### Investigations into the Linear Algebra Concepts used in the XYZ and Quadray Language Games

by Kirby Urner

First posted: November 20, 1997
Last updated: October 1, 2005

The quadray basis vectors go to the corners of a regular tetrahedron from the origin at (0,0,0,0). They're basis vectors in the sense that no three algebraically add to give the fourth using the closed, canonical notation suggested in many texts: no negative numbers, at least one of the four coordinates zero.

What is true is at least one basis vector is always not needed for any given point, whereas Cartesians posit three mutually orthogonal numberlines in R**3 (Python notation) and require the involvement of each for all points but those directly on the axes.

However, if we count positive and negative rays separately, per the quadray practice (we'll get to negatives in a moment), then the Cartesian actually has six basis vectors, not three.

The question is how do we treat the operation of reversing direction -- should this be considered a form of scaling (e.g. multiplication by -1), or another operation in its own right, i.e. that of negation.

 Scalar multiplication is something of a hybrid concept, because it includes what might be considered entirely separable operations: scaling a vector (expanding or contracting it) without changing its orientation, versus rotating a vector by 180 degrees, which is what the negative sign accomplishes if the scalar is negatively valued. What quadrays permit is something xyz does not: 4-tuple addresses consisting entirely of one sign (either positive or negative). Even if we normalize by some other scheme, such as to make the sum of the coordinates (a,b,c,d) equal zero, or one, we always have access to an all-same-sign 4-tuple isomorphism. This means we can afford to take only the first notion of scaling (expanding and contracting) and let go of the second (180 degree reversal) and still use vector addition to span volume.
 So our quadray vectors might be considered basis vectors if we adopt a notion of "strict" scaling. Whereas the three xyz basis vectors, i,j and k, will not natively add to reach points outside of their all-positive octant without scaling of the vector reversal type (negation), quadrays natively will. This stricter definition of scalar multiplication doesn't preclude having the negative operator in the picture, but it does suggest in what sense we might define the quadrays (1,0,0,0) (0,1,0,0) (0,0,1,0) and (0,0,0,1) as "basis vectors": in combination with our "strict" definition of scaling. Negation, or vector reversal, is by this definition considered a separate (though of course still doable) operation.