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As I understand it, there are three undefined terms (alternatively they are sometimes called primitive notions) in Geometry:

  1. Point: A point has 0 dimensions and merely denotes a location.
  2. Line: A line has only 1 dimension (that of length, but no others) and spans infinitely in both directions.
  3. Plane: A plane has 2 dimensions (length and width, but no height) in which it extends indefinitely in all directions.

However, it seems incongruous with my understanding of the world. A graphical representation of a line for example appears to also have width, however negligible (and my observations about points and planes are similar). And then, these elements are supposed to combine to form 3D solids! I suppose my question is then:

How do you understand and/or think about these concepts? How robust should your understanding be to utilise them effectively?

Additionally, I am also wondering what purpose it serves to describe lines and planes as limitless. Everywhere I have looked prescribes this same quality to them, but it doesn't explain why.

Thank you guys so much!

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  • $\begingroup$ "A graphical representation of a line" is not a line. Ceci n'est pas une pipe. $\endgroup$
    – Blue
    Nov 9, 2015 at 0:32
  • $\begingroup$ Of course! After having studied Magritte, I can't believe I didn't make that connection. But of course, that begs the question: What constitutes a 'true' line? Is it a set of real numbers? Something else? How about a 'true' point or plane for that matter? It seems for now we shall have to make do with imperfect real world analogues and graphical representations of these concepts, while we ponder how they may truly exist theoretically. Anyway, It's all very interesting to think about. Thanks for weighing in! :D $\endgroup$ Nov 10, 2015 at 4:41

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In short, yes, the lines and planes of physical space all appear to have at least some "thickness" to them (we can see them), and they all have finite extent (in our finite universe). The mathematical concept of lines, planes, etc. differ in that they are logical ideals, created to help us reason about geometric relationships. We usually call such concepts Platonic - they may never be realized in nature, but are easily conceived and manipulated in the mind's eye.

To think about a one-dimensional line, you imagine a regular line passing through space (perhaps fishing line) which is thick enough to be visible, but should you zoom in on it you find that its width does not grow. It is thick enough to see, but closer inspection does not render it any thicker! Should you find yourself skewered by a line you could zoom in and see it passing between the atoms making up your torso (for the moment surrendering to the Bohr model of the atom). Conversely, you can zoom out and still see it from outer space, apparently glancing off the surface of the earth. The properties of being limitless and without thickness allow us to use lines as models when plotting the course of a space shuttle, as well as when describing the structure of a molecule. There is no smallest or longest line imaginable; the concept can be applied on any scale.

Your question of how to build a two-dimensional object out of one-dimensional (or zero-dimensional) objects is a very good one, and neighbors many other subtle questions. A course on measure theory may help you get your answers, but I don't think it's necessary for you to think productively about geometry.

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  • $\begingroup$ Thank you for your answer! If I'm reading it correctly it would imply that these ideas are abstract entities that operate by their own set of laws, separate from our world. It would also seem that some intuition or the suspension of disbelief is necessary: we needn't understand it so completely and thoroughly (or perhaps even wholeheartedly agree with it!) for it to be of use to us. $\endgroup$ Nov 8, 2015 at 12:23
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    $\begingroup$ There is a quote from Einstein - "As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." So yes, we usually consider them separate. However, they often appear to be "unreasonably effective." "en.wikipedia.org/wiki/…" You don't need to understand everything, but it is rare for deep understanding to prove ineffectual. $\endgroup$
    – Titus
    Nov 8, 2015 at 23:19

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