field!). This might be worth discussing at some point.
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even crazy things like Lurie's infinity-stacks) are more what I was thinking of. These are objects that ostensibly don't look almost anything like geometric objects even from last century, but because the tools people use to study them feel very geometric in nature (or are inspired by analogy with quote-unquote "actual geometry") they are viewed as geometric subjects. This seems to be the way that geometry is viewed nowadays at least within some parts of the pure maths community, but as you say perhaps the fundamentals of this viewpoint aren't as modern as I said! I guess the point of my comment was really a query about whether its worth going into this in the lead of the geometry page, or if it is best relegated to a section about non-commutative spaces or stacks in the contemporary geometry section (I think the latter).
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points/shapes, or coarser rigidifying structures such as holomorphic or algebraic structures (such as in complex or algebraic geometry), and also includes examples where the underlying object is not a set of points, but is an algebra (like C-star algebras in noncommutative geometry). It might make sense to clear up the remarks later in the lead along these lines (I agree that it reads a bit as though geometry is about taking euclidean geometry and removing some of the properties, but some effort has been made to steer away from that viewpoint), but I'm not sure how to make the first paragraph reflect this ultramodern viewpoint, or even if that should be the goal of the first few sentences (after all, the lead should be readily understandable to anyone).
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fluid dynamics, quantum dynamics, pattern recognition programming in computing, etc., for their fair shake of "Geometry" if that makes sense. These are all fields that deserve a fair path to consideration of their people being geometers per their having evolved from the ancient geometry of Euclid, who began geometry with allowable spatial movements and exercises prompting readers to reach QED.
1677:) 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.
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1540:– 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.
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None of the formulations is a mathematical definition, as the terms that are used (flat, indefinitely, surface, two-dimensional) are not defined in this context. So, it is somehow pointless to discuss here the case where they apply or not. Moreover, I have not exactly reverted your edit, since I have
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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
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2nd question is where you say "not about patterns". I guess I'd just like clarification on why geometry is not a study of spatial patterns. Take topography for instance. If we declare geometry is on object of pattern, I feel the way is not prepared for topography, spatial analysis, tensor geometry,
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The statement is not false in general. It is at worst a bit ambiguous what "plane" is referring to, although a
Euclidean geometry plane is a perfectly reasonable assumed referent. Nevertheless, the section "Main concepts" is already strewn with "according to Euclid" and "in Euclidean geometry", so
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I make no claim of having a precise reference for these remarks, but at least as I tend to view it, one way of summarising the modern viewpoint might be to say that geometry is concerned with objects, usually sets (but sometimes rings, algebras, and so on) that have been rigidified by imposing some
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that geometry was synonymous with
Euclidean geometry up until the development of hyperbolic geometry and the work of Gauss (evidenced by how shocked everyone was that you could have geometries without the parallel postulate) but in fact they were doing non-Euclidean geometry without realising, for
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My point was more that nowadays things that are very far removed even from what the
Erlangen program would think of as geometry are treated as geometry, because of a further shift in perspective. Non-commutative spaces (objects that don't actually literally exist), p-adic geometry, and stacks (and
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It seems difficult to write a lead which will be easily understandable to the average reader while also faithfully representing the (ultra)modern viewpoint that "geometry is anything you can study using geometric ideas or methods." I think the first paragraph reasonably represents what geometry is
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and is can be highly dynamic. The previous opening sentences "Geometry is concerned with properties of space that are related with distance, shape, size, and relative position of figures" seem to me to be more accurate and more understandable to a layperson, so I don't understand why they've been
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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
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First, I want to apologize for the etiquette error of updating it before making a new section in the talk page. In math, it seems the answers are self apparent, and the 'previous v. current' look at that proposed change (I thought) seemed to display that clarity. Perhaps I have been found to be
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if circles have no sides, how is that possible? if it had no sides it wouldnt exist cuz that not possible. so circles have infinite sides cause the sides are so tiny they dont even exist and youll never see them bc it has infinite sides. am i missing something??? how does it have no sides? :/
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Until the 19th century, geometry was dominated by the assumption that all geometric constructions were
Euclidean. This was challenged by the development of hyperbolic geometry by Lobachevsky and other non-Euclidean geometries by Gauss (etc.etc.) around this time. In fact it was realised that
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Classically the objects are sets of points, and the extra structures are things like specifying lengths, angles, incidences, and so on, but one could be more modern extra structure such as just volumes (like in symplectic geometry), more discrete relationships of relative position between
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My 1st question in light of the response to my error is: What is an object? In object-oriented logic, for example, one would be talking about variables (X,Y,Z, etc). In that sense, Algebra is more appropriately the study of "objects"... If I could I'd like to prompt clarification on that
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and such are more modern than
Synthetic Geometry and the Erlanger Program, but after half a century they are very much mainstream. I agree that there's no need to discuss them in the lead; my concern is that the current text is wrong. We should either remove or qualify the statement
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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
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I find "omits continuity" a bizarre description of discrete geometry, which is not generally formulated around disregarding properties of the spaces its objects live in. I think discrete fits better into the topics grouped by underlying methods. Maybe you were thinking of
241:, and this alternative name seems to refer to the underlying methods. However the alternative name is much less common, and it does not refer to the methods but on the nature of the problems that are studied. So, it belongs to a third category of extensions of geometry,
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before "omits continuity", but it is not clear for me whether finite geometry is sufficiently important to appear here. On the other hand, discrete geometry is clearly sufficiently important to appear here, but its place is unclear. Discrete geometry is also known as
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Does anyone know what the phrase "spatial (static) patterns" means. I am a professional research geometer and have absolutely no idea what this means, and it certainly doesn't seem to capture most of the geometry I've ever seen, which is among other things about
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implicitly non-Euclidean geometry had appeared throughout history, including the work of
Desargues in the 17th Century, all the way back to the implicit use of spherical geometry to understand the Earth (geodesy) and to navigate the oceans since antiquity.
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current: A plane is a flat, two-dimensional surface that extends infinitely far. Planes are used in many area of geometry. suggestion: A plane is a flat, two-dimensional surface that extends infinitely far. Planes are used in many areas of geometry.
982:. Also, if one would discuss here non-Euclidean geometries, one should also discuss finite geometries, for which "indefinitely" and "infinitely" are both nonsensical. This does not mean that I agree with the tone of the section, but
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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.
99:. Also, it answers to the issue mentioned in the heading of this thread. In any case, it is easier to edit for improvements. Be free to improve it, and, if there are possibly controversial issues, to discuss them in a new section.
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Within the ‘points’ subsection, the use of the word ‘moderm’ is incorrect and should be modern. Also within that section, the year range is given for a mathematician’s name as a 5 digit number, it likely is correct as 1919-xxxx.
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I think the statement in the lead is a bit disingenuous. For example, mathematicians were working with spherical geometry in ancient greece and even before that (through navigation and geodesy). It is true that mathematicians
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in the 17th
Century involved points at Infinity, and the resulting Geometry does not satisfy Euclid's axioms; in fact, it is no longer possible to talk of lengths for all segments and there are no longer any parallel lines.
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It is certainly correct to say that
Geometry is "concerned with properties of space that are related with distance, shape, size, and relative position of figures", because it is. Saying that doesn't imply that those are the
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While Affine Geometry can certainly be considered a Euclidean Geometry without, e.g., angles, lengths, much of modern Geometry consists of new structures rather than just ignoring some properties of old structures, e.g.,
184:(in the eyes of the non-specialist) as it is, and the rest of the lead does a fairly good job of representing this more modern idea (if not in name) by describing how classical geometry relates to its more modern form.
442:'s suggestion as a footnote, after having slightly edited it (mainly by adding wikilinks). The reason for a footnote is that, included in the text, this explanation would disrupt the flaw of reading, and would be too
859:. Compass and straightedge may have been preferred, but other methods were known, used, and allowed. As Chatul says above, making compass-and-straightedge into an absolute requirement is a more modern invention. —
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I have written a new version of the lead, and implemented it boldly. I am misplaced for judging it, but it is certainly much better than the previous version, and much closer to the prescription of
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Really? I thought the word was "geometer". Google ngrams agrees, with "geometrist" far lower in word frequency. Do you have any evidence of "geometrist" being more popular anywhere? —
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removed "far" from the original sentence, for taking your objection into account. The mention of non-Euclidean geometries is out of scope here, since this would make the section too
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That is false; classically, the ancient Greeks discussed constructions using other instruments. The restriction to compass and straightedge is more recent. I changed the text to
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Why (static)? Well, because that paves the way for what a shape is. It's a pattern in space that is static enough to yield itself to analysis. (Can be said to be in "stasis").
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could appear also in this category. Thus, it seems that this (already too long) sentence deserves to be split for being expanded. Further discussion on this point is needed.
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least Euclidean geometry, Differential geometry, Non-Euclidean geometry, Algebraic geometry, Complex geometry, Discrete geometry, Computational geometry, Convex geometry.
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would not have been important problems for them. So, if you disagree with the current formulation, you must find reliable sources supporting your favorite formulation.
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The broadening of the field didn't start with the 19th Century; Projective Geometry was a couple of centuries earlier, with antecedents long before the 17th century.
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As it seems that there is a consensus that the new lead is better than the oler one, and as it is now the new lead that is discussed, I have split this long thread.
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A person who studies geometry is commonly called a 'geometrist' worldwide beyond the USA. Should this not be added to the end of tte first paragraph in the Lead?
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is a flat, two-dimensional surface that extends infinitely." This would be unambiguous and it avoids discussing non-Euclidean geometry at an inappropriate time.
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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.
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Both versions of the lead have issues, but the previous lead strikes a better balance among, e.g., accuracy, brevity, clarity. I too have a problem with, e.g.,
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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
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Add me to the list of people baffled by the attempted new phrasing "spatial (static) patterns such as ". I don't think the addition was an improvement. —
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Perhaps an instance of this discussion's importance is in the works of Mathematicians Ralph Abraham and Robert Shaw "Dynamics--the Geometry of Behavior"
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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
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To me, I hear: Geometry is similar to Arithmetic in being a study the ancients did. When you do Geometry, you will work with terms like , , , ...etc
573:, and the answer depends on what definition you choose. Depending on the definition, it has 0, 1 or 2 sides. With no reasonable definitition is
119:? Finite and discrete are two different things here, although (like many of these subcategorizations) there is occasional overlap between them.—
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are shoved into the "symmetry" subsection also seems like a poor choice. These should probably both be elevated to (separate) sections. –
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Many branches are still concerned with properties of space that are related with distance, shape, size, and relative position of figures.
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Classical Geometry is concerned with properties of space that are related with distance, shape, size, and relative position of figures.
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It underwent periodic growth---implying that the growth of "Algebraic geometry" was "happening repeatedly over a period of time" (see
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section reads "It underwent periodic periods of growth...". The use of the adjective "periodic" to describe the noun "periods" seems
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Inre (Grassmann/Clifford/Hestenes style) geometric algebra I think it would be better to instead add 'vectors' and 'multivectors' to
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It doesn't matter whether they are mathematical formulations; the fact that the statement is incorrect is still relevant. Changing
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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
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1114:'s version is not an improvement. As it introduces new concepts, it should have been discussed here first. According to
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It is concerned with properties of space that are related with distance, shape, size, and relative position of figures.
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It is concerned with properties of space that are related with distance, shape, size, and relative position of figures.
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distinction.... In terms of Physics as well, mass (object) v. energy has also caused a lot of fuss in the field...
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It is possible that ancients Greeks knew other instruments for geometric constructions, but they did not use them
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Given the context and subsequent text in the sentence, the first option seems more appropriate than the second.
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It underwent periods of growth---implying that "Algebraic geometry" underwent "a length of time" of growth (see
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is "only a tool used in Euclidean geometry". For more context, you may perhaps be interested in Hestenes (2002)
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Making articles more understandable does not necessarily mean that detailed technical content should be removed.
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example in Desargues work, and in spherical geometry. Perhaps the line should be changed to something like
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to be too technical and too specialized for having more than a single mention in this general article.
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I agree with your concern. However, I am not sure of the best way for fixing it. It is easy to replace
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How common is "common enough"? nGrams shows it far behind. This is an encyclopedia, not a thesaurus. —
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Hello, it's not that "geometrist" is more popular just that it's common enough here in the UK:
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moved to the second paragraph of the lead and replaced by something obscure and non-standard.
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is finite but not discrete. It is not, of course, derived from an affine or projective space.
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has updated the lead from the previous version which was discussed fairly extensively here
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as written promises too much, and conflicts with the cited reference. How about changing
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and therefore hampers readability. I suggest changing the sentence to open with either:
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The Contemporary Geometry section describes ten different fields of Geometry. Shouldn't
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Elliptic plane geometry is essentially spherical geometry with antipodes identified.
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does not take my objection into account; the statement is still false in general.
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If you wish to start a new discussion or revive an old one, please do so on the
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is a flat, two-dimensional surface that extends infinitely far or indefinitely.
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Classically, the only instruments used in most geometric constructions are the
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Classically, the only instruments allowed in geometric constructions are the
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the study of specific classes of problems occurring in Euclidean geometry
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Knowledge:Redirects for discussion/Log/2023 March 12 § Geometric space
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I agree that the sentence as quoted is wrong, and should be changed.
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Removal of pleonasm "periodic periods" in Algebraic geometry section
1360:. Readers of this page are welcome to comment on this redirect at
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The ancient Greeks had some constructions using other instruments.
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I have changed "exclusively" into "almost exclusively", and added
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It is false that the Greeks disallowed other constructions. See
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https://www.collinsdictionary.com/dictionary/english/geometrist
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is a flat, two-dimensional surface that extends infinitely far.
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1494:. For use beyond Euclidean geometry, see e.g. Hestenes (1991)
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why not add one more. How about: "In Euclidean geometry, a
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Until the 19th century, geometry was exclusively devoted to
986:'s edit is not an improvement, since it adds to confusion.
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In Euclidean geometry it extends infinitely, but in, e.g.,
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to determine whether its use and function meets the
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Geometry is the study of spatial (static) patterns.
1429:Euclidean geometry can be defined through axioms (
133:The new lead is better, but still has some issues.
1427:. This should be done along the following lines:
678:Semi-protected edit request on 10 November 2021
593:Semi-protected edit request on 7 October 2021
504:change ((Escher)) to ((M. C. Escher|Escher))
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1496:"Projective Geometry with Clifford Algebra"
857:The Ancient Tradition of Geometric Problems
1554:. While we're at it, the current way both
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1239:Shmuel (Seymour J.) Metz Username:Chatul
1033:Shmuel (Seymour J.) Metz Username:Chatul
922:Shmuel (Seymour J.) Metz Username:Chatul
800:Shmuel (Seymour J.) Metz Username:Chatul
633:Minor single letter change under Planes
579:Shmuel (Seymour J.) Metz Username:Chatul
330:Shmuel (Seymour J.) Metz Username:Chatul
284:Shmuel (Seymour J.) Metz Username:Chatul
167:Shmuel (Seymour J.) Metz Username:Chatul
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529:. Also a similar change for
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1600:has been answered. Set the
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700:has been answered. Set the
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549:shapes cant have no sides
282:was in the 19th century.
1354:redirects for discussion
1336:Redirects for discussion
831:quadrature of the circle
759:Compass and straightedge
320:, e.g., replace it with
1160:https://g.co/kgs/gbq8ce
827:Duplication of the cube
1412:Nevertheless, section
239:combinatorial geometry
1237:static or dynamic? --
896:geometry, but not in
745:ScottishFinnishRadish
42:of past discussions.
1514:does not imply that
1512:§ Euclidean geometry
1414:§ Euclidean geometry
920:reverted my edit. --
821:), and they did not
798:revertd the edit. --
1675:commutative algebra
1671:algebraic varieties
1667:birational geometry
1663:projective geometry
1647:Kyle F. Hartzenberg
1556:projective geometry
1358:redirect guidelines
1352:has been listed at
1235:Lorentzian manifold
770:The article claims
163:Symplectic geometry
1620:Algebraic geometry
1552:Geometry § Objects
1431:synthetic geometry
1387:be in this list?
1118:, I'll revert it.
968:, it wraps around.
379:Euclidean geometry
265:pseudometric space
148:Classical Geometry
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1516:geometric algebra
1488:geometric algebra
1469:Geometric algebra
1452:Geometric Algebra
1443:analytic geometry
1425:Geometric algebra
1419:Geometric Algebra
1405:
1393:comment added by
1385:Geometric Algebra
1379:Geometric Algebra
966:Elliptic geometry
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230:discrete geometry
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1332:Geometric space
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1191:Ok, I yield...
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1314:David Eppstein
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1178:David Eppstein
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18:Talk:Geometry
1644:
1617:
1609:
1598:edit request
1515:
1451:
1449:in his book
1418:
1389:— Preceding
1382:
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1334:" listed at
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1013:WP:TECHNICAL
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853:Wilbur Knorr
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790:straightedge
778:straightedge
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577:meaningful.
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444:WP:TECHNICAL
413:
390:— Preceding
374:
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78:
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37:
1465:dual spaces
1461:quaternions
1435:coordinates
1366:fgnievinski
1297:Billsmith60
1261:Billsmith60
1052:Danstronger
877:NPOV: Plane
815:in geometry
349:Paul August
276:Felix Klein
272:ultramodern
36:This is an
1624:pleonastic
1602:|answered=
1447:Emil Artin
1255:Geometrist
1082:recently.
1023:; section
1015:. Section
1006:infinitely
894:hyperbolic
702:|answered=
617:|answered=
488:|answered=
311:Certainly
1564:jacobolus
1500:jacobolus
1097:Tazerenix
556:1fractal4
440:Tazerenix
426:Tazerenix
385:Desargues
299:Tazerenix
270:The term
206:Tazerenix
66:Archive 2
60:Archive 1
1697:D.Lazard
1683:D.Lazard
1639:periodic
1521:D.Lazard
1474:D.Lazard
1391:unsigned
1295:Regards
1120:D.Lazard
1092:patterns
988:D.Lazard
918:D.Lazard
898:Elliptic
884:D.Lazard
855:'s book
835:D.Lazard
829:and the
819:Elements
796:D.Lazard
766:D.Lazard
698:Geometry
613:Geometry
535:D.Lazard
531:da Vinci
484:Geometry
448:D.Lazard
404:contribs
392:unsigned
324:or with
251:D.Lazard
101:D.Lazard
97:MOS:LEAD
82:D.Lazard
75:New lead
1457:vectors
1229:. Is a
1223:pattern
1219:dynamic
1134:wrong.
1088:objects
1027:states
1019:states
786:compass
774:compass
414:thought
313:schemes
222:editor
39:archive
1673:, and
1632:period
1545:WP:DUE
1421:(book)
1227:static
1116:WP:BRD
1090:, not
984:Chatul
849:neusis
464:Escher
396:Chatul
1606:|ans=
1596:This
1233:in a
1048:plane
935:Notes
912:plane
904:plane
890:plane
823:allow
706:|ans=
692:This
621:|ans=
607:This
492:|ans=
478:This
16:<
1701:talk
1687:talk
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1558:and
1525:talk
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1437:and
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1031:. --
992:talk
926:talk
916:and
865:talk
839:talk
804:talk
794:but
788:and
776:and
749:talk
741:Done
725:talk
668:talk
659:Done
643:talk
583:talk
571:side
560:talk
539:talk
526:Done
510:talk
452:talk
430:talk
400:talk
345:only
334:talk
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105:talk
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1604:or
1567:(t)
1503:(t)
1498:. –
1004:to
908:to
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490:or
482:to
278:'s
232:by
146:to
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1634:).
1610:no
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910:A
902:A
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220:To
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