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

orthojoin

Expand Messages
  • Steve Sigur
    In ETC points 1512 through 1568 are the orthojoin of various points This whole section seems irrelevant to anything (most are what I call lonely points
    Message 1 of 4 , Mar 3 8:51 AM
    • 0 Attachment
      In ETC points 1512 through 1568 are the "orthojoin" of various points


      This whole section seems irrelevant to anything (most are what I call
      "lonely points" which little connection to other points or lines).

      For some points (such as the Gossard point) their lonliness is a
      descriptive quality. For these points it seems to indicate their
      irrelevance.

      I call them "silly points."

      I plan to drop them from my next pictures of the distribution of
      triangle points.

      Would anyone out there like to defend these points or the orthojoin
      operation.

      Steve
    • Jeff Brooks
      ... points ... call ... orthojoin ... Dear Steve, The orthojoin stems from the orthopole which seems a neat geometric construct. I am hesitant to dismiss the
      Message 2 of 4 , Mar 3 10:01 PM
      • 0 Attachment
        > In ETC points 1512 through 1568 are the "orthojoin" of various
        points
        >
        >
        > This whole section seems irrelevant to anything (most are what I
        call
        > "lonely points" which little connection to other points or lines).
        >
        > For some points (such as the Gossard point) their lonliness is a
        > descriptive quality. For these points it seems to indicate their
        > irrelevance.
        >
        > I call them "silly points."
        >
        > I plan to drop them from my next pictures of the distribution of
        > triangle points.
        >
        > Would anyone out there like to defend these points or the
        orthojoin
        > operation.
        >
        > Steve
        >

        Dear Steve,

        The orthojoin stems from the orthopole which seems a neat geometric
        construct. I am hesitant to dismiss the orthojoin. The coordinates
        given in ETC are quite ugly, but that should not detract from the
        possible uses of the operation.

        Sincerely, Jeff
      • Steve Sigur
        ... Jef, So what are the possible uses? A clear thought process is based on good fundamentals. I suspect that the orthojoin is not one of these. Look at how
        Message 3 of 4 , Mar 4 9:17 AM
        • 0 Attachment
          On Mar 4, 2006, at 1:01 AM, Jeff Brooks wrote:

          > The orthojoin stems from the orthopole which seems a neat geometric
          > construct. I am hesitant to dismiss the orthojoin. The coordinates
          > given in ETC are quite ugly, but that should not detract from the
          > possible uses of the operation.
          >
          > Sincerely, Jeff

          Jef,

          So what are the possible uses?

          A clear thought process is based on good fundamentals. I suspect that
          the orthojoin is not one of these.

          Look at how algebra has developed. There are only a few operations
          out of which more complicated operations can be expressed. When we
          wish a complicated operation, we invent a function on an ad hoc
          basis. Then we forget that function as in other contexts.

          But I am suspicious that the orthojoin dismisses itself. Except for
          ad hoc uses, I suspect I will not miss much by not learning this
          operation. The Steiner inverse is fundamental. One misses a lot by
          not adding it to one's repetoire, but, again except for ad hoc uses,
          I suspect that orthojoin is not.

          I would love to be convinced otherwise.

          Regards to OK from the Eastern US,

          Steve





          [Non-text portions of this message have been removed]
        • Jeff Brooks
          ... geometric ... coordinates ... that ... for ... uses, ... Hey Steve, Please let me digress a bit before turning back to the orthojoin operation. I ve been
          Message 4 of 4 , Mar 5 2:09 AM
          • 0 Attachment
            --- In Hyacinthos@yahoogroups.com, Steve Sigur <s.sigur@...> wrote:
            >
            >
            > On Mar 4, 2006, at 1:01 AM, Jeff Brooks wrote:
            >
            > > The orthojoin stems from the orthopole which seems a neat
            geometric
            > > construct. I am hesitant to dismiss the orthojoin. The
            coordinates
            > > given in ETC are quite ugly, but that should not detract from the
            > > possible uses of the operation.
            > >
            > > Sincerely, Jeff
            >
            > Jef,
            >
            > So what are the possible uses?
            >
            > A clear thought process is based on good fundamentals. I suspect
            that
            > the orthojoin is not one of these.
            >
            > Look at how algebra has developed. There are only a few operations
            > out of which more complicated operations can be expressed. When we
            > wish a complicated operation, we invent a function on an ad hoc
            > basis. Then we forget that function as in other contexts.
            >
            > But I am suspicious that the orthojoin dismisses itself. Except
            for
            > ad hoc uses, I suspect I will not miss much by not learning this
            > operation. The Steiner inverse is fundamental. One misses a lot by
            > not adding it to one's repetoire, but, again except for ad hoc
            uses,
            > I suspect that orthojoin is not.
            >
            > I would love to be convinced otherwise.
            >
            > Regards to OK from the Eastern US,
            >
            > Steve
            >
            Hey Steve,

            Please let me digress a bit before turning back to the orthojoin
            operation.

            I've been thinking a lot lately about how different geometric
            constructions yield the same function and conversely how different
            functions can provide the same geometric meaning. As you know, there
            was some discussion recently about Keith Dean and Floor van Lamoen's
            paper on "Geometric Construction of Reciprocal Conjugations." I read
            with great interest the different approaches each person took in
            trying to help me to better understand the development of this
            generalized conjugation. One person pointed out that a particular
            step was used in the authors' paper to simply `motivate' the
            development of the subsequent material. I was also provided some
            direction on why I should not bother to learn more about this
            particular step. I truly understand these sentiments especially in
            light of my poorly worded questions and comments.

            To quote Henri Poincarẻ "Mathematicians do not study objects, but
            relations among objects; they are indifferent to the replacement of
            objects by others as long as relations do not change. Matter is not
            important, only form interests them."

            And, perhaps, this is where I messed up. I did not explain that I
            was trying to explore a particular form. I honestly don't care
            whether a proof is given analytically or synthetically because I know
            that I can work from either. Having said this, let me also state
            that it is typically easier for me to follow along with pictures of
            some sort.

            For example, I have constructed the inverse in the circumcircle using
            basic geometric methods and I now have a `feel' for this operation
            that I can relate to similar problems or situations. I know of
            several different formulas for the inverse in the circumcircle, but I
            don't know why Clark Kimberling chose to use Peter Moses'
            formulations to highlight some common inverses given in the
            Encyclopedia of Triangle Centers. I suspect he has a reason.

            Likewise, I am confident that I can use Antreas' theorem given in
            Hyacinthos message #11327 to construct points of the form P/(X U)
            because I have a lot of evidence to support this conjecture. This
            seems more general than the definition of reciprocal conjugate. But
            even if I prove this, I would still be left wondering about that
            basic step I missed earlier in Floor and Keith's paper. I would also
            be left wondering how this might relate to the discussions you and
            Peter have had regarding conics.

            Steve, I've not explored the orthojoin operation and I don't know
            what motivates it. I too would love to have a better understanding
            of the operation, where it comes from, how it is used, etc.

            "To see what is general in what is particular and what is permanent
            in what is transitory is the aim of scientific thought." Alfred North
            Whitehead

            Sincerely, Jeff
          Your message has been successfully submitted and would be delivered to recipients shortly.