dream_weaver Posted August 14, 2010 Report Share Posted August 14, 2010 (edited) A sphere is defined as the set of all points in three-dimensional Euclidean space that are located at a distance (the "radius") from a given point (the "center"). Twice the radius is called the diameter, and pairs of points on the sphere on opposite sides of a diameter are called antipodes. <snip> Regardless of the choice of convention for indexing the number of dimensions of a sphere, the term "sphere" refers to the surface only, so the usual sphere is a two-dimensional surface. Edited August 14, 2010 by dream_weaver Quote Link to comment Share on other sites More sharing options...
Plasmatic Posted August 14, 2010 Report Share Posted August 14, 2010 I suspect from the above that you are failing to distinguish between the words "bounded" and "boundary". A thing can be both bounded and boundary-less. Consider a circle, an example of a closed curve. Where is the boundary? Where is the beginning of the curve? The end? Finiteness does not entail a boundary. fi·nite /ˈfaɪnaɪt/ Show Spelled[fahy-nahyt] Show IPA –adjective 1. having bounds or limits; not infinite; measurable. 2. Mathematics . a. (of a set of elements) capable of being completely counted. b. not infinite or infinitesimal. c. not zero. 3. subject to limitations or conditions, as of space, time, circumstances, or the laws of nature: man's finite existence on earth. –noun 4. something that is finite. Use finite in a Sentence See images of finite Search finite on the Web Synonyms 1. bounded, limited, circumscribed, restricted lim·it /ˈlɪmɪt/ Show Spelled[lim-it] Show IPA –noun 1. the final, utmost, or furthest boundary or point as to extent, amount, continuance, procedure, etc.: the limit of his experience; the limit of vision. 2. a boundary or bound, as of a country, area, or district Synonyms 2. confine, frontier, border. 8. restrain, bound. bound·a·ry (bo̵un′drē, -də rē) noun pl. boundaries -·ries any line or thing marking a limit; bound; border bound·a·ry /ˈbaʊndəri, -dri/ Show Spelled[boun-duh-ree, -dree] Show IPA –noun, plural -ries. 1. something that indicates bounds or limits; a limiting or bounding line. 2. Also called frontier. Mathematics . the collection of all points of a given set having the property that every neighborhood of each point contains points in the set and in the complement of the set. 3. Cricket . a hit in which the ball reaches or crosses the boundary line of the field on one or more bounces, counting four runs for the batsman. Compare six ( def. 5 ) . Use boundary in a Sentence See images of boundary Noun 1. boundedness - the quality of being finite finiteness, finitude quality - an essential and distinguishing attribute of something or someone; "the quality of mercy is not strained"--Shakespeare Quote Link to comment Share on other sites More sharing options...
Jake Posted August 14, 2010 Report Share Posted August 14, 2010 Absolutely rediculous. The place where one thing ends and another begins is a boundary."Surface" of what? You are divorcing the attribute from the entity and asking "where is the center". This discussion seems to have been made more difficult due to: (1) confusion about the definitions of the various forms of "bound", and (2) sloppiness of context when discussing boundedness. (1) I don't have access to the OED, so you'll have to accept Merriam-Webster Main Entry: boundary : something that indicates or fixes a limit or extent Main Entry: bounded : having a mathematical bound or bounds <a set bounded above by 25 and bounded below by −10> Main Entry: bound 1 a : a limiting line : boundary —usually used in plural b : something that limits or restrains <beyond the bounds of decency> 3 : a number greater than or equal to every number in a set (as the range of a function); also : a number less than or equal to every number in a set One can see that the boundary of an entity is the limit of its extent, regardless of whether or not another entity is at or abutting that boundary. Imagine a droplet of water floating in the emptiness of space. If you were to travel outward from the center of the droplet, you would cross the boundary when you are no longer inside the droplet and start being outside. The boundary is where the water stops and nothing (no thing) starts. (2). The surface of a sphere (or of Earth) is bounded in the sense that the entire surface can be described by a finite coordinate system (i.e. Lat/Long), and it is unbounded in the sense that the distance one can travel in a straight line is potentially infinite. One must be clear about the context of the property "bounded." Finiteness of a certain aspect most certainly entails boundedness in that same aspect. The size of a circle is bounded and the numbers required to describe a position on a circle are bounded (0 < s < 2pi). The unbounded aspect of a circle is travel along the curve, which is also potentially infinite. The meaning of "finite and boundless" may be clear to mathematicians and physicists when discussing circles/spheres, but it involves sloppy context - It's the equivalent of saying I am "brown and green," because my hair is brown and my eyes are green. Quote Link to comment Share on other sites More sharing options...
Mindy Posted August 14, 2010 Report Share Posted August 14, 2010 Your "meanings" are not consistent with each other by the standard of having the same referent. Right. I was asking which meaning was correct. I did that because they were not equivalent/consistent. Mindy Quote Link to comment Share on other sites More sharing options...
Jake Posted August 14, 2010 Report Share Posted August 14, 2010 Another comment about circles/spheres... The metric on the curve of the circle / surface of the sphere is finite (bounded by R*pi in both cases). As shown by plasmatic's dictionary pull, "finite", "bounded", and "limited" are synonymous with respect to a given quantity. Quote Link to comment Share on other sites More sharing options...
Plasmatic Posted August 14, 2010 Report Share Posted August 14, 2010 Before anyone claims it, I know concepts are not their definitions but their referents. Quote Link to comment Share on other sites More sharing options...
Mindy Posted August 14, 2010 Report Share Posted August 14, 2010 The center of a circle does not reside on the circle, so why does the center of a sphere need to? Mindy Quote Link to comment Share on other sites More sharing options...
Trebor Posted August 14, 2010 Report Share Posted August 14, 2010 There once was posted, several years ago, an interesting essay by Alex titled, "The Unbounded, Finite Universe," but the old link (www.geocities.com/rationalphysics/Unbounded_Finite.htm) for it no longer works, and I haven't been able to find it with a Google search. (If anyone knows if it's still available online, please post a link.) Even though that essay is no longer available, that I know of, Harry Binswanger made some comments on the essay, comments which are related to this discussion and which I thought others here might find interesting food for thought. Originally Mr. Binswanger made his comments on his own email list, but Stephen Speicher requested his permission to post it on his forum, The Forum for Ayn Rand Fans. If you're interested, here are Mr. Binswanger's comments (on Alex's essay). Quote Link to comment Share on other sites More sharing options...
Jake Posted August 14, 2010 Report Share Posted August 14, 2010 The center of a circle does not reside on the circle, so why does the center of a sphere need to? Maybe another definition-pull will help here (this time from Wolfram's MathWorld) Center A special point which usually has some symmetric placement with respect to points on a curve or in a solid. The center of a circle is equidistant from all points on the circle and is the intersection of any two distinct diameters. The same holds true for the center of a sphere. The standard "center" of a circle/sphere is right where you think it is, and it defines the circle/sphere. Alfred Centauri was discussing the curve of the circle / surface of the sphere. The definition above does not say the center must be unique, so any point on the circle's curve or sphere's surface can be considered a center, since every point has "symmetric placement" with respect to the metric on the curve/surface (i.e. the sum of the distances from the chosen point to all other points is zero). Mindy 1 Quote Link to comment Share on other sites More sharing options...
brian0918 Posted August 14, 2010 Report Share Posted August 14, 2010 (edited) There once was posted, several years ago, an interesting essay by Alex titled, "The Unbounded, Finite Universe," but the old link (www.geocities.com/rationalphysics/Unbounded_Finite.htm) for it no longer works, and I haven't been able to find it with a Google search. (If anyone knows if it's still available online, please post a link.) The Internet Archive has some archived copies of that page: http://web.archive.org/web/*/http://www.geocities.com/rationalphysics/Unbounded_Finite.htm Edited August 14, 2010 by brian0918 Quote Link to comment Share on other sites More sharing options...
Alfred Centauri Posted August 14, 2010 Report Share Posted August 14, 2010 Compact Manifold A compact manifold is a manifold that is compact as a topological space. Examples are the circle (the only one-dimensional compact manifold) and the n-dimensional sphere and torus. Compact manifolds in two dimensions are completely classified by their orientation and the number of holes (genus). It should be noted that the term "compact manifold" often implies "manifold without boundary," which is the sense in which it is used here. When there is need for a separate term, a compact boundaryless manifold is called a closed manifold. That a circle, sphere, or torus is bounded is, as far as I can tell, not being questioned in this thread. That they are boundaryless evidently is. I hope that the above puts that misconception to rest. @Jake, It's clear that the center of a circle is not in the circle, right? And it's clear that it is special, i.e., unique, right? Thus, how can you say that any point in the manifold can be considered a center? Moreover, AFAIK, distance is a magnitude, i.e., positive (we're not talking about intervals here). Thus, the sum of the distances to all other points cannot be zero unless all "other" points are the chosen point. Quote Link to comment Share on other sites More sharing options...
Mindy Posted August 14, 2010 Report Share Posted August 14, 2010 Maybe another definition-pull will help here (this time from Wolfram's MathWorld) The standard "center" of a circle/sphere is right where you think it is, and it defines the circle/sphere. Alfred Centauri was discussing the curve of the circle / surface of the sphere. The definition above does not say the center must be unique, so any point on the circle's curve or sphere's surface can be considered a center, since every point has "symmetric placement" with respect to the metric on the curve/surface (i.e. the sum of the distances from the chosen point to all other points is zero). I'm not sure I understand your conclusion. A circle is defined as the group of points equidistant from a point in a plane. That point is its center. A circle is not a disc, but the circle that is the circumference of a disc shares centers with the center of the disc. In the case of the disc, the center is a part of the disc itself. In the case of the circle, it is not. I realize we are talking about the surface of a sphere, but, as an analogy to a circle proper, doesn't the center of a sphere's surface coincide with the center of the solid sphere of which it is the surface? Mindy Quote Link to comment Share on other sites More sharing options...
dream_weaver Posted August 14, 2010 Report Share Posted August 14, 2010 The Internet Archive has some archived copies of that page: http://web.archive.org/web/*/http://www.geocities.com/rationalphysics/Unbounded_Finite.htm Thanks Brian0918. Quote Link to comment Share on other sites More sharing options...
Mindy Posted August 14, 2010 Report Share Posted August 14, 2010 (1) Where is the center of the surface of the Earth? (2) If the universe has a boundary, what is it that is ending and what is it that's beginning? While a bound implies that something is ending, does it also require that something is beginning? Mindy Quote Link to comment Share on other sites More sharing options...
Alfred Centauri Posted August 14, 2010 Report Share Posted August 14, 2010 While a bound implies that something is ending, does it also require that something is beginning? Mindy Consider a sphere, the boundary of a ball, does the ball end at the boundary or does it begin at the boundary? Quote Link to comment Share on other sites More sharing options...
Mindy Posted August 14, 2010 Report Share Posted August 14, 2010 The more I read of this the more confused I get. I went back and searched for earlier threads on bounded or unbounded universe, and came across some great stuff. mroctor, 1-16-2010, wrote that "unbounded" means "can increase without limit' but that that doesn't imply infinity, because an unbounded variable always has a particular, finite value. That makes sense to me as a description of the finite universe. What happens as the javelins of the "Two javelin experiment" travel to (or, say, one of them reaches) the furthest point of the universe? Well, if it has enough energy to continue (its progress would be slowed by the gravity of the whole universe, now opposing its motion, and, of course, it started out with a finite amount of energy,) if it has the energy to continue, it continues, and, in doing so, it expands the universe. Presumably it could continue indefinitely, though it would always have traveled a finite distance, and the universe would remain finite in extent. This leaves me struggling for the proper language. Some things are unbounded and lack a boundary, some are bounded but lack a boundary. Some are unbounded and have a boundary, and some are unbounded and have no boundary. If I've got it sorted out correctly, I want to say the universe is unbounded, in that it can expand, and without a boundary, in that there is nothing "outside" or apart from the universe to represent a limit to its extension. Yet, it is always finite. Add all the extended things up, and you get a size, and that is the size of the universe. Same for time, though the highly relative nature of measurement of time needs to be respected. Space is an aspect of existing things. Time is an ordering of changes of all sorts. Extended things undergoing changes make up the universe, so it has a sum of extension and time "within" it. That doesn't contradict Rand's statement that we can't ascribe space or time ...to the universe as a whole. If I understand correctly, she was saying we can't ask where the universe is in space, or when in time. Mindy p.s. Being finite means it has an actual extent. But there is no metaphysical significance to what its present extent is. If there is nothing that limits the expansion of the universe, are the "infinite universe" people satisfied? Quote Link to comment Share on other sites More sharing options...
dream_weaver Posted August 14, 2010 Report Share Posted August 14, 2010 Compact Manifold Interesting distinction. When you mentioned 'center' earlier, obviously the 'center point' of the origin of the radius of the sphere came to mind. To place a point that is coincident with the surface is easily enough described. On a two dimensional circle, the center of the circle is where you stick the compass to strike the circular line. It appears you are inquiring where the center point of the spherical surface is, analgous to if you were to inquire where the middle point on the circular line. Quote Link to comment Share on other sites More sharing options...
dream_weaver Posted August 14, 2010 Report Share Posted August 14, 2010 . . . If there is nothing that limits the expansion of the universe . . . Expand? Into what? The universe is all that is. If the new area it 'expanded' into existed, it was already part of the universe to begin with by definition. Sounds a little pardoxical, does it not? Quote Link to comment Share on other sites More sharing options...
Grames Posted August 14, 2010 Report Share Posted August 14, 2010 I realize we are talking about the surface of a sphere, but, as an analogy to a circle proper, doesn't the center of a sphere's surface coincide with the center of the solid sphere of which it is the surface? Mindy Yes it does coincide. Note that the center of the sphere's surface is not on the surface. By analogy with the "universe", the center of the universe would be outside the universe. But that is impossible. Quote Link to comment Share on other sites More sharing options...
Trebor Posted August 14, 2010 Report Share Posted August 14, 2010 The Internet Archive has some archived copies of that page: http://web.archive.org/web/*/http://www.geocities.com/rationalphysics/Unbounded_Finite.htm Thank you! Quote Link to comment Share on other sites More sharing options...
Alfred Centauri Posted August 14, 2010 Report Share Posted August 14, 2010 Interesting distinction. When you mentioned 'center' earlier, obviously the 'center point' of the origin of the radius of the sphere came to mind. To place a point that is coincident with the surface is easily enough described. On a two dimensional circle, the center of the circle is where you stick the compass to strike the circular line. It appears you are inquiring where the center point of the spherical surface is, analgous to if you were to inquire where the middle point on the circular line. A circle is the one-dimensional boundary of the two-dimensional disk. The center of a circle is actually the center of the disk and is not in the circle. The circle has no boundary. As Wheeler famously said, "the boundary of a boundary is zero". IOW, the circle has no boundary because it is itself a boundary. If one were to consider only the 1-D space that is the circle, there is no center to speak of as there is no center in that space; the center is elsewhere. Similarly, the sphere is the 2-D boundary of the 3-D ball. The center of a sphere is actually the center of the ball and is not in the sphere... Quote Link to comment Share on other sites More sharing options...
Alfred Centauri Posted August 15, 2010 Report Share Posted August 15, 2010 ... and the universe would remain finite in extent. I'm don't think that one can meaningfully talk about the extent of the universe unless it is qualified, e.g., the extent of the observable universe. Surely it's true that the distance between any two entities in the universe is finite and surely it's true that the extent of any entity is finite. I think, in this sense, it is correct to say that the universe is not infinite, i.e., that there are no infinite distances, infinite extents, etc. Nonetheless, there is no logical reason that distances and extents cannot be arbitrarily large, i.e., have no upper bound. Quote Link to comment Share on other sites More sharing options...
dream_weaver Posted August 15, 2010 Report Share Posted August 15, 2010 A circle is the one-dimensional boundary of the two-dimensional disk. The center of a circle is actually the center of the disk and is not in the circle. The circle has no boundary. As Wheeler famously said, "the boundary of a boundary is zero". IOW, the circle has no boundary because it is itself a boundary. If one were to consider only the 1-D space that is the circle, there is no center to speak of as there is no center in that space; the center is elsewhere. Similarly, the sphere is the 2-D boundary of the 3-D ball. The center of a sphere is actually the center of the ball and is not in the sphere... We may be talking past each other here. What you are expressing, I understand. Quote Link to comment Share on other sites More sharing options...
Alfred Centauri Posted August 15, 2010 Report Share Posted August 15, 2010 Expand? Into what? The universe is all that is. If the new area it 'expanded' into existed, it was already part of the universe to begin with by definition. Sounds a little pardoxical, does it not? What if the universe were "shrinking" instead? Which case is more paradoxical? Quote Link to comment Share on other sites More sharing options...
dream_weaver Posted August 15, 2010 Report Share Posted August 15, 2010 What if the universe were "shrinking" instead? Which case is more paradoxical? Since neither sound credulous, it would amount to Galileo's observation of comparing infinities to one another for greater than or less than as essentially meaningless. Quote Link to comment Share on other sites More sharing options...
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