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[mathlearn] Quadray Coordinates
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 3space, 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 4tuple address to every point in space, and these tuples will feature allpositive 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 tiptotail 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 (tiptotail  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 interrelated 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 4tuple 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 quadray coordinates thread, about which I have a lot more material at my website: http://www.teleport.com/~pdx4d/quadrays.html
Kirby
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