Results 1 to 9 of 9

Math Help - A Triangle With Points Infinitely Distant

  1. #1
    Newbie
    Joined
    Jul 2011
    Posts
    5

    A Triangle With Points Infinitely Distant

    This may sound strange, and of course completely paradoxical, but what exactly are the properties of a triangle which has its points an infinite distance apart?

    I have been toying with dimensions recently, kick started by an idea I had for a puzzle game using more than 3 dimensions, and being the sort of person that always needs a "how", I think that I'm in the process of developing a model for dimesions 3 and above.

    I'll explain that when I think it works.

    On a normal triangle, of the three vectors between the three points, only two of them are needed to pinpoint the location of a point inside its area. Is this true on such a paradoxical shape?

    And please, keep it in laymans terms, I have only just finished my GCSEs...
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Super Member
    Joined
    Dec 2009
    From
    1111
    Posts
    872
    Thanks
    3

    Re: A Triangle With Points Infinitely Distant

    Quote Originally Posted by Splashkay View Post
    This may sound strange, and of course completely paradoxical, but what exactly are the properties of a triangle which has its points an infinite distance apart?

    I have been toying with dimensions recently, kick started by an idea I had for a puzzle game using more than 3 dimensions, and being the sort of person that always needs a "how", I think that I'm in the process of developing a model for dimesions 3 and above.

    I'll explain that when I think it works.

    On a normal triangle, of the three vectors between the three points, only two of them are needed to pinpoint the location of a point inside its area. Is this true on such a paradoxical shape?

    And please, keep it in laymans terms, I have only just finished my GCSEs...
    Hi Spashkay,

    If the three points are infinite distance apart the three sides of the triangle will never meet with each other. Hence the three sides of the triangle (if you call that a triangle) will be parallel to each other.
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Newbie
    Joined
    Jul 2011
    Posts
    5

    Re: A Triangle With Points Infinitely Distant

    Quote Originally Posted by Sudharaka View Post
    Hi Spashkay,

    If the three points are infinite distance apart the three sides of the triangle will never meet with each other. Hence the three sides of the triangle (if you call that a triangle) will be parallel to each other.
    Yes, I know, hence it being a paradox. Im just wondering if using trigonometry, there is any way to prove that the two vectors will never equal the third.
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member
    Joined
    Dec 2009
    From
    1111
    Posts
    872
    Thanks
    3

    Re: A Triangle With Points Infinitely Distant

    Quote Originally Posted by Splashkay View Post
    Yes, I know, hence it being a paradox. Im just wondering if using trigonometry, there is any way to prove that the two vectors will never equal the third.
    The magnitude of the three vectors in this case would be infinite (does not have a bound). Hence the equality or inequality of the three vectors (or their algebraic manipulations) is meaningless, since infinity is a term that does not refer to a exact magnitude.
    Follow Math Help Forum on Facebook and Google+

  5. #5
    Newbie
    Joined
    Jul 2011
    Posts
    5

    Re: A Triangle With Points Infinitely Distant

    Quote Originally Posted by Sudharaka View Post
    The magnitude of the three vectors in this case would be infinite (does not have a bound). Hence the equality or inequality of the three vectors (or their algebraic manipulations) is meaningless, since infinity is a term that does not refer to a exact magnitude.
    I was thinking more in direction than length, as I am trying to see if it is possible to represent three dimensions on a two dimensional shape.
    In three dimensional space, lets say a cube, you need a maximum of three different vectors to be able to get from one point to another. That means lenth is not that important, just the direction. If trigonometry can prove that the same is true for a two dimensional shape, then my model might just work..
    Follow Math Help Forum on Facebook and Google+

  6. #6
    Super Member
    Joined
    Dec 2009
    From
    1111
    Posts
    872
    Thanks
    3

    Re: A Triangle With Points Infinitely Distant

    Quote Originally Posted by Splashkay View Post
    I was thinking more in direction than length, as I am trying to see if it is possible to represent three dimensions on a two dimensional shape.
    This is done in Isometric projection.
    Follow Math Help Forum on Facebook and Google+

  7. #7
    Newbie
    Joined
    Jul 2011
    Posts
    5

    Re: A Triangle With Points Infinitely Distant

    Quote Originally Posted by Sudharaka View Post
    This is done in Isometric projection.
    ...Not what I meant.
    I mean an actual two dimensional shape. Not a three dimensional shape drawn in a two dimensional format.
    I just want to check if my suspicions of the three vectors on said triangle can not equal each other are true, with a proof. If not, it would be helpful to have a proof of it being flase, but of course this is using infinity, so at most im hoping for some insight I can use.
    Follow Math Help Forum on Facebook and Google+

  8. #8
    MHF Contributor

    Joined
    Apr 2005
    Posts
    16,237
    Thanks
    1795

    Re: A Triangle With Points Infinitely Distant

    What geometry are you using? Your "I mean an actual two dimensional shape" seems to imply Euclidean geometry but inn Euclidean Geometry there are NO "points at infinity" so the whole question is meaningless. In elliptic geometry or projective geometry, there are "ideal points" (points at infinity) but then the postulates you can use to prove things are, of course, different.
    Follow Math Help Forum on Facebook and Google+

  9. #9
    Newbie
    Joined
    Jul 2011
    Posts
    5

    Re: A Triangle With Points Infinitely Distant

    Quote Originally Posted by HallsofIvy View Post
    What geometry are you using? Your "I mean an actual two dimensional shape" seems to imply Euclidean geometry but inn Euclidean Geometry there are NO "points at infinity" so the whole question is meaningless. In elliptic geometry or projective geometry, there are "ideal points" (points at infinity) but then the postulates you can use to prove things are, of course, different.
    ...I think I know what you mean, but I think I may have wrapped my head around it anyway.

    If two vectors meet at infinity, then they must be parallel, and so in this case, each of the three vectors in the triangle must be simultaneously parallel: a paradox.
    This means, that each pair of vectors is simultaneously parallel, and the vector which connects them is the third vector, and so all three vectors act as a separate dimension, and so has the properties of a 3D shape.

    At least, thats how it goes in my head. I think I may have posted this in the wrong forum, its not really a problem you can prove with triganometry XD

    If you don't understand what I mean, I'll try and explain it a bit better.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Show that every open interval has infinitely many points
    Posted in the Differential Geometry Forum
    Replies: 5
    Last Post: June 4th 2011, 10:26 AM
  2. Replies: 1
    Last Post: September 9th 2010, 02:33 PM
  3. Replies: 0
    Last Post: September 9th 2010, 02:15 PM
  4. Points on a triangle
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: April 1st 2008, 05:55 AM
  5. triangle points
    Posted in the Trigonometry Forum
    Replies: 2
    Last Post: February 4th 2008, 05:06 AM

Search Tags


/mathhelpforum @mathhelpforum