Talk:Point (geometry)
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[edit] Postulates?
Since points are included in the postulates of Euclidean geometry, should we include those with a special note to the Playfair Axiom?Mrchapel0203 (talk) 05:56, 5 September 2008 (UTC)
[edit] Dimension?
Since we know that a line has length and curvature, it is a 2 dimensional entity. We also know that a point has no length, so therefore it must have infinite curvature, making it a 1 dimensional entity. --anon
- Your definition of "dimension" is wrong. It is not about length or curvature, rather, about how many variables control the object in question. For a line, one variable is enough, so it is one dimensional. The curvature and length have to do with where the line is embedded. If you take it that way, a line has three dimensions, which are length, slope, and curvature. No? Oleg Alexandrov (talk) 10:03, 26 October 2005 (UTC)
Doesn't it seem strange that a point can have 0 dimensions ? When you think of it, it's like it's nothing. I think I challenge that fact. This doesn't mean that I am going to change the article.--Granpire Viking Man 22:20, 14 October 2006 (UTC)
- Strange, but true. It is nothing. Quite simply, it's something with no volume, area, or even length, which makes it a nonexistent physical entity (or at least an infinitely small one). If you challenge the existence of points, by a similar notion you could challenge the existence of many limits, and by extension all of calculus. Disbelieving points is like disbelieving math. Liempt 14:29, 8 October 2007 (UTC)
- Equally strange to me that a line segment can have one dimension. A line is nothing also in the 3 dimensional world. Well, I just think of degrees of freedom - no where to go if you're stuck on a point in space. Tom Ruen 01:20, 15 October 2006 (UTC)
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- This is not really strange when you realize the important difference between an entity and a concept. --Profero 11:59, 14 November 2006 (UTC)
[edit] Random, but must ask...
I'll admit, I've been puzzling about this. Could it be that a line has dimensions surrounding a coordinate, such as the Height of a point is equal to the limit of (any) function of X as the change in (any values) X approaches zero? Sure, it doesn't really say anything except that its height is really really small, but does it seem at all significant? I was just thinking about this along the applications of single-point-energy, but the only person I've ever asked of yet has been my Calc. teacher.. --Kazarian 24.176.171.123
[edit] points
Points are most often considered within the framework of Euclidean geometry, where they are one of the fundamental objects. Euclid originally defined the point vaguely, as "that which has no part". In two dimensional Euclidean space, a point is represented by an ordered pair, (x,y), of numbers, where the first number conventionally represents the horizontal and is often denoted by x, and the second number conventionally represents the vertical and is often denoted by y. This idea is easily generalized to three dimensional Euclidean space, where a point is represented by an ordered triplet, , with the additional third number representing depth and often denoted by z. Further generalizations are represented by an ordered tuplet of n terms, where n is the dimension of the space in which the point is located.
Many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. This is usually represented by a set of points; As an example, a line is an infinite set of points of the form L={}, where through and are constants and n is the dimension of the space. Similar constructions exist that define the plane, line segment and other related concepts.
In addition to defining points and constructs related to points, Euclid also postulated a key idea about points; he claimed that any two points can be connected by a straight line, this is easily confirmed under modern expansions of Euclidean geometry, and had grave consequences at the time of its introduction, allowing the construction of almost all the geometric concepts of the time. However, Euclid's axiomatization of points was neither complete nor definitive, as he occasionally assumed facts that didn't follow directly from his axioms, such as the ordering of points on the line or the existence of specific points, but in spite of this, modern expansions of the system have since removed these assumptions. —Preceding unsigned comment added by Gon56 (talk • contribs) 07:24, 14 June 2008 (UTC)
