Posted by: Alexandre Borovik | April 7, 2008

Altitudes of a triangle and the Jacobi identity

It is many years that I know the expression which belongs to Arnold and which sound something like that:

Altitudes of a triangle intersect in one point because of the Jacobi identity.

What is meant here is the defining identity of Lie algebras which is known in undergraduate mathematics mostly as an identity for cross product of vectors in three dimensional space:

$(A \times B)\times C + (B \times C)\times A + (C\times A)\times B = 0$

I even produced a crude computational proof of that link; later Hovik Khudaverdyan showed me a streamlined proof. Finally, I found in the literature a really elegant proof. Interestingly, it is done with the help of spherical geometry and observation that cross product gives a polarity on the real projective plane (that is, on the sphere with identified antipodal points). My conjecture is that a more careful analysis should show that this is the same as a “calculus of reflections” proof originating in Hjelmslev’s paper of 1907 and developed into an impressive theory by Friedrich Bachman.

After all, $\mathbb{R}^3$ with cross product is the Lie algebra of the group $PSO_3(\mathbb{R})$ which preserves the polarity, and reflections are half-turns around axes which could be conveniently identified with the points of projective plane.

As usual, references, further discussion, etc. can be found in my book.

Responses

1. I prefer to think about 3-space with cross product as the Lie algebra of SU(2) which is homeomorphic and isomorphic to the 3-sphere. The SO(3) bit is just SU(2) double covering the real projective 3-space.

A soccer ball can be used to explain the correspondence between SO(3) and RP(3) t an interested child. [still afraid that the word processors might eat my carrots-^]. Of course, you have to wave your hands that a rigid motion of space has an axis. That is where the soccer ball comes in handy (or footy).

I haven’t tried Arnold’s exercise yet.