--- 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
> > construct. I am hesitant to dismiss the orthojoin. The
> > given in ETC are quite ugly, but that should not detract from the
> > possible uses of the operation.
> > Sincerely, Jeff
> So what are the possible uses?
> A clear thought process is based on good fundamentals. I suspect
> 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
> 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
> I suspect that orthojoin is not.
> I would love to be convinced otherwise.
> Regards to OK from the Eastern US,
Please let me digress a bit before turning back to the orthojoin
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
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