Determining Areas of Lines & Points: Suggestions Welcome

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Discussion Overview

The discussion centers around the concept of determining the area of lines and points, exploring whether such areas can be defined or quantified. Participants engage with theoretical implications, mathematical definitions, and proofs related to dimensionality and area.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants suggest that the area of a straight line and a point is defined to be zero, as they are one-dimensional and zero-dimensional shapes, respectively.
  • Others argue that the area for any line or point does not exist (DNE) because area requires at least two dimensions.
  • A participant introduces the concept of fractal curves, questioning whether such curves, which can have a fractal dimension greater than 1, might exhibit characteristics of width.
  • Space-filling curves are mentioned as an interesting topic, with a request for proof regarding their ability to fill a plane.
  • Some participants discuss the use of geometric shapes, such as rectangles and triangles, to illustrate the concept of area approaching zero as dimensions reduce.
  • There is a challenge to provide proofs using triangles to demonstrate that the area of a line is zero, with references to trigonometric functions.

Areas of Agreement / Disagreement

Participants generally agree that the area of a line and a point is considered to be zero or DNE, but there is ongoing debate about the implications of higher-dimensional curves and the validity of different proofs.

Contextual Notes

Some discussions hinge on definitions of area and dimensionality, with participants expressing varying levels of certainty about the applicability of certain geometric proofs. The conversation reflects a mix of formal definitions and exploratory reasoning.

Who May Find This Useful

This discussion may be of interest to those studying geometry, mathematical definitions of area, or fractal geometry, as well as individuals curious about the implications of dimensionality in mathematical contexts.

abia ubong
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i was thinking can anyone findv the area of a straight line and also can anyone determine the area of a point or dot as the case maybe ,any suggestions wiil be appreciated
 
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Areas of such things are declared to be zero, as I suspect you know.
 
The area for any line or point is DNE. Area requires exactly two dimensions.
 
Last edited:
Using any reasonable definition of area, the area of a 1 dimensional set is 0.
 
What happens when the dimension of a curve exceeds 1? An example is the fractal curves of Weierstrass. This curve zig-zags so much that it has infinite slope at every point in its domain. Doesn't this sound like the curve is taking on the "character" of "width" as exhibited by it's fractal dimension which is greater than 1?
 
hypermorphism said:
Also, check out the fun things that are called space-filling curves.

I'm familiar with the curves. Someone here in the group is suppose to be I think proving that the curve fills the plane. I'd like to see that proof.
 
hey hallsofivy is there a proof that their ares are zero if there is send me a private pls same goes 2 anyone who beleives the area is zero.for u jon f wats the meaning of d.n.e?
 
dne is does not exist.

abia, the area is "declared" to be zero, ie it is a deinition that the area of a point is zero, as is the area of a straight line. This makes sense: the area of a rectangle of sides a and b is ab. A line can be thought of as a rectangle of sides a and 0, so the area is zero. If you want to think about an infinitely long line, then we need to invoke some other theory of what areas are, but in any reasonable sense a straight line has zero area. Areas are usually integrals over the set whose area you want to find.
 
  • #10
matt ,can this be proven using a triangle,at least u used a rectangle,pls tell me if a triangle can be used to prove the value
 
  • #11
Area is a property of a 2 dimensional shape. Lines are one-dimensional shapes. So to talk about the area of a line is without meaning.

It’s the same idea as this question being meaningless “what color is loud”.
 
  • #12
given a rectangle c long (constant) and x wide ...

lim c*x
x--> 0

rectangle becomes line and area becomes 0
 
  • #13
or hey sin0 = 0

triangle proof

did i just blow your mind?
 
  • #14
not really noslen i still need more proof,my friend says he has a proof that its 0 using a triangle,so can u help
 
  • #15
abia ubong said:
not really noslen i still need more proof,my friend says he has a proof that its 0 using a triangle,so can u help
noslen answered this.

The angle between the two long sides of this "triangle" is zero degrees.
sin(0) = 0, meaning the opposite side of the "triangle" is of length zero.
Measure the area of the "triangle" (1/2L*H).
The height is zero, thus the area is zero.
 

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