Talk:Geometry
From Academic Kids
This doesn't strike me as a very good way to introduce geometry to a general audience: "Geometry is a branch of mathematics concerning spaces together with some sort of symmetry relations defined on it. Objects are said to be congruent if there is a symmetry mapping one to the other, and quantities unaffected by all symmetries are called invariants. The study of invariants is the main part of geometry." Those sentences should come about eight paragraphs into the article, I think. -- Larry Sanger
Really meant as a stop until something better comes along...
The current definition is I think inaccurate as well. Theatetus certainly didn't invent geometry, I don't know if he is so credited (e.g. in Plato's dialogues?) Many authorities quote Thales as having proved theorems about isosceles triangles (this would be some 200 years before Theatetus), much earlier "ruler and compass" constructions are credited to Egyptian surveyors or "rope-stretchers", marking fields after the annual Nile flooding. Eudoxus, another of Plato's circle, contributed rather more that Theatetus (as far as we can tell, though this sort of work is largely reconstruction from material in Euclid's Elements coupled with contemporary claims about who did what). Theatetus is one of the possibile inventors of the dodecahedron, that was purportedly discovered by one of Plato's school, and it's right that he be credited, but the current definition saying he invented the subject is an anachronism.
I too would be happier with a definition or description of what geometry is, before a history of how geometry arose. If this is a better structure I'd be willing to work on it, though will not change the original article unless there is concensus. Dominic Widdows 19 Apr 2005
What is the relationship to trig? Is trig a subfield of geometry in general? Of analytic geometry? Do "real mathematicians" not say "trigonometry"?
The discussion of congruency was apparently ripped out of this page, and poorly done, as a later paragraph refers to "figures" which are not present on this page, but the context is compatible with the figures on the congruency page. Brent Gulanowski 17:59, 30 Nov 2003 (UTC)
Should there perhaps be some mention of geometric series here? There is a redirect from geometric to geometry and I suspect it does get linked when talking about things like geometric growth. --fvw 11:46, 2004 Jul 27 (UTC)
"definitions"
In the opening paragraph to this article I find the wording "in conjunction with mathematical definitions for points, straight lines[...]" problematicsince points and lines, among other things, are usually left as undefined terms in Euchlidean geometry. Servais 17:08, 27 Nov 2004 (UTC)
Though these definitions are often absent from a course in Euclidean geometry, Euclid himself defines them (see [1] (http://aleph0.clarku.edu/~djoyce/java/elements/bookI/bookI.html#defs)), though it has often been argued that his definitions aren't that much good! However, in the context of being only 3 or 4 generations after Leucippus and 2 or 3 after Democritus, Euclid's "a point is that which has no parts" could have been a reasonable stance in stating Euclid's position with respect to atomic theory. Again, it may be useful to have a "history of geometry" section because it is sometimes hard to distinguish between Euclid's own geometry and the framework that became "Euclidean geometry".
It might also be useful to cite Felix Klein's Erlangen Program since this definition was seminal in balancing Euclidean and non-Euclidean geometry back into a single subject. (Sorry, I couldn't get this link to 'take', but there is a good article there already.) Dominic Widdows 19 Apr 2005
A better introduction
Algebra and Geometry are the 2nd largest branches of mathematics (after analysis). It's possible to divide geometry into six main branches:
1) Euclidean geometry (the basic geometry)
2) Analytic geometry (could also be an offshoot of algebraic geometry but is more basic)
The more advanced and complex branches are:
- Probably Riemannian geometry and symplectic geometry and other geometries that live on manifolds should be sub-branches of differential geometry. -Lethe | Talk 19:32, May 30, 2005 (UTC)
6) Non-Euclidean geometries; this includes:
6a) Riemannian geometry (also called elliptic geometry) and 6b) Lobachevsky-Bolyai-Gauss Geometry (also called hyperbollic geometry). Basically the Riemannian metric can account for an infinite number of non-Euclidean geometries.
- Why are these seperate branches? Can't those hyperbolic geometries be modeled on Riemannian manifolds? -Lethe | Talk 19:29, May 30, 2005 (UTC)
7) Topology which is the youngest and most sophisticated branch of geometry which also meets modern algebra.
- I don't agree with you that Topology should be considered a branch of geometry. -Lethe | Talk 19:29, May 30, 2005 (UTC)
8) Noncommutative geometry is currently an active area of research, especially in the area of advanced theoretical physics. -- Orionix 04:36, 13 Mar 2005 (UTC)
- And where are symplectic geometry and Kähler geometry on your list? -Lethe | Talk 19:32, May 30, 2005 (UTC)
Topology is no longer young. Noncommutative geometry isn't really geometry - it has a geometric language, but so does (for example) Boolean algebra theory. It might be better to concentrate on Euclidean geometry, algebraic and differential geometry, and geometric topology, as the basic classification.
Charles Matthews 22:04, 22 Mar 2005 (UTC)
Everything which explores structure & spatial relationships has a 'geometric language'. -- Orionix 16:40, 8 Apr 2005 (UTC)
the mention of pick's theorem on this page is highly inappropriate. (and suffers from spelling issues). It should go somewhere else. Dmharvey 14:45, 30 May 2005 (UTC)
Fans of Pick's theorem would be disappointed. Seriously, it isn't far from geometry of numbers, which might be worth a mention. But the whole page probably should be reconsidered, since it has probably been given little attention. Charles Matthews 15:46, 30 May 2005 (UTC)
