A quadratic form $q$ on a vector space $V$ is isotropic if $q(v) = 0$ admits a nonzero solution. What was this terminology originally intended to evoke?

There is some prior discussion on Math SE, Etymology of the word "isotropic", which traces early uses of the term, but it doesn't seem to arrive at an answer as to why the term was chosen.


1 Answer 1


This is an example of how a term migrates from the original context by broken telephone through various generalizations and transfers. It started with Poncelet introducing "imaginaries", i.e. complex coordinates, into projective geometry in Traité des propriétés projectives des figures (1822) based on his earlier work. He discovered circular points at infinity, the two points on the line at infinity through which all circles pass. Through them pass the lines with slopes $\pm i$. These were to become the "isotropic" points and lines, see Bourbaki, Elements of the History of Mathematics, p. 131.

Laguerre observed in 1853 that one can use those lines to define angles projectively through the cross-ratio by combining them with the two given lines, see Reid's review. In Sur l’emploi des imaginaires en la géométrie (1870) he calls them "isotropic", and also has "isotropic" coordinates, etc. Why "isotropic" he did not say, perhaps because circles are isotropic. And then he finally links them to quadratic forms:

"An essential property of isotropic lines, and which can be used to define them, is the following: the distance between any two points of an isotropic line situated at a finite distance in the plane is zero. In other terms, these lines satisfy the differential equation $ds^2 = 0$. On an arbitrary surface one can study curves that satisfy this differential equation; these curves are the geodesic lines of the surface, and we also call them isotropic lines."

Thus, the isotropic curves are tangent to vectors on which the first fundamental form of the surface vanishes. The terminology then spread in the work on Kleinian geometries derived from Poncelet's projective geometry by using various quadrics as absolutes, and the work on quadratic forms and geometric groups more generally. Under the influence of Poincare and Minkowsky, it also made its way into the jargon of relativity (isotropic = null = lightlike vectors, cones, etc).

  • $\begingroup$ Thanks, this is a wonderful answer $\endgroup$ Oct 27, 2021 at 17:33

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