Associated Topics || Dr. Math Home || Search Dr. Math

### The Second Octant

```
Date: 04/03/2002 at 09:37:32
From: Kjetil
Subject: Geometry

Hi,

Where is the second octant?

We are always talking about the first octant, where x, y, and z are
positive. But no one seems to know how to count the next octants.
Is it when x > 0, y < 0 and z > 0 or x < 0, y > 0 and z > 0 ?

Kjetil
```

```
Date: 04/05/2002 at 14:10:49
From: Doctor Douglas
Subject: Re: Geometry

Hi, Kjetil,

thanks for submitting your question to the Math Forum.

By analogy with the two-dimensional case (x,y), we have the first
coordinate being inverted first:

(-,-)  third quadrant    [not (+,-) as in binary counting,
(+,-)  fourth quadrant    because we want a continuous path]

For the three-dimensional case:

(x,y,z) = (+,+,+)  first octant
(-,+,+)  second octant
(-,-,+)  third octant   [convention is same as in 2D case]
(+,-,+)  fourth octant
(+,+,-)  fifth octant   [here's a natural choice - for
(-,+,-)  sixth octant    fifth through eighth just repeat
(-,-,-)  seventh octant  first through fourth for negative
(+,-,-)  eighth octant   z values]

I think that this is the most common convention, and thus the second
octant is identified with {x < 0 and y,z > 0}. However, there are
other conventions that could be adopted, particularly for the fifth
through the eight octants. For example, if it is important to preserve
the "continous path" character among all eight octants (e.g., the
fourth octant touches the third and the fifth octant), then the
sequence might go like this:

+++  :  -++  :  --+  :  +-+  :  +--  :  ---  :  -+-  :  ++-

In this last sequence, we see that we flip exactly one sign in going
from one octant to the next.

- Doctor Douglas, The Math Forum
http://mathforum.org/dr.math/
```
Associated Topics:
College Definitions
College Higher-Dimensional Geometry
High School Definitions
High School Higher-Dimensional Geometry

Search the Dr. Math Library:

 Find items containing (put spaces between keywords):   Click only once for faster results: [ Choose "whole words" when searching for a word like age.] all keywords, in any order at least one, that exact phrase parts of words whole words

Submit your own question to Dr. Math
Math Forum Home || Math Library || Quick Reference || Math Forum Search