• TiredThinker
    819
    If there were a finite number of things on 3 axises, could that same information be represented in a single line? Do higher dimensions exist only for when values go on forever in an axis?
  • jgill
    3.5k
    If its any help, the "number" of points inside a cube is the same as found on one of its defining edges (lines).
  • TiredThinker
    819


    I don't mean number of points in a math object. Lets say miles or representation of number of countable things.
  • Pie
    1k
    If there were a finite number of things on 3 axises, could that same information be represented in a single line? Do higher dimensions exist only for when values go on forever in an axis?TiredThinker

    It sounds like you'd want an invertible function , with finite. That should be easy (an infinity of choices.) One boring but easy approach is ordering the points in the domain set ('alphabetically') and just counting them off like .

    I think what you are really interested in is bases for vector spaces.
    https://en.wikipedia.org/wiki/Basis_(linear_algebra)
  • alan1000
    175
    "If there were a finite number of things on 3 axises, could that same information be represented in a single line?"

    Yes indeed. You might read up on Georg Cantor, who did a lot of work in this area.
  • alan1000
    175
    "If its any help, the "number" of points inside a cube is the same as found on one of its defining edges (lines)."

    Can you develop that a bit further? Points in any continuum are infinitely many.
  • jgill
    3.5k
    There's a one to one correspondence between points on such a line and points in the interior of the cube. You could look it up, it's a simple arithmetic trick involving manipulating the digits of the numbers representing the points.
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