Why x,y,z?

  1. if the first 3D object has 4 points. why use x,y,z to describe 3D? am i confused about one or more concepts? does a tetrahedron skip 3D? a plane plus another point some where off that plane would immediately provide 4 directions to move. one for each point. could these directions not be used to describe all 3D space?
     
  2. jcsd
  3. as of now 52 people have viewed this thread. not one could be bothered to give me the slightest hint. someone could atleast tell me these questions are goofy. it seems pretty clear i have no background in geometry. i just want to know why we use six directions instead of four, or if my reasoning is off.
     
  4. DaveC426913

    DaveC426913 16,128
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    I think no one can understand your assumptions.
    What does this mean? A 3D object has an infinite number of points. Even a 2D or 1D object has an infinite number of points.

    It requires three axes of measurement define a point's location in 3D space: x, y and z.
    It also requires three axes of measurement to define a direction movement within 3D space (plus another datum to define magnitude of movement.)

    Or simply two sets of coordinates in 3D space to define a transpostition from one point to another. to wit: A point at x=1, y=-1, z=-2 is distinct from a point at x=0, y=0, z=0.
     
  5. micromass

    micromass 18,674
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    I really don't understand what you're trying to say... Maybe the best thing I can do is to translate your statement to the 2D case (where it also applies), I'll let you figure out where your misconception lies:

    In the above, I translated your statement to 2D. Do you still agree with the above quote?
     
  6. DaveC426913

    DaveC426913 16,128
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    Upon rereading your post several times it sounds like you're using a coordinate system that defines a plane with 3 points, then a fourth point off that plane. That can be done but it is excessive. You have one more dimension that necessary.

    Any point on your plane can be uniquely identified with two coordinates (let's call them x and y.), then you need only one more point (let's call it z) to define how far from that plane your point is.
     
  7. i was under the impression a line was defined by two points and a plane three. way back in high school this was the case. i seem to have been misinformed. the first object with hight width and depth derived by points i can fathom is a tetrahedron. each point would mark a direction and moving in any direction u could map 3D. i apologise for my ignorance.
     
  8. micromass

    micromass 18,674
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    Ah no, that makes perfect sense. Indeed, a line is defined by two points. However, you can only move in 2 directions. A plane is defined by three points, and a 3D space is defined by 4 points. However, the last statement doesn't make much sense in our intuition.

    In fact, a n-dimensional space is defined by n+1 points. These n+1 points are called the affine basis of the space. However, it is a curious fact, that we only need n coordinates to represent any point in n-dimensional space. So for 3D-space, we need 4 points to describe the space, but we can give any point by just three coordinates. It's something you need to get used too. If you ever take linear algebra, then this will become very clear!
     
  9. DaveC426913

    DaveC426913 16,128
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    You are correct. Who said otherwise?

    No. That's excessive.

    Let's plot the points of a tetrahedron. A is the top, then B C and D are around the base.

    I could move in the direction of A by, say, three feet, right? B, C and D stay at zero. But look what happens when I try to move in the direction of B: While I am moving away from the B apex, I'm actually heading somewhat downward. I find myself also moving a short distacne in the direction of -A. (To exaggerate: a movement of a mile in the B direction would carry me several hundred yards in the -A direction at the same time.) A single movement in one direction of your coordinate system overlaps with movement in other coordinates. I cannot, if I choose to, move only in the B direction.

    So, to move in the B direction also affects my movement in the A direction. And that indicates that your system has more coordinates than it needs to have.


    Contrarily, in a 3 coordinate system, I am free to move in any of the three directions while having absolutely no effect on the other two. I could travel 10 miles in direction X while my Y and Z movement remains perfectly zero.
     

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    Last edited: Apr 12, 2011
  10. i was hoping some one could set me straight. my mentor said he didn't know why we chose one over the other. he has me reason out most things before he fills in the gaps. i don't see how i could have come to that conclusion on my own. thank you
     
  11. i cant get the multi quote to work, but i can move 3 ft in the A direction without changing B,C, or D. how is that different than moving ten miles in the x direction without changing y or z? barring distance. make a 3d movement say xn yn zn. to do this i would have to move An -Bn -Cn -Dn or something to that effect. it is just simpler to describe with x y z. that is what i get from your explination. i am just using an inefficient coordinate system.
     
  12. DaveC426913

    DaveC426913 16,128
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    1] Select [ Multiquote ] from different posts.
    2] Finally, select [ Quote ] from (any) one of them.

    No. Moving 3 feet in the A direction does move you a short distance in all of the B, C and D directions.

    If you put tick marks along the B, C and D axes, and extend those tick marks both positive and negative, you will see that, as you move along the axis, your B coordinate is also changing. So is your C and D coordinate.
     
  13. Maybe the "first 3D object" is 1/8th of an octahedron? Then you'd have your 4 points and x,y,z axis.

    Although its a lot easier to work with a whole octahedron which looks like a square from any of its axis points.

    FYI, if you take a twisted circle, duplicate and rotate 90 degrees around each axis, these twisted circles form a sort of "blueprint" with which you can derive the octahedron and cube. Where they intersect you can get more shapes. Its a sequence that keeps going. Within this sequence, the tetrahedron comes in pairs. It does not appear as a standalone shape.
     
  14. DaveC426913

    DaveC426913 16,128
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    I've looked through every textbook I have and have not come across a official definition or diagram of a twisted circle.
     
  15. I don't work from textbooks.

    Here's a picture: [​IMG]

    And here's another perspective showing the platonics and how they connect to it: link to image.

    The length of a twisted circle is (pi+(pi*2sqrt))/2.
     
  16. DaveC426913

    DaveC426913 16,128
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    That is apparent. Leading to being unable to communicate your ideas to others. :tongue:

    You've used a phrase of your own making without describing what it means. No one knows what a twisted circle is except you.

    I still don't know. Your diagram seems to show four circles mapped onto a square.

    Or, I suppose if I get imaginative, it could be two figure eights. A figure eight could be a twisted circle I suppose... :rolleyes:
     
  17. isnt a mobius strip a twisted circle?
     
  18. DaveC426913

    DaveC426913 16,128
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    That was one of my guesses.

    I was giving circlemaker enough credit that, if he meant moebius strip, he would have said moebius strip. Still, it has nothing to do with the diagram he posted, nor can I see how a moebius strip could be applied to our discussion.
     
  19. micromass

    micromass 18,674
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    I'm also not aware of something called a twisted circle. I did hear of a twisted cubic, though. This is a very interesting curve in projective geometry. I doubt, however, that this is what is meant with twisted circle...

    So, I welcome circlemaker to clarify his ideas! :smile:
     
  20. Lol, sorry should have clarified. Twisted circle is my shorthand for a circle twisted 180 degrees. It looks like 2 circles from one angle, and a square from another. I might've said figure-8 but I've seen different geometries represent that.
     
  21. DaveC426913

    DaveC426913 16,128
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    Frankly, I still don't follow. I can see 'circle twisted 180 degrees' being a figure 8, but I don't get the 'looks like a square from one angle'.

    Also, does this twisted circle cross at a tangent or does it cross at 90 degrees?
    [​IMG]
    [​IMG]
    (I guess it must be the latter, since the circumference of the former is trivial.)
     
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