A sphere results when a number of lines, all of equal length, with no two lines parallel, are drawn from a single point in three dimensional space. Is there a necessary number of such lines needed to produce a sphere? Is it possible to figure out what the maximum number of such lines could be in this designation of a sphere? Is any attempt to find the maximum number of such lines necessary to describe a sphere yet another example of Zeno’s paradoxes?
Gee, man, I don’t know where you learned math, but that’s the strangest definition of a sphere yet. Standard definitions involve a locus of points equally distant in three-dimensional space from a given point, named the [b]centre[/b] of the sphere.
en.wikipedia.org/wiki/Locus_%28mathematics%29
If you can prove the equivalence of your definition, then maybe we can flesh some things out, but I really thing something’s missing there. 
M.S.,
Are you saying a sphere could not be produced in the way I described?
I take it you imagine the sphere together with its interior, but even so, your definition lacks clarity.
What does it mean to “draw lines from a single point” ? Is that point the centre ? Do you, by that, mean: trying to populate the space around the given point with segments equal in length, so that these accumulate into a homogenous volume, which will form a sphere together with its interior ?
Aside from the WMD of mathematical axioms, look at the simple rules given here for construction of a sphere. An increasing number of lines of the same length, drawn from a single point in 3D space, with no two lines parallel, will eventually produce a sphere. The point from which the lines emege will become the sphere’s center. The ends of the lines at a distance from the point will describe the sphere’s exterior surface. Any tyro at computer graphics can produce a sphere using these rules. Here’s the problem. In the real world lines have width. In an abstract world they may not. Following the computer graphics constuction, is there a limit of the increasing number of lines (1 to infinity), a saturation point of density of lines? If you use a simple prototype consisting of a clay ball for the point and toothpicks for the lines, you will get to a number of usable toothpicks that cannot be exceeded.
The real problem here is that, although a sphere can be constructed in the manner I prescribed using computer graphics, such a sphere must include physical properties that are both real and imaginary. At the point (center) where all possible lines meet, the lines cannot have width; otherwise they would amount to two or more objects occupying the same space at the same time. At the sphere’s surface the lines must have width in order to describe the surface. How much of our logic contains both real and imaginary propositions?
Any segment of a sphere (hemisphere, round-bottomed cone, and so on) could be produced in the exact same way- and since the segments don’t have width, there’s no need for them to ever be drawn outside of the segment, even if an infinite number of lines are drawn and none of them are paralell.
So no, as far as I can see, your description will not ever necessarily result in a sphere- a sphere could be produced that way, overcoming some of the obstacles you’re talking about, but the definition isn’t enough to say ‘this process WILL result in a sphere’.
I think that what Ierrellus is trying to say here is that we attribute abstract properties to real life principals, perhaps, too easily.
We can’t even make a perfect sphere in real life, out of actual materials, because we don’t live in a perfectly zero gravity environment. And then, on the flipside we can make a perfect sphere in a computer environment – but it has no tangible attributes.
Zeno’s paradoxes are simply word games which play on the interchange between the abstract perfections, and real world applications. At the fundamental level, yeah, sure we took a bit of a jump; but like with Descartes, it seemed to pay off.
Thanks, O G,
You understand what I’m saying. May whatever gods you believe in bless you!
Zeno’s paradoxes (paradoxii?!?) are also interesting from the language point of view. We have the language to pose the paradox but not to satisfactorily unravel it. They demonstrate that we have not yet clearly understood the philosophy of language (ooops! bit of a vague and sweeping statement there … sorry).
They remind me of the visual paradoxes of Escher … how can you draw something that looks possible and yet is not? As another thread points out, this leads to theories of perception.
Late to the gathering again, humph! Anyway, abe1ng, can you elaborate on drawing something that looks possible, but isn’t? Anything can be materialistically made to contour exactly to a drawing from my knowledge.
-Imp.,
Thanks for the Escher. I love his work. Right now I’m reading “Godel, Escher, Bach” with much pleasure.