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17-Point Cubic

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  • xpolakis@otenet.gr
    The 17-Point cubic (aka Thomson cubic) passes through the (3) vertices of the reference triangle, the (3) vertices of the medial triangle, the (3) midpoints of
    Message 1 of 16 , Sep 23, 2000
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      The 17-Point cubic (aka Thomson cubic) passes through the (3) vertices of
      the reference triangle, the (3) vertices of the medial triangle, the (3)
      midpoints of the altitudes, the (4) in/excenters, and the two dyads (2x2)
      of isogonals (H,O), (G,K), in total: 3+3+3+4+4 = 17 points.

      Kimberling, with his program CUBIC, found four more:
      X_9, and its isog. X_57; X_223, and its isog. X_282 (TCCT, p. 240).

      Are there other known notable points lying on this curve?

      Antreas
    • Steve Sigur
      ... Antreas, this is one of the canonical cubics that we have discussed here, the isogonal cubic with pivot G. Additional point that I know of are The four
      Message 2 of 16 , Sep 23, 2000
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        on 9/23/00 11:24 AM, xpolakis@... wrote:

        > The 17-Point cubic (aka Thomson cubic) passes through the (3) vertices of
        > the reference triangle, the (3) vertices of the medial triangle, the (3)
        > midpoints of the altitudes, the (4) in/excenters, and the two dyads (2x2)
        > of isogonals (H,O), (G,K), in total: 3+3+3+4+4 = 17 points.
        >
        > Kimberling, with his program CUBIC, found four more:
        > X_9, and its isog. X_57; X_223, and its isog. X_282 (TCCT, p. 240).
        >
        > Are there other known notable points lying on this curve?

        Antreas, this is one of the canonical cubics that we have discussed here,
        the isogonal cubic with pivot G.

        Additional point that I know of are

        The four Mittenpunkts and their isogonic conjugates and the ex-extras of the
        isogonic conjugates of the Mittenpunkts (and their conjugates).

        That's 16 more.

        Where does the name Thompson cubic come from?

        Steve
      • Steve Sigur
        ... I checked Kimberling, and his points are the central parts of the quadrangles I quoted above. Remember that points on a cubic come in fours, so in this day
        Message 3 of 16 , Sep 23, 2000
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          on 9/23/00 6:00 PM, Steve Sigur wrote:

          > on 9/23/00 11:24 AM, xpolakis@... wrote:
          >
          >> The 17-Point cubic (aka Thomson cubic) passes through the (3) vertices of
          >> the reference triangle, the (3) vertices of the medial triangle, the (3)
          >> midpoints of the altitudes, the (4) in/excenters, and the two dyads (2x2)
          >> of isogonals (H,O), (G,K), in total: 3+3+3+4+4 = 17 points.
          >>
          >> Kimberling, with his program CUBIC, found four more:
          >> X_9, and its isog. X_57; X_223, and its isog. X_282 (TCCT, p. 240).
          >>
          >> Are there other known notable points lying on this curve?
          >
          > Antreas, this is one of the canonical cubics that we have discussed here,
          > the isogonal cubic with pivot G.
          >
          > Additional point that I know of are
          >
          > The four Mittenpunkts and their isogonic conjugates and the ex-extras of the
          > isogonic conjugates of the Mittenpunkts (and their conjugates).
          >
          > That's 16 more.


          I checked Kimberling, and his points are the central parts of the
          quadrangles I quoted above.

          Remember that points on a cubic come in fours, so in this day and time it is
          not possible to know "17 points" on a cubic.

          I reproduce the group table below. It is a much simpler group table than the
          Neuberg cubic. The work of Barry and me over the summer guarantees that we
          can _construct_ an infinite number more. About two weeks ago I summarized
          our work with a posting to Hyacinthos about constructing points on cubics.

          Many of the cubics we have discussed extensively can be thought of as having
          two operations that generate points. For this cubic, it is the isogonic
          conjugate and the ex-extra operation (aka, Cevian quotient with respect to
          G). Using these we can generate as many points (with their corresponding
          colinearities) as we wish.

          For the Darboux cubic the two operations are the isogonic conjugate and
          reflection in the circumcenter.

          Another relevant comment is that the odd/even-ness inherent in the group
          table guarantees that, if the desmon is a strong point, an infinite number
          of weak, quartile points (such as the Mittenpunkts) will be on the cubic,
          occupying every other row of the group table. The other rows will be strong
          points.

          Here is the group table, listing 36 points. By the way we would love to know
          a geometric interpretation of the third intersection of AO, BO, and CO with
          the cubic.


          Thompson centroidal isogonic cubic
          precevian of R = pedal of P
          isogonal

          o a b c
          -------------
          -V
          -IV
          -III gM'M'o :gM'M'b:
          -II H ***
          -I M'o M'a M'b M'c
          0 K A_G B_G C_G
          I Io Ia Ib Ic
          II G A B C
          III Mo Ma Mb Mc
          IV O H+A H+B H+C
          V M'M'o :M'M'b:

          pivot G
          center K
          constant O = row IV

          Notation: Mo = Mittenpunkt = X(9), M'o = mate of Mo = isogonal conjugate of
          Mo = X(37), A_G = a-trace of G , H+A = midpt of HA, M'M'o= ex-extra pf M'o
          = M'o/G = X(223).

          *** - the points in this row are the third meets of lines AO, BO, CO with
          the cubic.

          The I row are the incenters, which are self conjugate. Corresponding rows
          above and below row I are isogonally conjugate.
        • Bernard Gibert
          Dear Steve and friends, [Steve] ... they are the isogonals of the midpoints of the altitudes. I just want to add a well-known fact that might give other points
          Message 4 of 16 , Sep 24, 2000
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            Dear Steve and friends,

            [Steve]
            >By the way we would love to know
            > a geometric interpretation of the third intersection of AO, BO, and CO with
            > the cubic.

            they are the isogonals of the midpoints of the altitudes.

            I just want to add a well-known fact that might give other points on this
            cubic :
            the homothecy h, center G, ratio -1/2, maps the Lucas cubic (L) to the
            Thomson cubic (T) therefore (T) is isotomic wrt the medial triangle.

            For example, X253 = isotomic of X20 (de Longchamps) being on (L), the point
            h(X253) is on (T) but maybe it's one of the points already mentionned.

            BTW, dear Steve, it's called Thomson and not Thompson but I don't know who
            Thomson was... [Antreas, any idea ?]

            The name 17-point cubic comes from the very extraordinary fact that this
            cubic is (probably) the only one meeting 22 usual lines in only 17 points :
            3 sidelines, 3 altitudes, 3 medians, 3 symmedians, 3 perp. bisectors, 6
            bisectors and the Euler line.

            Denote that the tangents in A,B,C are the symmedians, the tangents in the
            midpoints of the sides are the perp. bisectors, the tangents in Ia,Ib,Ic are
            the internal bisectors.
          • xpolakis@otenet.gr
            ... Dear Bernard, There were some geometers named Thomson but I don t know which one is of the Thomson cubic. Most likely one the following: James Thomson
            Message 5 of 16 , Sep 24, 2000
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              Bernard Gibert wrote:

              >BTW, dear Steve, it's called Thomson and not Thompson but I don't know who
              >Thomson was... [Antreas, any idea ?]

              Dear Bernard,

              There were some geometers named Thomson but I don't know which one
              is of the Thomson cubic.

              Most likely one the following:

              James Thomson (1786-1849), the author of:
              Thomson, James: Elements of plane and spherical trigonometry, with the first
              principles of analytic geometry . 2nd ed.
              Belfast : Simms & McIntyre, 1830. 108 p.

              Thomson, James: First six and the eleventh and twelfth books of Euclid's
              elements : with notes and illustrations and an appendix in five books.
              Edinburgh : A. & C. Black, 1834. v+386 p.

              Herman Ivah Thomson author of:
              Thomson, Herman Ivah: The Osculants of Plane Rational Quartic Curves.
              Reprint from _American Journal of Mathematics 32:3 (1910) 207-234_
              Baltimore, Johns Hopkins Univ., Diss., 1909.

              I searched Zbl for "Thomson cubic" and here are two related articles
              (which probably contain references to Thomson, but haven't them handy):

              ___________________________________________

              Cundy, Henry Martyn - Parry, Cyril Frederick: Some cubic curves associated
              with a triangle.
              J. Geom. 53, No.1-2, 41-66 (1995).

              Keywords: trilinear coordinates; isogonal conjugates; auto-isogonal cubics;
              cubic curves; polar conic; circular cubics; Euler pencil; Darboux cubic;
              McCay cubic; Thomson cubic; Neuberg cubic; Neuberg group; Moebius
              involution

              -------------------------------------------

              Pinkernell, Guido M.: Cubic curves in the triangle plane.
              J. Geom. 55, No.1-2, 141-161 (1996).

              The term ``triangle plane'' in the title means a plane in which a triangle
              ABC is singled out. The author studies two pencils of cubic curves that are
              the result of certain constructions in the triangle plane. The two pencils
              contain nearly all the important special cubics, for example the Darboux
              cubic, the 17-point cubic (also known as Thomson's cubic), the Neuberg
              cubic, and the Lucas cubic.
              [ E.J.F.Primrose (Leicester) ]

              ___________________________________________

              BTW, which is McCay cubic ?

              PS: I will post a query on Thomson to Historia Matematica list.

              Antreas
            • xpolakis@otenet.gr
              ... Here is Julio s informative response: FWD MESSAGE ---------------------------------------------------------- Date: Thu, 28 Sep 2000 22:05:18 -0300 To:
              Message 6 of 16 , Sep 29, 2000
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                [Bernard Gibert]:

                >>BTW, dear Steve, it's called Thomson and not Thompson but I don't know who
                >>Thomson was... [Antreas, any idea ?]

                [APH]:

                >PS: I will post a query on Thomson to Historia Matematica list.

                Here is Julio's informative response:

                FWD MESSAGE ----------------------------------------------------------

                Date: Thu, 28 Sep 2000 22:05:18 -0300
                To: "Antreas P. Hatzipolakis" <xpolakis@...>
                From: Julio Gonzalez Cabillon <jgc@...>
                Subject: Re: [HM] Thomson
                Cc: historia-matematica@..., guido.pinkernell@...


                On 24 Sep 2000, Antreas P. Hatzipolakis asked:

                "Who was Thomson of the well-known triangle cubic
                named after him (Thomson cubic aka 17point cubic)?"


                -------

                Dear Antreas,

                I am sorry to say that, as far as I can tell, virtually nothing
                has been recorded of this Thomson, other than the issue that he
                posed his geometrical locus as a problem in the enjoyable British
                _Educational Times_ journal (August 1864).

                F. D. Thomson ( as he used to sign most of his short notices [*])
                was a habitue of that journal, having submitted many solutions of
                questions posed there.

                [*] Thomson, F. D.
                "Notes on the Geometry of a Cubic Curve", _Messenger_ volume V
                (1869) pp 27-30.
                "The Equation to the Axes of a Conic",_Messenger_ volume V (1869)
                pp 65-66.

                In 1865, Thomson's problem appeared also in the well-known French
                journal _Nouvelles annales de mathematiques_:


                _Question 735_

                "Le lieu des centres des coniques tangentes aux cotes d'un
                triangle, et telles que les normales menees par les points
                de contact, se rencontrent en un meme point est une courbe
                du troisieme degre qui passe par les sommets du triangle, le
                point de rencontre des hauteurs, le centre de gravite, les
                centres incrit et ex-inscrits, les milieux des cotes, les
                milieux des hauteurs." (2a. serie, t. IV, p 144)

                Interestingly enough, this "Question 735" was solved by a junior
                named Arthur Poussart, a student of 'Lycee de Douai' (class of Mr
                Painvin), and his demonstration appeared on pages 469-473 of the
                same journal (2a. serie, t. IV).

                Joseph Neuberg (1840--1926) is responsible for the popularization
                of term THOMSON CUBIC [ "La cubique de THOMSON" ], in his article
                "Bibliographie du triangle et du tetraedre" (Mathesis 1923).

                There are at least two interesting and brand new articles on the
                subject of cubic curves on the plane in which the given triangle
                is considered. For instance, Guido Pinkernell studies two pencils
                of cubic curves which contain basically the main special cubics;
                namely, Darboux's, Lucas's, Neuberg's & Thomson's.

                Cf. Pinkernell, Guido M:
                "Cubic Curves in the Triangle Plane",_Journal of Geometry_ vol 55
                (1996), no 1-2, pp 141-161.

                See also:
                Cundy, Henry Martyn; Parry, Cyril Frederick:
                "Some Cubic Curves Assoc. with a Triangle", _Journal of Geometry_
                vol 53 (1995), no 1-2, pp 41-66.


                Best wishes from Montevideo, Julio

                PS Antreas... I apologise for having forgotten an important event
                in mid August. Better late than never... they say.

                END ------------------------------------------------------------------

                Very many Thanks, dear Julio!

                Greetings from Athens

                Antreas
              • Jean-Pierre.EHRMANN@wanadoo.fr
                Dear Antreas, Julio and other Hyacinthists, ... Very interesting, indeed. So if we look at the locus of the centers of the conics tangent to the sides and such
                Message 7 of 16 , Sep 29, 2000
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                  Dear Antreas, Julio and other Hyacinthists,
                  Antreas quoted :

                  > "Le lieu des centres des coniques tangentes aux cotes d'un
                  > triangle, et telles que les normales menees par les points
                  > de contact, se rencontrent en un meme point est une courbe
                  > du troisieme degre qui passe par les sommets du triangle, le
                  > point de rencontre des hauteurs, le centre de gravite, les
                  > centres incrit et ex-inscrits, les milieux des cotes, les
                  > milieux des hauteurs." (2a. serie, t. IV, p 144)
                  >

                  Very interesting, indeed.
                  So if we look at the locus of the centers of the conics tangent to
                  the sides and such as the three lines going through the contact
                  points and making a given angle Phi with the corresponding side
                  concur, we get a cubic C(Phi) member of the pencil generated by the
                  Thomson cubic and the union of the three sides : all those cubics go
                  through the midpoints of the sides and are tangent at A, B, C to the
                  corresponding symedian.
                  The isogonal conjugate of C(phi) is C(-phi) and, in the particular
                  case of the Thomson cubic, the common point of the three normals
                  should lie on the Darboux cubic.
                  A lot of work for Bernard !!

                  Friendly from France. Jean-Pierre.
                • John Conway
                  ... May I ask where this quotation s from? I gather that lieu in such contexts means locus - is the Latin word not used in French? ... I agree that this
                  Message 8 of 16 , Sep 29, 2000
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                    > > "Le lieu des centres des coniques tangentes aux cotes d'un
                    > > triangle, et telles que les normales menees par les points
                    > > de contact, se rencontrent en un meme point est une courbe
                    > > du troisieme degre qui passe par les sommets du triangle, le
                    > > point de rencontre des hauteurs, le centre de gravite, les
                    > > centres incrit et ex-inscrits, les milieux des cotes, les
                    > > milieux des hauteurs." (2a. serie, t. IV, p 144)

                    May I ask where this quotation's from? I gather that "lieu"
                    in such contexts means "locus" - is the Latin word not used in
                    French?

                    > So if we look at the locus of the centers of the conics tangent to
                    > the sides and such as the three lines going through the contact
                    > points and making a given angle Phi with the corresponding side
                    > concur, we get a cubic C(Phi) member of the pencil generated by the
                    > Thomson cubic and the union of the three sides : all those cubics go
                    > through the midpoints of the sides and are tangent at A, B, C to the
                    > corresponding symedian.

                    I agree that this is very interesting. It's a pity that for
                    general Phi these cubics involve a term XYZ and so aren't
                    either isotomic or isogonal ones (unless I misunderstand something?).

                    However, the fact that they go through the midpoints mA,mB,mC
                    of the sides means that their "dilated" forms have a chance of being
                    isogonal or isotomic. Are they? I'd check myself, but of course
                    can't remember which one the Thomson cubic is - can somebody please
                    remind me (preferably by giving its equation)?

                    Regards, John Conway
                  • conway@math.princeton.edu
                    ... I m not terribly surprised, since of course all these words mean the same thing, but this does make me wonder why English (alone?) continues to use the
                    Message 9 of 16 , Sep 29, 2000
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                      On Fri, 29 Sep 2000, Antreas P. Hatzipolakis wrote:

                      > Yes, in the French math. literature "lieu" = locus.
                      >
                      > In the Greek, by the way, topos = locus.

                      I'm not terribly surprised, since of course all these words
                      mean the same thing, but this does make me wonder why English
                      (alone?) continues to use the Latin word for this specialised
                      sense.

                      > > However, the fact that they go through the midpoints mA,mB,mC
                      > >of the sides means that their "dilated" forms have a chance of being
                      > >isogonal or isotomic. Are they? I'd check myself, but of course
                      > >can't remember which one the Thomson cubic is - can somebody please
                      > >remind me (preferably by giving its equation)?
                      >
                      > bcx(y^2 - z^2> +....(cyclically) = 0

                      Thanks. This looks like the form in (orthogonal) trilinears,
                      so I'll rescale it to abbccx(yy-zz) + &c = 0, and replace
                      ax,by,cz by X,Y,Z to get X(ccYY-bbZZ) + &c = 0, or equivalently

                      | X Y Z |
                      |aaYZ bbZX ccXY| = 0,
                      | 1 1 1 |

                      showing that this is the isogonal cubic with pivot/perspector G.

                      Oh - I've just realised that I didn't need to ask, since the
                      fact that it passes through the midpoints of the sides forces G
                      to be the pivot. But thanks anyway!

                      OK So the general cubic of the pencil is

                      X(ccYY-bbZZ) + ... + kXYZ = 0

                      which I dilate by putting Y+Z,Z+X,X+Y for X,Y,Z:

                      (Y+Z)[cc(XX+2ZX+ZZ) - bb(XX+2XY+YY)] + ... + k(Y+Z)(Z+X)(X+Y) = 0.

                      I sort terms, using term' and term" for the 2nd and 3rd copies
                      of the first bracket here.

                      Coeff(ZZZ) = cc - (bb)' = 0.

                      Coeff(XYZ) = (2cc-2bb)+ that' + that" + 2k = 2k

                      Aha - so this kills my hope that perhaps all these (dilated) cubics
                      would be isotomic/isogonal. But let's continue with the Thomson one
                      (k=0), looking for terms in things like XXY

                      (cc-bb)XXY 2ccZZX -bbYYZ ccYZZ (cc-bb)ZXX -2bbXYY
                      ': (aa-cc)YYZ 2aaXXY -ccZZX aaZXX (aa-cc)XYY -2ccYZZ
                      ": (bb-aa)ZZX 2bbYYZ -aaXXY bbXYY (bb-aa)YZZ -2aaZXX
                      --------------------------------------------------------------
                      (cc-bb+aa)XXY + &c (-bb+aa-cc)XYY &c

                      giving SA.ZZX + SB.XXY + SC.YYZ = SA.XYY + SB.YZZ + SC.ZXX

                      or SA.X(YY-ZZ) + &c = 0, or

                      | SA SB SC |
                      | YZ ZX XY | = 0
                      | X Y Z |
                      for the dilated form, ie., the isotomic cubic with perspector
                      dK = (SA:SB:SC).

                      I recognize this as what I used to call "the antipodal cubic",
                      since it's equal to its reflection in O. So (good) the Thomson
                      cubic is one of the three great rationally equivalent cubics I
                      wrote about some time ago. (I doubt if any two of the other
                      well-known cubics are rationally equivalent - does anyone know
                      of any such pairs?)

                      John Conway
                    • Bernard Gibert
                      Dear John, ... A cubic with a term in xyz can be isogonal (or isotomic) but, in this case, there s no pivot. These cubics are very interesting but difficult to
                      Message 10 of 16 , Sep 29, 2000
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                        Dear John,

                        > I agree that this is very interesting. It's a pity that for
                        > general Phi these cubics involve a term XYZ and so aren't
                        > either isotomic or isogonal ones (unless I misunderstand something?).
                        >
                        A cubic with a term in xyz can be isogonal (or isotomic) but, in this case,
                        there's no pivot.

                        These cubics are very interesting but difficult to deal with.
                        I know a (small ) number of them and I still have a lot of work to do to
                        understand them a little bit better.

                        Regards

                        Bernard
                      • Bernard Gibert
                        Dear Jean-Pierre and friends, ... It does, JP, and the perspector is on the isogonal cubic with pivot X69 = isotomic of H. What about the locus of the foci ?
                        Message 11 of 16 , Sep 29, 2000
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                          Dear Jean-Pierre and friends,

                          > The isogonal conjugate of C(phi) is C(-phi) and, in the particular
                          > case of the Thomson cubic, the common point of the three normals
                          > should lie on the Darboux cubic.

                          It does, JP, and the perspector is on the isogonal cubic with pivot X69 =
                          isotomic of H.
                          What about the locus of the foci ? It seems it's a much higher degree curve
                          (12, I think) which decomposes into the Lucas cubic and ???...)

                          Regards

                          Bernard
                        • John Conway
                          ... [...] ... This is a matter of how we use language. The term isogonal cubic is usually used to mean a cubic that s not only self-isogonal but also passes
                          Message 12 of 16 , Sep 29, 2000
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                            On Fri, 29 Sep 2000, Bernard Gibert wrote:

                            > Dear John,

                            [...]

                            > A cubic with a term in xyz can be isogonal (or isotomic) but, in this case,
                            > there's no pivot.

                            This is a matter of how we use language. The term "isogonal cubic"
                            is usually used to mean a cubic that's not only self-isogonal but also
                            passes through the incenters, or equivalently "has a pivot" P (meaning
                            that it's the locus of Q for which Q,gQ.P are collinear). Similarly
                            "isotomic cubic" means one that's not only self-isotomic but also
                            passes through G and its associates, or "has a pivot" in the sense
                            that Q,tQ,P are collinear.

                            I agree this convention isn't universal, but we've discussed it
                            before, and the convention is a good one because there's no point
                            in making "isogonal" and "self-isogonal" mean exactly the same thing!

                            > These cubics are very interesting but difficult to deal with.
                            > I know a (small ) number of them and I still have a lot of work to do to
                            > understand them a little bit better.

                            My feeling as far as the Triangle Book is concerned is that the
                            isotomic and isogonal cubics in the above sense will be as far as
                            we go, because their (P,Q,tQ) and (P,Q,gQ) definitions are quite
                            geometrical, whereas other high-degree curves really involve more
                            algebra than geometry.

                            Despite that, I've done some investigations of particular ones,
                            most notably the locus of P whose pedal and prepedal triangles
                            are at a given angle theta. This generalized Thomson cubic sounds
                            very similar.

                            Regards,
                            John Conway
                          • Steve Sigur
                            ... Bernard. By my way of thinking the cubics are not bad at all, rather it is the conics that are confusing. The reason I think they are not confusing is that
                            Message 13 of 16 , Sep 29, 2000
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                              on 9/29/00 3:36 PM, Bernard Gibert wrote:

                              > These cubics are very interesting but difficult to deal with.
                              > I know a (small ) number of them and I still have a lot of work to do to
                              > understand them a little bit better.


                              Bernard.

                              By my way of thinking the cubics are not bad at all, rather it is the conics
                              that are confusing. The reason I think they are not confusing is that to me
                              they are more fundamental. I take points and lines to be the most
                              fundamental objects and the cubics organize them both.

                              Geometrically they are more simple; by themselves, they only use projective
                              properties of concurrence and colinearity. Mathematically this is expressed
                              by a group law between these points. Unfamiliar and abstract, yes, but not
                              difficult. We have picked up a trick or two in the last century and perhaps
                              the best way to understand the ancient mathematical creation of the Greeks
                              is to use the less ancient mathematical creations of the French and Germans.

                              What we get from this is that points are arranged in quadrangles (whence the
                              triangle has 4 "sides," the in-side, the a-side, the b-side, and the
                              c-side--this comment is supposed to be as much a pun as geometry). These
                              quadrangles are those incredible desmic quadrangles that often relate the
                              weak and strong points. If the cubic has a pivot that is strong, then there
                              is a powerful organization and inter-relation between the strong and weak,
                              quartile points.

                              So cubics, by themselves, are actually easy and profound to understand.

                              But of course their interactions with other geometric structures may be
                              harder to understand, which is what you were referring to.

                              Friendly from Princeton,

                              STeve
                            • Bernard Gibert
                              Dear Jean-Pierre and friends, ... please correct a typo : it s not isogonal but isotomic cubic (Lucas cubic = locus of points P such that a cevian triangle of
                              Message 14 of 16 , Sep 29, 2000
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                                Dear Jean-Pierre and friends,
                                >
                                >> The isogonal conjugate of C(phi) is C(-phi) and, in the particular
                                >> case of the Thomson cubic, the common point of the three normals
                                >> should lie on the Darboux cubic.

                                I wrote :

                                > It does, JP, and the perspector is on the isogonal cubic with pivot X69 =
                                > isotomic of H.

                                please correct a typo : it's not isogonal but isotomic cubic (Lucas cubic =
                                locus of points P such that a cevian triangle of P is a pedal triangle of Q.
                                That's why Q is on Darboux)

                                It's nice to observe [I didn't prove it yet but it shouldn't be very
                                difficult] that the line PQ goes through L=X20 and the line through Q and
                                the center of the inconic [which is on Thomson] goes through H.

                                Another little thing : the in-Steiner ellipse has its foci isogonal and on a
                                line through G therefore those foci are on Thomson and on the locus below.

                                > What about the locus of the foci ? It seems it's a much higher degree curve
                                > (12, I think) which decomposes into the Lucas cubic and ???...)

                                I don't think now this is correct... I have to redo this part if I can...
                                The vertices A,B,C are at least triple, the in/excenters at least double,
                                and the feet of the altitudes are on it : that makes at least 7 points on
                                each side !!??

                                Any idea, JP ?

                                Sorry for so much carelessness.

                                Regards

                                Bernard
                              • xpolakis@otenet.gr
                                Steve Sigur wrote: the cubics ... ^^^^^^ ^^^^^^^ ...rather of francophones (= French, Belgians etc) and germanophones (Germans, Austrians, etc) And of the
                                Message 15 of 16 , Sep 30, 2000
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                                  Steve Sigur wrote:

                                  the cubics
                                  >Geometrically they are more simple; by themselves, they only use projective
                                  >properties of concurrence and colinearity. Mathematically this is expressed
                                  >by a group law between these points. Unfamiliar and abstract, yes, but not
                                  >difficult. We have picked up a trick or two in the last century and perhaps
                                  >the best way to understand the ancient mathematical creation of the Greeks
                                  >is to use the less ancient mathematical creations of the French and Germans.
                                  ^^^^^^ ^^^^^^^
                                  ...rather of francophones (= French, Belgians etc) and germanophones (Germans,
                                  Austrians, etc)

                                  And of the British/Irish as well!
                                  (It was Newton who first classified the cubics)
                                  There was also a great Dutch geometry tradition (which is still alive!)

                                  Antreas
                                • Steve Sigur
                                  ... and the Greekophone tradition is still alive, if all those books you quote are an indication! In the above paragraph I was thinking of the development of
                                  Message 16 of 16 , Sep 30, 2000
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                                    on 9/30/00 7:30 PM, xpolakis@... wrote:

                                    > Steve Sigur wrote:
                                    >
                                    > the cubics
                                    >> Geometrically they are more simple; by themselves, they only use projective
                                    >> properties of concurrence and colinearity. Mathematically this is expressed
                                    >> by a group law between these points. Unfamiliar and abstract, yes, but not
                                    >> difficult. We have picked up a trick or two in the last century and perhaps
                                    >> the best way to understand the ancient mathematical creation of the Greeks
                                    >> is to use the less ancient mathematical creations of the French and Germans.
                                    > ^^^^^^ ^^^^^^^
                                    > ...rather of francophones (= French, Belgians etc) and germanophones (Germans,
                                    > Austrians, etc)
                                    >
                                    > And of the British/Irish as well!
                                    > (It was Newton who first classified the cubics)
                                    > There was also a great Dutch geometry tradition (which is still alive!)
                                    >

                                    and the Greekophone tradition is still alive, if all those books you quote
                                    are an indication!

                                    In the above paragraph I was thinking of the development of abstract algebra
                                    and group theory, rather than of cubics.

                                    While we are listing strong geometric traditions, there is an influx of
                                    Eastern European mathematicians into the US, and their geometry is very
                                    strong. While geometry fades into obscurity here in the US, their tradition
                                    is still taught to their young and is vibrant.

                                    Steve
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