Etymology of the word "isotropic"

Question

Given a quadratic form $q : V \rightarrow k$, a nonzero vector $v \in V$ is said to be isotropic if $q(v) = 0$. Any subspace of $V$ containing such a vector is also said to be isotropic, and the quadratic form itself is also said to be isotropic. The word "isotropic" is Greek for "the same in every direction." What does this have to do with being a zero of a quadratic form?

Answer 1

EDIT: Mariano found this LECTURE

I have borrowed Quadratische Formen und Orthogonale Gruppen by Martin Eichler, Springer-Verlag 1952.

On page 3

Ist $\xi$ zu sich selber senkrecht und $\xi \neq 0,$ so nennt man $\xi$ einen isotropen Vektor. Enthält $R$ keinen isotropen Vektor, so soll $R$ anisotrop heißen, im anderen Fall isotrop.

Then section 3 of chapter one begins on page 12, with the title

Die Automorphismengruppe eines isotropen Raumes

Varied other German endings occur throughout; the index says pages 3, 12ff, 57ff, 99ff.

It is difficult to say whether Eichler feels he is introducing new terminology. The word isotropic comes from physics, materials science, biology and other places in mathematics as well. If this is important, the thing to do is get on MathSciNet and look up occurrences of the word isotropic in reviews under subject headings related to quadratic forms, integral lattices, and so on. That takes you back to slightly before 1940. Do the same with Zentralblatt. Also look up the collected works of Hasse, then of Minkowski.

In the Vorwort, he does seem to be acknowledging a debt to the book(s) of P. Bachmann, Arithmetik der quadratischen Formen, part I 1898, part II 1923.

Answer 2

I am becoming pretty confident that this goes back to Ernst Witt. German forms of the word include "isotrop," "isotrope," and "isotropen." You can download a 1970 article using these by Winfried Scharlau for free at

Link

Pretty much everyone refers to an article in Crelle's journal, which is now Journal für die reine und angewandte Mathematik, by Ernst Witt, Theorie der quadratischen Formen in beliebigen K$\ddot{o}$rpern, volume 176 pages 31-44, (1937).

Alright, I was able to read the Witt article on the Goettingen archives of the journal. Every place the concept came up, Witt uses the word Null as a noun.

Note that Leonard Eugene Dickson always referred to zero forms. Most of the terminology in English was standardized by 1963, in Introduction to Quadratic Forms by O. Timothy O'Meara. So perhaps it moved from English back to German.

Another really groundbreaking book was Martin Eichler in 1952, Quadratische Formen und orthogonale Gruppen. Eichler introduced the spinor genus, which codifies behavior of psitive ternary forms found by Jones and Pall in 1939.

I tried. I really did.