An impossible circle?

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Main Question or Discussion Point

A circle is an infinite series of points equidistant from the center.


but can a circle be composed of an infinite series of lines?
if you draw a square & connect the 4 side, that are equidistant from the center, then draw draw and infinite series of points on those lines. each one of the points on the line is not the same distance from the circle.
know draw a trillion points all equidistant from the center, and draw lines between those points & draw points on those lines. those points are not equidistant.
so a circle cannot be composed of an infinite series of lines. because each would be composed of an infinite series of points each not equadistant from the center.
which says making a circle with an infinite series of lines is impossible.
am i thinking right?
 

Answers and Replies

  • #2
tiny-tim
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Hi yyttr2! :wink:
A circle is an infinite series of points equidistant from the center.


but can a circle be composed of an infinite series of lines?
if you draw a square & connect the 4 side, that are equidistant from the center, then draw draw and infinite series of points on those lines. each one of the points on the line is not the same distance from the circle.
know draw a trillion points all equidistant from the center, and draw lines between those points & draw points on those lines. those points are not equidistant.
so a circle cannot be composed of an infinite series of lines. because each would be composed of an infinite series of points each not equadistant from the center.
which says making a circle with an infinite series of lines is impossible.
am i thinking right?
Yes, that's fine … if any line contains two points, then it contains all points between them, and they can't all be the same distance from the centre :smile:
 
  • #3
Is there a difference between "an infinitesimal line" and "a point"?
 
  • #4
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Is there a difference between "an infinitesimal line" and "a point"?
I think a line ('infinitesimal' or not) always contains an infinite number of points.
 
  • #5
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I'm not sure infinitesimal line really exists. Lines are infinite in extent, by definition.

Now, are you asking whether a line segment with length zero is the same thing as a point? That's an interesting question.
 
  • #6
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I'm not sure infinitesimal line really exists. Lines are infinite in extent, by definition.

Now, are you asking whether a line segment with length zero is the same thing as a point? That's an interesting question.
I guess it's a matter of definitions. If it's a line segment, then its not a point. I don't think you can transform a line segment into a point analytically.
 
  • #7
Now, are you asking whether a line segment with length zero is the same thing as a point? That's an interesting question.
What I had in mind was a limit question, but I guess it does boil down to a line of zero length.

I always thought: Lim(Line) = Point as length ---> 0.

Similar to: Lim([a,a+e]) = a as e ---> 0.

I'm not sure infinitesimal line really exists. Lines are infinite in extent, by definition.
I know infinitesimal is not the right word, but I did not have a better alternative word.

The OP asked whether we can cram an infinite number of lines into a finite length (= the circumference of a circle). We can if we define "Lines" as a closed set, but not otherwise.
 
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  • #8
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I always thought: Lim(Line) = Point as length ---> 0.
I'm not sure that's a proper use of a limit. You can say that with the Taylor series expansion around points (b,a) ON an analytic curve, the error b-a will converge to 0 as a limit for increasing n. The tangent line or line segment at that point will share exactly one point with the curve (or specifically a circle in this case), but that's a different proposition than saying a line segment can contain exactly one point.

A curve like the space filling Peano curve, which is composed of line segments, contains singular points of zero measure. I don't think you can define a line segment as simply a closed set and expect the words 'line segment" to be meaningful. I think of a line segment in terms of the minimum Euclidean distance between two points and a line as the infinite (in at least one direction) extension of a line segment.
 
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  • #9
By "line," the OP clearly meant "line segment" and later posts adopted this convention.

So, I have a simple question: what is the length of the shortest line segment that is not a point?
 
  • #10
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There doesn't have to be a shortest. That's like asking what the real number right below 3 is.
 
  • #11
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By "line," the OP clearly meant "line segment" and later posts adopted this convention.

So, I have a simple question: what is the length of the shortest line segment that is not a point?
I don't know how to specify the length of an arbitrarily short line segment, but it contains an infinite number of points.
 
  • #12
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Unless it has length equal to zero... ?

x = 1 + 2t
y = 3 + 3t
0 <= t <= 1

That defines a line segment from (1, 3) to (3, 6). It has length sqrt(13).

The question is this: does this constitute a line segment of length zero, or is a point, or is it both?

x = 1 + 2t
y = 3 + 3t
t = 1/2
 
  • #13
DaveC426913
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Well, a line segment must contain at least two points, one on each end. According to Wiki (so it must be true), the points must be distinct. That means the endpoints cannot be coincident. And that means there is no such thing as a zero length line.

Only thing left to do to find a more trustworthy source for the definition of a line segment.
 
  • #14
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Huh. That seems to make sense. Perhaps Euclid has something to say about lines (and line segments as well)...
 
  • #15
tiny-tim
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Euclid

Perhaps Euclid has something to say about lines (and line segments as well)...
This is the html version of the file http://voteview.ucsd.edu/logical_systems.pdf.
Google automatically generates html versions of documents as we crawl the web.


Euclid’s Postulates​

1. A straight line segment can be drawn joining any two points.

2. Any straight line segment can be extended indefinitely in a straight line.

3. Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.

4. All right angles are congruent.

5. Parallel Postulate: For every line l and for every point P that does not lie on l there exists a unique line m through P that is parallel to l.
 
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  • #16
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I don't think Euclid is of much help here. This is a problem of a continuum vs a point. It's clear Euclid considered a line as distinct from a point. You can certainly have singular points of zero measure on a continuum as with the Peano curve, but if an isolated point can be a line segment of zero measure, then this raises problems as to what a continuum really is. A line segment is a continuum. An isolated point is not. (Also see post 8.)
 
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  • #17
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Well, two points...

When Euclid says "two points", I suppose he means two distinct points. However, nothing he says really requires that... does it?

And SW VandeCarr:
It would be a simple enough matter to let zero-length line segments be the exception to the rule, call them degenerate line segments, or something. In graph theory, for instance, you often neglect graphs on zero or one vertex because they provide silly counterexamples for more general theorems. But nobody would argue they're graphs. Maybe the same thing is going on here?
 
  • #18
If we define line segments not to include points, wouldn't that create a problem with defining the zero vector, for example?
 
  • #19
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Well, two points...

SW VandeCarr:

It would be a simple enough matter to let zero-length line segments be the exception to the rule, call them degenerate line segments, or something. In graph theory, for instance, you often neglect graphs on zero or one vertex because they provide silly counterexamples for more general theorems. But nobody would argue they're graphs. Maybe the same thing is going on here?
I suppose we are free to define special cases, but is it necessary? I think it's better to define a circle from first principles: the set of all points equidistant from a given point in a plane. The equilateral n-gon can then approximate the circle through the Taylor series expansion such that the limit is the circle.
 
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  • #20
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If you define the circle as all the points at a given distance from the center you do not need the line segments anymore, the circle will be continuous.
If you use line segments you can argue if a line can consist of one point. Then a point is also a line, but these lines have lost the most important properties of lines, like a slope. You have to be very careful when comparing circles and approximating polygons you quickly get into the theory of measures. For example if you take all regular polygons which have a corner on the coordinate line and subtract the corner points from the circle, it would still be a closed circle in a physical sense, although you have taken away an infinite number of points.
 
  • #21
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If you define the circle as all the points at a given distance from the center you do not need the line segments anymore.
Of course. We have a perfectly good way of defining a circle. Why define a problematic special case of a line segment (which is not a line segment at all) when there is no need to do so?

If you consider the plane as a point set and define a regular polygon on the plane, then you have also defined a central point on the plane and an implicit circle circumscribing the polygon. With standard analytic techniques the n-gon will merge with that circle at the limit. There is no need to define strange beasts like a "line segment" that consists of exactly one point.

The point (no pun intended) is that you need at least an implicit continuous function in order to take a limit using standard analytic techniques.
 
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  • #22
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In some situations it may be convenient to include the zero-length line segment as a valid line segment, just so you don't have to include special cases. For instance:

What is the locus of points swept out by line segments with endpoints p and q where p = (2t + 7, 3t^3) and q = (ln(|t| + 1) + 7, exp(t) - 1) and -1 < t < 1.

Notice that when t=0, p and q will be equal. So you'd have to say something very awkward in this case in order to even ask the question properly. Unless, of course, you let the reader figure this out on their own, as is done in Graph Theory already.
 
  • #23
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In some situations it may be convenient to include the zero-length line segment as a valid line segment, just so you don't have to include special cases.
OK. There are certainly zero scalars, vectors and tensors of higher rank. It seems in your example t is actually a scalar quantity. In terms of the problem at hand, we don't have a definite metric. That's why questions about length are not appropriate here. We need a (conceptual) tangent line (not point) to a continuous (differentiable) function in order to take a limit. A point doesn't have a slope (and you don't need a coordinate system to evaluate a slope; just a general orientation.)
 
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