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Re: desmics again
Posted:
Jan 2, 2000 1:56 AM
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on 1/1/00 9:10 AM Floor van Lamoen wrote
>Steve Sigur wrote:
>>
>Steve,
>
>I see no reason to divide the geometry we use in triangle geometry into
>camps.
>
>You wrote: ``I do not think that projective geometry in its traditional
>form says much about triangles and it is triangle geometry that I am
>interested in. More precisely, I am interested in the structures in and
>around one triangle, the one we designate ABC. To do this we
>prioritize.'' And to be honest, I find it hard to believe that you were
>writing this as a comment on what I was saying about very triangle
>linked structures. The only thing I wanted to stress above is that the
>structure is a little more rich than we realize when we mention desmic
>quads (see below).
>
>>
>> The desmic quadrangles do for me what I want. They give order to the weak
>> quartile points, which formerly seemed to have not had much order at all.
>>
>
>It turns out that a nice bunch of weak points can be put into this
>structure, and structure is good. But when you only mention weak points,
>when talking about desmic quads, then you are missing the point imho.
>The structure is all over the place, and in fact any triangle
>perspective to ABC can be extended to a desmic structure.
>One of the places where this structures appears too, is in the 'isogonal
>triangles' (triangle A'B'C' is an isogonal triangle if <BAC' = <CAB'=
>alpha etc.). This triangle is perspective to ABC. We can make the
>'isogonal conjugate' A"B"C" of this triangle (<BAC' = <CAB'= A - alpha).
>The three triangles together with their perspectors make three desmic
>quads.
>
>In triangle geometry we very often work with notions from projective
>geometry. For instance, we often mention perspective triangles. And
>then, most often, we think of this as AA', BB', CC' being concurrent
>giving a perspector. But, the projective theorem of Desargues tells us
>that there is a perspectrix, too. Perspectivity is self-dual.
>
>The desmic quads can be seen as three triangles being pairwise
>perspective, with three collinear perspectors (forming the fourth
>vertices of the quads). The big surprise for me was that, for example,
>ABC/NaNbNc/GaGbGc are desmic triangles, and that their perspectors are
>No, Go and isoH. So indeed it becomes worthwile to consider NoNaNbNc and
>GoGaGbGc as quadrilaterals. Yes, I am still quite excited about this way
>of organizing the weak points!
>Dually, we know that ABC/NaNbNc/GaGbGc have perspectrices. In fact, it
>is only one. In fact, not dually, we find this perspectrix fit in a
>desmic structure arising from triangles ANaGa/BNbGb/CNcGc. Thier
>perspectors lie on the persectrix of ABC/NaNbNc/GaGbGc. Did you realize,
>Steve, that the trilinear pole of this perspectrix is the harmonic
>conjugate of isoH w.r.t. No and Go? For this harmonic conjugate Conway
>coined the name Harmon. Let us consider A'B'C' as the Cevian triangle of
>this Harmon. Let C" be the harmonic conjugate of C' w.r.t. A and B. This
>C" is the perspector of ANaGa and BNbGb. In other words, A"B"C" forms
>the common perspectrix of ABC/NaNbNc/GaGbGc.
>What I find particulary funny is the type of symmetry: We can build the
>two desmic structures on ABC/NaNbNc/GaGbGc, but also on
>ANaGa/BNbGb/CNcGc.
>In this way we also see the ambigious nature of desmic structures: when
>we consider NoNaNbNc/GoGaGbGc/ABCiso-H then it is natural to consider
>these as quadrilaterals, when we consider ANaGa/BNbGb/CNcGc/A"B"C", then
>it is natural to consider these as triangles.
>
>> In the world of projective structures, this is very restricted; but in
>> the world of triangle geometry, this is an important and deep structure.
>
>But we should cross-over to really gain knowledge, and not see triangle
>geometry and projective geometry as two seperated planets. My main
>interest is, other than readers of your comments might think, Euclidean
>geometry, and especially triangle geometry. And I think that is close to
>yours.
>
>> These discussions are very helpful to me.
>
>That's nice. Quite often they are also helpful to me, and I hope in the
>years to follow that we can have many good discussions.
>
Floor,
Again, this discussion is very helpful because I think that something new
has come into existence in the last few years. I am going to try to make
my points in a different way. You will see that there is nothing you say
that I disagree with. But rather there is something else that I am trying
to grab hold of. I am trying to understand for myself what if truely
fundamental in triangle geometry. I am beginning to think that projective
structures come from deeper principles. So be patient with my argument as
I will be struggling to make sense of these thoughts.
But first, I agree with both what you say and my understanding of the
spirit with which it is said. I of course think that we should explore
whatever shows itself as far as we can. And I also know that desmic
situations are really larger parts of, potentially, interesting
structures. Previously I have tried to point out some of these extra
structures and their significance and I have recently tried to apply
desmic structures to the Fermat-like points, a very different situation
from the quartile weak points
........but.........
I see the desmic structures of the quartile weak points as a
fundamentally algebraic statement. I am scared that perhaps the more
important part, the algebra, gets lost.
The extraversion relation is an algbraic one, about what types of
solutions certain equations can have, as well as an expression of an
abstract symmetry of the triangle.
When I see the colinearities that make up the desmic structure for three
quadrangles, I see that it is the same for the group structure of points
on a cubic. It is also clear to me that the structures we call desmic are
often, but not always, related to the existence of a cubic equation.
Hence I want to explore the desmic relation by using two different
algebraic techniques, extraversion and cubics. I tried to give my
alternative view of quartile desmics as extraversion in the last post.
From the point of view of triangle geometry, we are faced with an
overwhelming number of structures in the triangle. Kimberling has 400
points but we could easily add 4000 more. Same for lines, circles,
subtriangles and conics. The goal is to make sense of this abundance. We
are like the particle physicists 30 years ago who had hundreds of
particles but no explanations for any of them.
It is the imposition of essentially algebraic structures and ideas that
is taming this jungle. The division into strong, quartile weak, and
pairwise weak is the essential first step here. Most strong points are
organized by what I consider affine techniques into Conway's triangles of
centers. The desmic structures effectively organize much of the quartile
weak points, and our recent discussion of the pairwise weak points may
solve those (as well as some of the remaining strong points).
What is ultimately really interesting is that these algebraic techniques
lead to pairs of points that are harmonic conjugates, and hence to
projective stuctures. Perhaps these can be thought of as an origin of the
projective structures.
I once thought that projective geometry was the most fundamental with
affine geometry next. But I began to think, why all the circles? So much
geometry, particularly triangle geometry, is about circles, which do not
exist in pure projective or affine geometries. Circles are the main
elements of inversive geometry; perhaps that is the more fundamental? I
also suspect that Conway's abstract theory of the triangle is more
fundamental still.
I have no doubt that there are important projective structures in
triangle geometry, but more and more I think that there are deeper
origins for these.
Well, that was difficult, but I think I got it out.
Steve
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