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 Kirby Urner Posts: 803 Registered: 12/4/04
Posted: Jul 17, 2001 2:49 PM

On the subject of vectors and coordinate system concepts, I like to
expose students to alternatives to the XYZ orthogonal apparatus,
and not just the spherical or cylindrical options but something
closer to the Cartesian in flavor, yet different (I call in Neo-
Cartesian). Sometimes it's through comparison and contrast that
we come to more deeply understand the concepts involved.

What I call the Quadray Coordinate System may be referred to by
others as "simplicial coordinates" because it centers around the
tetrahedron, called the simplex by many (because it's the simplest
container -- that spatial structure with the fewest edges, vertices
and faces which we might logically say divides and inside space from
an outside space -- spheres consisting of unboundly high populations
of these primitives, i.e. have gazillions of edges, facets, going
in the direction of infinity (was Gauss who called "infinity"
a "direction"?)).

Quadray Coordinates are cousins to the barycentrics (a favorite of
Mobius), but not identical to them.

Start with a tetrahedron, and put a point at the origin. Call that
(0,0,0,0). Make it a regular tetrahedron. Vectors to its four
corners, from the origin, will now be labeled (1,0,0,0) (0,1,0,0)
(0,0,1,0) and (0,0,0,1). You may not want to call these "basis
vectors" since there are too many for 3-space, and therefore
you won't consider them to be linearly independent either (even
though their dot product, if defined in the usual fashion, is zero).
However, you can nevertheless assign a unique 4-tuple address to
every point in space, and these tuples will feature all-positive
numbers, with at least one of them always zero.

To see how this works, consider that the four elementary vectors
partition space into quadrants, each bracketed by three of the
four. So vector addition is defined in the usual fashion, as the
tip-to-tail placement of successive vectors. Clearly, any point in
space will be reachable by adding at most 3 positively scaled
(expanded or shrunk) elementary vectors. A fourth, pointing away
from the current quadrant, will not be needed.

When you add two vectors, you do the usual:

(1,2,3,0) + (4,0,1,1) = (5,2,4,1)

But then there's the added step of casting out any tetrahedron
comprised of four scaled elementaries, as these net to 0. In other
words, (1,1,1,1) is the zero vector, as is any (n,n,n,n). So with
(5,2,4,1), we subtract (1,1,1,1) to get (4,1,3,0) -- which is the
canonical expression for this particular vector -- or point, if we
consider vector tips as points.

When we negate a vector, we flip it to point in the opposite
direction. -(4,1,3,0) = (-4,-1,-3,0). But then we can rephrase
this in canonical form by adding (4,4,4,4) to get (0,3,1,4), which
will be the same vector, and pointed 180 degrees opposite (4,1,3,0).

So vector addition and negation, and therefore subtraction, are
defined in the usual fashion (tip-to-tail -- with subtraction =
addition of the negated vector). We have an Abelian group with
respect to addition (vectors being the elements, as scaled by
the field of reals).

One useful feature of this coordinate system is you can assign
whole number coordinates to several inter-related geometric shapes,
e.g. the tetrahedron we've already covered. The cube, comprised of
this tetrahedron and its inverse, has all vertices with either
three 1s and a 0, or three 0s and a 1. The 6 permutations of
{0,0,1,1} i.e. (0,0,1,1) (0,1,0,1) (1,0,0,1)... and so on,
define the octahedron. The 12 permutations of {2,1,1,0} give the
corners of the cuboctahedron.

I have a Vector module, written in Python, which freely interconverts
between XYZ and Quadray formats, as well as Spherical format. So
you can express your vectors in any of these ways. If you go into
it more deeply, you'll find a distance formula, a method analogous
to dot product, and even rotation matrices and volume formulae,
expressible in 4-tuple format.

A final interesting feature: if we take the tetrahedron defined by
the 4 quadrays to be our unit of volume, then it may be proved that
*any* random tetrahedron with corners at the centers of closest
packed spheres (FCC), where sphere diameter = elementary vector
length, will have whole number volumes.

Thanks to Bob Gray, Tom Ace, Gerald de Jong and of course David
Chako (and others) for contributing all of their research into the

Kirby

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