18 September 2007

In two-dimensional Cartesian geometry, it's conventional to refer to the area where x and y are both positive as the "first quadrant", to that where x is negative and y is positive as the "second quadrant", to that where y is negative and x is negative as the "third quadrant", and to that where y is negative and x is positive as the "fourth quadrant". In tabular form, we have

 x < 0 x > 0 y > 0 2nd 1st y < 0 3rd 4th

There seems to be no conventional way to extend this to the eight octants in three-dimensional Cartesian geometry. The only thing that people seem to do consistently is to call the octant where x, y, and z are all positive the "first octant", or sometimes the "positive octant". And I've noticed that a few of my students want to call the octant where x, y, and z are all negative the "eighth octant"; I actually want to do the same. It's far from the first octant; it should have a number that is far from 1. But we don't call the quadrant where x and y are negative the "fourth quadrant".

Actually, the "next step" in this recursion would lead to a scheme in which the (-, -, -) octant is called the fifth octant, and in general the octant obtained by reflecting the nth octant over the xy-plane "should be" (9-n)th octant. This preserves the property that the kth octant and the (k+1)st octant are adjacent; there's no a priori reason why this is a good condition to have, but it's the same recurstion that says that the 3rd and 4th quadrants are obtained by reflecting the 2nd and 1st quadrants, respectively, over the x-axis.

More generally, in d+1 dimensions, the point (ε1, ε2, ..., εd, 1), where each ε is ±1, is in the (d+1)-dimensional orthant with the same number as the d-dimensional orthant containing (ε1, ε2, ..., εd); call this number k. The point (ε1, ε2, ..., εd, -1) is in the orthant with number (2d+1+1-k).

This means that if we traverse the octants (or, more generally, the "orthants" in d-dimensional space) in numerical order, we get a Hamiltonian tour of the vertices of the d-dimensional cube, which is kind of cute.

This page from Math Forum seems to advocate calling the all-negative octant the "seventh octant", which comes from a hybrid of my convention and the convention that octants which border each other over the plane z=0 have numbers differeing by 4. I suppose it's okay in three dimensions, but my sense of aesthetics demands that we have a definition which recurses to higher dimensions.

Still, there's really no reason to be able to number the orthants other than the all-positive one (often called the "first orthant"), and anyone who needs to refer to a particular orthant really ought to just specify it in the form x1 * 0, x2 * 0, ... where each * is < or > in order to avoid confusion.

1 comment:

Harald Hanche-Olsen said...

One might wish to maintain the property of quadrant numbering that quadrants n and n+1 (mod 4) are neighbours. That leads naturally to using a Gray code to number octants. Just replace 0 and 1 in the Gray code by + and -.