Definitions of quaternions, complex quaternions and octonions

Subject: Re: Quaternions
From: sweetser@alum.mit.edu (Doug B Sweetser)
Date: 1997/05/02
Message-Id: <E9JD91.4qq@world.std.com>
Newsgroups: sci.physics.electromag,sci.physics.relativity

Maybe I should define explicitly the three players on the stage:
quaternions, complex quaternions (or biquaternions) and octonions (or
the Cayley algebra).

Quaternions are like the three vectors of vector calculus plus a scalar.
They are written

  q =  t + x i  +  y j + z k  

  where i^2 = j^2 = k^2 = -1
  and ij = -ji = k
  and jk = -kj = i
  and ki = -ik = j

When programming, I usually use the matrix representation which is exactly
equivalent:

  | t   -x  - y  -z |
  | x    t  - z   y | = q
  | y    z    t  -x |
  | z   - y   x   t |

Importantly, t, x, y, and z are all real numbers. What makes this always
invertible (a division algebra) if these are not all zero? The inverse of
this matrix is the transpose divided by the norm. The norm is
t^2 + x^2 + y^2 + z^2. The norm will not be zero unless t, x, y and z are
zero.

An aside: Multiplying two quaternions is not that tough, honest! It is
just like doing complex numbers, but there is in addition thecross
product. Let

   q1 = a + B  and q2 = c + D  where B and D are 3 vectors

   q1 q2 = a c + B . D + a D + B c + B x D

If q1 is the differential operator, d/dt del, then this is the divergence,
the gradient, the curl and two time derivatives. Compact notation!

Complex quaternions obey all of the above rules. The only difference is
that t, x, y, and z can take on complex values, a + b I. Note that I is
not i! Capital I commutes with everyone, just as a normal complex number
does. The inverse of a complex quaternion is the same as a quaternion.
Sometimes, though, it does not exist. For example, 3 + 3 I i would not
have an inverse because the norm 3^2 + (3 I)^2 = 9 - 9 = 0, and one can't
divide by zero.

Octonions have an 8 dimensional basis: {1, i, j, k, e, ei, ej, ek}. e is
just like the I above, but it does _not_ commute with i j and k.

   e^2 = -1  ie = -ei  je = -ej  ke = -ek

This has an important consequence.

   (ij)e = ke but i(je) = -ke

Where those parentheses go matters!

Quaternions have been used to examine rotations and Julia sets. Complex
quaternions have been neatly tied into group theory, particularly the
issue of spin. Octonions might have to do with the standard model, but
they are very tough to work with.


Doug Sweetser
http://world.std.com/~sweetser

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