Comments on De la beauté et symétrie de la configuration de Desargues

Les dégénérescences, comme potentialités aristotéliciennes, sont hélas parfois véritablement incarnées…

Mon antique mémoire m'indique qu'un de mes professeurs de Maths appelait "triangle véritable" un "vrai" triangle (trois points non tous sur une même droite).
Après, je ne sais pas à quel point cette terminologie est un standard (ou répandue).
Un cercle véritable (i.e. de rayon strictement positif) n'est, dans ce même tonneau, pas une conique véritable (cf. un cas "à la limite" de l'excentricité).
"May the right side of the Force be with you !"

I would like to mention the book Reiman István: “A geometria és határterületei” (1986), Gondolat, Budapest. I was fortunate enough that this book was available to me when I was a young student, so it was what first introduced me to many of the concepts mentioned here. Chapter 17 even explains how if you have a projective plane with the Pappus theorem you can introduce coordinates from a skew-field, and with the Desargues theorem the coordinates come from a (commutatitve) field, though it omits some of the proofs. If you were to want to acquire this book, you're more likely to find the updated edition with the same title, published (1999) Szalai, Kisújszállás, though I am not familiar with that edition.

A few years later around 2000, I met the author professor Reiman István, when he was holding a practical course intended to prepare high school students to the mathematics olympiad. Every Friday afternoon, a dozen gifted students who studied outside of Budapest took the train for this course (though in later years professor Reiman was leading the course only on alternating weeks). I lived in Budapest so luckily I didn't have to travel – I probably wouldn't have been able to attend the course otherwise. As you may know, a lot of the olympiad style problems practiced there involve Euclidean geometry, but as projective geometry isn't among the required knowledge, that almost never came up. Professor Reiman was otherwise working in ELTE, and passed away in 2012.

I'll also mention another book. H. S. M. Coxeter, “A geometriák alapjai” (1973), Műszaki kvk., Budapest, translated by Sztrókay Katalin, original “Introduction to geometry”, John Wiley & Sons, Inc. Chapter 13 explains a set of axioms for the affine plane, and uses as axiom the special case of the Desargues theorem with the ideal line as the axis of perspectivity. 13.8 mentions that the axiom becomes redundant if you use similar axioms for the affine space. However, since the book doesn't expand on that remark with even a sketch proof, I hadn't been aware before of the nice symmetric arrangement of five planes before this blog post. I also have not realized before your blog post that this implied that octonionic projective planes and quaternionic projective spaces exist, but octonionic projective spaces don't, even though you explicitly say so in <URL: http://www.madore.org/~david/weblog/d.2016-11-20.2408.plan-projectif.html#d.2016-11-20.2408.autres-espaces.plan-quaternionique >.

> il faut comprendre (t:x:y) comme le point de coordonnées usuelles (= affines) (x/t, y/t)

I am surprised by your particular choice of embedding the Euclidean plane into the projective plane here. The usual convention is that the projective point (x: y: t) corresponds to the Euclidean point (x/t, y/t). That is the convention that the Richter-Gerbert book uses, just like many other books, and also the convention used by software libraries that I used. You have also been using the latter convention in <URL: http://www.madore.org/~david/weblog/d.2016-11-20.2408.plan-projectif.html#d.2016-11-20.2408.autres-espaces.dimensions-superieures >. Obviously the difference is irrelevant for this blog entry, where you are proving pure projective theorems.

Super entrée :)

dans le paragraphe … espace…:
"Le fait que les deux droites B₁C₁ et B₂C₂ se rencontrent (en U) montre qu'elles sont situées dans un même plan (disons Σu), donc B₁B₂ et C₁C₂ sont dans ce plan, donc se rencontrent : c'est le même raisonnement que ci-dessus : appelons Ou ce point de rencontre. De même, C₁C₂ et A₁A₂ se rencontrent, disons en Ov, et B₁B₂ et C₁C₂ se rencontrent, disons en Ow."
-> A la fin remplacer "B₁B₂ et C₁C₂" par "A₁A₂ et B₁B₂"!?
Sinon, merci pour cette entrée très claire!