Re: Is a line segment composed of points?

On Nov 9, 11:20 pm, vred...@xxxxxxxxx wrote:
On Nov 10, 8:37 am, Lester Zick <dontbot...@xxxxxxxxxxx> wrote:



On Fri, 09 Nov 2007 09:58:44 -0800, vred...@xxxxxxxxx wrote:
On Nov 8, 11:24 pm, Lester Zick <dontbot...@xxxxxxxxxxx> wrote:
On Wed, 07 Nov 2007 21:39:15 -0800, vred...@xxxxxxxxx wrote:
On Nov 8, 5:55 am, Virgil <vir...@xxxxxxxxxxx> wrote:
In article <1194471215.470102.253...@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
David R Tribble <da...@xxxxxxxxxxx> wrote:
venkat wrote:
A point seem to represent a "split" or absence of material and as well
as the content or existence of the material.

David R Tribble wrote:
Interesting analogy, which allows us to say that an uncountable
interval [a,b] consists of an uncountable set of points [a,c)
and another uncountable set of points (c,b] surrounding the
zero-width "crack" at point c. The distance between the two
intervals, i.e., the width of the crack, is exactly zero.

venkat wrote:
Yep, that's exactly what I'm saying. A location or point has no
magnitude or extent, and it is not even infinitesimally small, and
hence it can't be a building block for a finite extent line segment.
This "line is composed of points" concept was simply comforting to
people by relating spaces of different dimensions by seeing that an
higher space is "built" by the building blocks of lesser dimensional
spaces. I'm saying this understanding is incorrect. You can never
build a N dimensional object by using blocks of lesser dimension.

If, as you say, a line segment is not in fact composed of an
uncountable infinitude of zero-width points, then what /is/ it
composed of?

What is the thing where two intersecting lines intersect?

Maybe he has a new geometry in which lines can cross without
intersecting! Or maybe not.- Hide quoted text -

It doesn't need a different geometry. Since both lines and points are
imaginary objects of zero width, lines do intersect in points.
However, I don't think the fact that lines intersect in points is
enough to indicate that lines are composed of points. My problem was
the finite "length" of the line and its contents. Zero lengths simply
do not make up finite length whether it is adding up or union or
summing up - whatever you call. Infinitesimal lengths do build up a
finite length. So it is better to say a point has infinitesimally
small length, area, volume etc.

Points of zero length, lines of zero width, plane of zero thickness
simply do not exist, not even by imagination - because zero stands for
"nothing ", non-existence - or say don't exist.

In general I agree with you but you might try framing the analysis
this way:

Is a solid composed of surfaces or defined by surfaces?

Is a surface composed of lines or defined by lines?

Is a line composed of points or defined by points?

These questions make lot of sense to me. I would accept the second
part of each of these questions - that is, a line is defined by its
end points but not composed by points as its contents. I can see a
question to me here - can points exist as "end points"? According to
my earlier post, since a point has zero extent in all dimensions, it
should cease to exist. Then how can points exist as boundary of a
line? They can, because boundary is just a seperation of a region from
outside. A seperation or split doesn't have any extent, and points can
serve as boundary of a line segment.

As an example, if a measuring scale is cut into two pieces at 6 inch
mark, the mark now belongs to both pieces and it serves as boundary
point. So a point can be split into multiple points, but they can't
make up a fininte extent.

Well I can generally agree without getting into philosophy. If points
represent intersections of lines then we can alway take the definition
of line segments to correspond to intersections of lines with lines.

Time to make some basic observations now. Not sure if these are
correct, but I can't see why they should be incorrect either. Sorry
for not using formal mathematical terminology, but I can always
clarify anyway.

In the abstract geometrical world, all that we have is regions.
Regions of lesser dimension serve as borders for regions of higher
dimension (objects) if they enclose the objects. However, a boundary
is not part of an object. Although a boundary has an extent in its own
dimensions, its extent is zero in the dimensions of the object which
it encloses,.

A line segment is a region of a single dimension object enclosed by
two points as boundary. The end points of the line segment simply
separate what we see as line segment from what it is not. So, the end
points are not part of the line segment.

The same logic holds good for points of intersection for lines - the
intersection points are borders for line segments and are not part of
any line.

This seems to indicate that neither the end points nor the
intersection points are part of the line segments. May be because
there is no "is-part-of" relation between objects of different
dimensions.

Each object can only be cut into objects of same dimension but never
into objects of lesser dimension. This holds good even for uncountable
number of cuts, assuming that you can't cut an object at its boundary
(or end point), since boundary is not part of the object.

Take a line
segment [0,1]. For every natural number
N, make a cut at 1/N. Note that there is no cut omega.
Note further that no cut is at the boundary. How long is
the piece that contains the endpoint 0? What are its
endponts?

- William Hughes


.

Relevant Pages

  • Re: Is a line segment composed of points?
    ... interval consists of an uncountable set of points [a,c) ... hence it can't be a building block for a finite extent line segment. ... Regions of lesser dimension serve as borders for regions of higher ... The same logic holds good for points of intersection for lines - the ...
    (sci.math)
  • Re: Is a line segment composed of points?
    ... Typically only one additional dimension is needed. ... qualities which are far away that best take the meaning of zero and so ... does addressing a short line segment yield ... infinities actually exist on a finite continuum are questionable. ...
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  • Re: A-invariant
    ... corner of intersection of orthogonal basis x, y, z is 0. ... null space has any dimension from 0 to min(r, n-r). ... Mx is in the range space. ...
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