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?


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


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.

  • $\begingroup$ The paper doesn't include the word "isotropy" or any recognizable form of it. I'm reading it right now to see if the concept is mentioned and named. $\endgroup$ Dec 12 '11 at 7:17
  • $\begingroup$ You're correct. I'm just seeing the word Null. $\endgroup$
    – Will Jagy
    Dec 12 '11 at 7:20
  • $\begingroup$ The earliest document I have that definitely uses the word is the O'Meara book. He gives a ton of credit to the Eichler book, so it may be worth borrowing that. Sorry about Witt, it really seemed promising. $\endgroup$
    – Will Jagy
    Dec 12 '11 at 7:57

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.

  • 1
    $\begingroup$ \ss and \"a respectively. The other uses of "isotropic" in other areas mathematics seem to make sense, in that there's some sense of something being the same in all directions. What I can't see is how you get from there to here. Perhaps I should have asked for a motivation rather than an etymology. $\endgroup$ Dec 17 '11 at 2:23
  • $\begingroup$ All right, this question seems to be more about chickens than isotropic forms now, so I guess I'll go ahead and accept this answer and resign. $\endgroup$ Dec 18 '11 at 1:52
  • $\begingroup$ @DanielMcLaury , on the plus side, I was just walking around the nearby pond, and saw about a dozen egrets in the same tree, also a black crowned night heron. $$ $$ The point about isotropy is not about a single field, it is about combining information from all the $p$-adic fields. It is a matter of low dimension. In five or more variables, all integral forms are isotropic in every $p$-adic field, even when there are still restrictions, as for, say, $$ v^2 + 3 w^2 + 3 x^2 + 3 y^2 + 3 z^2 $$ which is never $2 \pmod 3.$ $\endgroup$
    – Will Jagy
    Dec 18 '11 at 3:42
  • $\begingroup$ @DanielMcLaury take a look at math.stackexchange.com/questions/89138/… and email me if you are still unhappy. I cannot tell what you want. $\endgroup$
    – Will Jagy
    Dec 18 '11 at 4:18
  • $\begingroup$ If you need to type non-math special characters that aren't on your keyboard, the easy way is to copy-and-paste them from the Wikipedia articles (e.g. umlaut and eszett). $\endgroup$
    – user856
    Jul 13 '12 at 23:44

I'm going to hazard an answer that is pure motivation and contains no historical work--but maybe it's not too hair-brained. A physicist probably was implicitly thinking of the quadratic form as the second fundamental form associated to a parametrized surface $z=f(x,y)$ in $\mathbb{R}^3$. Roughly speaking, this quadratic form defines the curvature at a given point. In this sense, if the quadratic form vanishes, then the curvature is zero, and indeed things look the same in all directions!

As far as post hoc rationalizations go, how bad could this be?


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