Loading ...
Sorry, an error occurred while loading the content.
 

[sl] Re: 3D geometry

Expand Messages
  • super_romeman
    Edwin A. Abbott s Flatland is on-line in various places, including here: http://nedwww.ipac.caltech.edu/level5/Abbott/Abbott_contents.html ... constitue ...
    Message 1 of 16 , May 14, 2002
      Edwin A. Abbott's "Flatland" is on-line in various places, including
      here:

      http://nedwww.ipac.caltech.edu/level5/Abbott/Abbott_contents.html

      --- In sacredlandscapelist@y..., "J Vincent Beall" <vincent@d...>
      wrote:
      > Well, our physical space-time is 4-dimensional, but time does not
      constitue
      > a whole dimension in the physical model, as I understand it. It is
      like a
      > half dimension because objects in our space-time don't seem to have
      > extension into negative time. Time in our universe seems only to be
      > positive.
      >
      > There is a book called Flatland that demostrates the intuitive
      chain of
      > logic that proposes the properties of dimesions higher than three.
      It is
      > assumed that all unfolding of dimension always adheres consistently
      to the
      > same set of rules. So, we need only consider the the unfolding of a
      simple
      > form like a tetrahedron from zero to three dimensions in order to be
      > confident of the rules which apply to unfoldment to four-dimensions.
      >
      > This contemplation of unfolding a tetrahedron from 0 to 4-D was
      explained to
      > me as Einstein's puzzle, but that might be just to make it a bit
      more
      > interesting. :)
      >
      > Mathematical dimensions are static and have no metaphysical
      consideration of
      > "substance" to complicate them. Physical space however depends in
      its being
      > on the mystery of "substance" even thought it is completely ignored
      in
      > scientific study, since all explanations of 'what the "substances"
      of our
      > experience are' would "turtle all the way down", so to speak.
      >
      > It might be said that we can know that we are substance, but we can
      not know
      > what that substance is. In otherwords there is a duality between
      description
      > and being. We are beings and not the descriptions.
      >
      > Vincent
    • mswaney
      ... Well, my views are just that, my views, I m not a physicist, just a lowly engineer. I don t want to be criticizing anyone s ideas here, these are not
      Message 2 of 16 , May 16, 2002
        > > time is not an ordinary dimension.
        >
        > I'm not sure I see that. Time is a measurable axis on which we move.
        > It's pretty much a dimension, like any other dimension. What's
        > interesting and unique are 1) how we perceive time, and 2) the ways
        > we move through time.
        >
        > > However, there is a big difference between spatial dimensions and
        > the
        > > temporal dimension. That's why we can't measure time in inches.
        >
        > On what do you base that statement? Why isn't time spatial? I am not
        > currently able to measure it in inches because I don't know how to
        > hold my tape measure. I think if you could figure that out, you could
        > not only measure time in inches, but it would be more accurate and
        > more relativistically sound than measuring in minutes.

        Well, my views are just that, my views, I'm not a physicist, just a lowly
        engineer. I don't want to be criticizing anyone's ideas here, these are not
        obvious questions and there are bound to be differing takes on the issue.

        But it seems to me that time can't be a spatial dimension, regardless of the
        commonly heard expression of time as the "fourth dimension". Dimensions
        don't have to have all the same qualities. They can be very different from
        each other, and the mathematicians call them dimensions because the
        variables that represent "dimensions" are treated in the same way. Time as
        the "fourth dimension" grew out of the theory of Relativity that basically
        says that you cannot be accurate (when dealing with speeds that are "close"
        to that of light) spatially without also including time in the calculation,
        and so space and time are connected.

        But that doesn't mean that space and time are the same. For one thing, as
        noted previously, it is possible to travel in all directions in space at any
        speed, while we are imbedded (it seems) in time and cannot travel in time in
        any direction except "foward", and cannot alter the flow of time (unless
        again, we travel in space at speeds approaching that of light, and then,
        relative to a fixed observer, in the observer's frame of reference we have
        apparently altered the flow of time).

        I think that time is not space, though both are "dimensions", and both are
        linked together in the way that relativity describes. But that's just my
        opinion, I like to hear other thoughts as well.

        Mark
      • Neil Fernandez
        In message , mswaney writes ... I think a good handle on how space-time theory treats time
        Message 3 of 16 , May 16, 2002
          In message <000e01c1fd2a$166cfe20$d9acb882@mtl2>, mswaney
          <mswaney@...> writes

          <snip>

          >But it seems to me that time can't be a spatial dimension, regardless of the
          >commonly heard expression of time as the "fourth dimension". Dimensions
          >don't have to have all the same qualities. They can be very different from
          >each other, and the mathematicians call them dimensions because the
          >variables that represent "dimensions" are treated in the same way. Time as
          >the "fourth dimension" grew out of the theory of Relativity that basically
          >says that you cannot be accurate (when dealing with speeds that are "close"
          >to that of light) spatially without also including time in the calculation,
          >and so space and time are connected.

          I think a good handle on how space-time theory treats time differently
          from the spatial dimensions (I'm not saying this is the 'correct' way!)
          is given by the formulae for distance. If the spatial dimensions are
          called x,y,z, and the temporal dimension is called t, then

          -distance (1D) from 0 to x is x
          -distance (2D) from (0,0) to (x,y) is sqrt(x^2 + y^2)
          -distance (3D) from (0,0,0) to (x,y,z) is sqrt (x^2+y^2+z^2)
          -distance (4D) from (0,0,0,0,) to (x,y,z,t) is sqrt(x^2+y^2+z^2-(ct)^2)
          where c is the velocity of light in a vacuum. The "-" is not a typo! :-)

          It is interesting that if you want to take a distance in time and do
          something to it so that it makes the fourth formula above an intuitive
          extrapolation from the first three, then you have to multiply the
          distance in time by ic, where i is the square root of minus 1 (or to be
          exact, it can be either of the two square roots of minus one), and c is
          the velocity of light in a vacuum.

          Whilst time is not the fourth Euclidean dimension (because the fourth
          Euclidean dimension is pulled out of the first three by standard rules,
          the main one being that it is at right angles to the other three - the
          fourth Euclidean dimension exists and it is not time), it is reasonable
          to call it a dimension, I feel, because you can get it by pulling in a
          direction...(point pulled to make a line segment; line segment pulled to
          make an area; area pulled to make a solid; solid pulled through time to
          make, well, the life history of a solid in time :-) ).

          That you can get distance in time to behave in some respects similarly
          to a distance in the Euclidean fourth dimension by multiplying it by i
          (a number invented and found useful in 2D, i.e. the application of L at
          right angles to itself; and multiplication by which, in 2D, defines
          rotation through a right angle), and by this weird constant c which has
          the compound dimensionality LT^(-1) (i.e. 1D-space divided by time)
          is...well...awesome... I do not share the physical worldview but it's
          worth noting that in that worldview c is not just *a* constant which has
          spatio-temporal dimensionality, it is *the* constant in that category.
          So you take i (the 'pure right-angle operator'), c (for physicists, the
          single 'universal' constant with space/time dimensionality), apply these
          to a distance in time...and you get something which behaves like a
          Euclidean dimension for the purposes of calculating distances in
          'space-time'. (The latter idea by assumption of course has conceived a
          priori of time as being in the same ballpark as space).

          The involvement of c in the relativity equations posits the question,
          what is a dimension, where 'dimension' is meant in the word's other
          sense, that of M,L,T, etc., the sense in which the constant c has the
          dimensions ML^(-1) - something about which I've long thought physics
          can't get its head round properly...

          Neil
          --
          Neil Fernandez
        • Neil Fernandez
          In message , Neil Fernandez writes ... [...] ... Apologies, I meant LT^(-1) :-) Neil -- Neil
          Message 4 of 16 , May 16, 2002
            In message <gLPSEKA$eE58EwE9@...>, Neil Fernandez
            <neil@...> writes
            >In message <000e01c1fd2a$166cfe20$d9acb882@mtl2>, mswaney
            ><mswaney@...> writes
            [...]
            >The involvement of c in the relativity equations posits the question,
            >what is a dimension, where 'dimension' is meant in the word's other
            >sense, that of M,L,T, etc., the sense in which the constant c has the
            >dimensions ML^(-1)

            Apologies, I meant LT^(-1) :-)

            Neil
            --
            Neil Fernandez
          • johnberger_x
            ... ^2) ... typo! :-) I am going to have to take your word for this. ;-) The hard math is above my head. My intuitive self urges me to think of time in more
            Message 5 of 16 , May 16, 2002
              --- In sacredlandscapelist@y..., Neil Fernandez <neil@b...> wrote:
              > In message <000e01c1fd2a$166cfe20$d9acb882@mtl2>, mswaney
              > <mswaney@e...> writes
              > -distance (4D) from (0,0,0,0,) to (x,y,z,t) is sqrt(x^2+y^2+z^2-(ct)
              ^2)
              > where c is the velocity of light in a vacuum. The "-" is not a
              typo! :-)

              I am going to have to take your word for this. ;-) The hard math is
              above my head. My intuitive self urges me to think of time in more
              spatial terms (which is a characteristic of and perhaps therefore
              derivative of consciousness according to Julian Jaynes), but my
              intuitive sense is not always right and can't hold up in this sort of
              debate. :D

              -j
            • johnberger_x
              ... lowly ... these are not ... issue. I m not even an engineer, but in fact a heedless writer/artist with a prediliction for stepping in over my head. ;-) No
              Message 6 of 16 , May 16, 2002
                --- In sacredlandscapelist@y..., "mswaney" <mswaney@e...> wrote:
                > Well, my views are just that, my views, I'm not a physicist, just a
                lowly
                > engineer. I don't want to be criticizing anyone's ideas here,
                these are not
                > obvious questions and there are bound to be differing takes on the
                issue.

                I'm not even an engineer, but in fact a heedless writer/artist with a
                prediliction for stepping in over my head. ;-) No criticism is
                intended -- but I do love a good arguement.

                > noted previously, it is possible to travel in all directions in
                space at any
                > speed, while we are imbedded (it seems) in time and cannot travel
                in time in
                > any direction except "foward", and cannot alter the flow of time

                Despite being mathematically cowed by Neil, I'll persist in tugging
                on this string for a little while longer. ;-)

                Time travel is seriously thought to be possible within the laws of
                physics as we know them today. Not everyone thinks so, but enough
                there are credible people out who believe it. Experiments using
                lasers have shown that information, at the very least, can be
                transmitted at least a small distance back through time through
                entangled photon pairs
                (http://www.biols.susx.ac.uk/home/John_Gribbin/quantum.htm).

                If you look solely at the big picture, you can't meaningfully alter
                your flow through the physical universe any more than you can
                apparently alter your flow through time. People can't stop the earth
                from orbiting the sun, the sun from revolving around the galaxy and
                the galaxy from zooming away from the site of the big bang. On the
                macro level, our scurrying around the planet is a stastistically
                insigificant variation in our big-picture physical trajectory.

                It is impossible with current technology to ever revisit the place
                in "space" that you occupied one year ago, because the entire solar
                system has moved something like 350-400 million miles since then.

                So.... If we're hurtling through time with the same momentum and on
                the same scale that we're hurtling through space, then it's not
                surprising that it seems impossible to change our trajectory in time,
                i.e. to stop travelling "forward" in time.

                But we can move physically through space in ways that are meaningful
                to us, despite our near motionlessness relative to our vast momentum
                as part of a galactic cluster. I suspect we will learn more about how
                to move through time in ways that are meaningful to us.

                -j
              • Neil Fernandez
                In message , Neil Fernandez writes ... [...] ... I forgot to add that with the fourth Euclidean
                Message 7 of 16 , May 17, 2002
                  In message <gLPSEKA$eE58EwE9@...>, Neil Fernandez
                  <neil@...> writes
                  >In message <000e01c1fd2a$166cfe20$d9acb882@mtl2>, mswaney
                  ><mswaney@...> writes

                  [...]

                  >I think a good handle on how space-time theory treats time differently
                  >from the spatial dimensions (I'm not saying this is the 'correct' way!)
                  >is given by the formulae for distance. If the spatial dimensions are
                  >called x,y,z, and the temporal dimension is called t, then
                  >
                  >-distance (1D) from 0 to x is x
                  >-distance (2D) from (0,0) to (x,y) is sqrt(x^2 + y^2)
                  >-distance (3D) from (0,0,0) to (x,y,z) is sqrt (x^2+y^2+z^2)
                  >-distance (4D) from (0,0,0,0,) to (x,y,z,t) is sqrt(x^2+y^2+z^2-(ct)^2)
                  >where c is the velocity of light in a vacuum. The "-" is not a typo! :-)

                  I forgot to add that with the fourth Euclidean dimension (calling the
                  fourth dimension w), distance from (0,0,0,0) to (x,y,z,w) is
                  sqrt(x^2+y^2+z^2+w^2), which follows on logically/intuitively from the
                  distance formulae for the first three.

                  Neil
                  --
                  Neil Fernandez
                Your message has been successfully submitted and would be delivered to recipients shortly.