- 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 > In ETC points 1512 through 1568 are the "orthojoin" of various

points

>

call

>

> This whole section seems irrelevant to anything (most are what I

> "lonely points" which little connection to other points or lines).

orthojoin

>

> 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

> operation.

Dear Steve,

>

> 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- On Mar 4, 2006, at 1:01 AM, Jeff Brooks wrote:

> The orthojoin stems from the orthopole which seems a neat geometric

Jef,

> 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

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] - --- In Hyacinthos@yahoogroups.com, Steve Sigur <s.sigur@...> wrote:
>

geometric

>

> 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

coordinates

> > given in ETC are quite ugly, but that should not detract from the

that

> > 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

> the orthojoin is not one of these.

for

>

> 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

uses,

> 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.

Hey Steve,

>

> I would love to be convinced otherwise.

>

> Regards to OK from the Eastern US,

>

> 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