
Article on Quadrays in March 1999 FoxPro Advisor
Posted:
Feb 6, 1999 6:33 PM


My article touching on quadrays in the context of 'Teaching ObjectOriented Programming with Visual FoxPro' is now in circulation, in the March 1999 issue of FoxPro Advisor, The Developer's Guide to Microsoft Visual FoxPro and FoxPro 2.x, ADVISOR MEDIA Inc., starting on page 48.
The primary web page reference is to:
http://www.teleport.com/~pdx4d/quadrays.html
 part of my Oregon Curriculum Network website (OCN).
This page links to numerous related papers and downloadable source code, including to a Java implementation which includes code for converting back and forth among xyz, polar and quadray coordinates.
For those unfamiliar with quadrays, this NeoCartesian system uses 4tuples instead of the familiar xyz 3tuples, labeling the origin as (0,0,0,0) and the 4 basis vectors to the vertices of a regular tetrahedron as (1,0,0,0)(0,1,0,0)(0,0,1,0) and (0,0,0,1).
These four vectors span volume (given vector addition and scalar multiplication of the usual sort), giving a unique, simplestterms 4tuple (a,b,c,d) for each point, with at least one term equal to zero and the rest positive.[1]
(n,n,n,n) is an additive identity i.e. Q + (n,n,n,n) = Q  because (n,n,n,n) simply represents four equal vectors to the corners of a regular tetrahedron with a zerovector net.
Given all basis rays are positive, no simplestterms 4tuple includes any negatives, although the negation operator does have a clear meaning (same as in xyz  changes a vector's orientation by 180 degrees). For example: (1,1,0,0) = (1,1,0,0) = (1,1,0,0) + (1,1,1,1) [identity] = (0,0,1,1).
My implementation of quadrays nests the "home base" tetrahedron in a unitradius spheres packing context (fcc) and also defines this tetrahedron to be the unit of volume.
The home base tetrahedron and its selfdual, a tetrahedron with quadray coordinates (0,1,1,1)(1,0,1,1)(1,1,0,1)(1,1,1,0)  the negatives of the four above  intersect to define the cube of volume 3.
This cube's dual, an octahedron with coordinates (1,1,0,0) (1,0,1,0)(1,0,0,1)(0,1,1,0)(0,1,0,1)(0,0,1,1), has a volume of 4.
The rhombic dodecahedron, with cube + octahedron vertices, has a volume of 6, and is a spacefiller within which the unitradius fcc spheres nest and "kiss" through the face centers (12 Kpoints where long and short diagonals of the 12 rhombic faces intersect, and where long diagonal is equal in length to the interspheric centertocenter prime vector of 1 interval = 2 radii).
The cuboctahedron with quadray coordinates {2,1,1,0}  where the curly braces signify "all 12 permutations of the enclosed terms"  has a volume of 20.[2]
The above cited article mentions all of this in passing, but is primarily about how to use Visual FoxPro in conjunction with freeware off the internet to (a) render colorful geometric shapes of intrinsic interest to students (including geodesic spheres  website) and (b) learn object oriented programming concepts in the process.
Those of you who have explored the interface between 20th century philosophy and geometry will recognize the above approach to volumetric mensuration, with the tetrahedron as primary, as essentially the same as the one pioneered by Dr. R. B. Fuller in his '4D geometry'  wherein '4D' refers neither to "3D + Time," nor to Cartesian hyperspace, but to the tetrahedron's paradigmatic status as the minimal volumetric system.[3]
In Fuller's '4D geometry', all objects are ab initio volumetric, even if we restrict their degrees of freedom to a plane or a point, i.e. no 'dimension ladder' with rungs labeled 0,1,2,3 (with fractional dimension interpolations) is natively defined in this particular philosophical language.
Traditional classroom geometry ports to this alternative definitional environment with only minor conceptual and terminological adjustments i.e. Euclidean and Cartesian language games coexist with 4D geometry without significant friction in this new curriculum context.
The quadrays apparatus has been developed by individuals working solo, at first unbeknownst to one another, but coming together via the internet to contribute various puzzle pieces. The article identifies some of the key players in passing, with more details at my website.[4]
As for myself, I am more an implementor than an inventor of the quadrays game, and do not push them at my website as any kind of "next big thing". I'm personally most interested in using quadrays in low key fashion as one more pedagogical device for assisting in philosophical investigations into the definitional framework (cultural) within which our mathematical style of thinking is embedded, in directions pioneered by the late philosopherengineer Ludwig Wittgenstein.[5]
Kirby
[1] although it makes sense to hyperlink quadrays to barycentric coordinates for pedagogical purposes, quadrays are not a subspecies of the barycentrics.
Quadrays also include analogous implementations as planar or linear games i.e.
Linear:
<*> (0,1) (0,0) (1,0)
and:
Planar:
(0,0,1)    * (0,0,0) / \ / \ / \ (1,0,0) (0,1,0) [2] Robert Gray has proved David Chako's conjecture that any tetrahedron with fcc vertices will have a whole number volume relative to the unitvolume tetrahedron comprised of 4 intertangent unitradius fcc spheres (i.e. the quadrays "home base" tetrahedron). It follows that any tetrahedralizable volume with fcc vertices is likewise wholenumber volumed.
[3] '4D geometry' is embedded within Fuller's "explorations in the geometry of thinking" or "synergeticenergetic geometry" or "synergetics" for short, a 20th century philosophy originally published in two volumes, by R. Buckminster Fuller, USA Medal of Freedom winner, and E.J. Applewhite, former deputy inspector general of the CIA. This work is now available on the web via my http://www.teleport.com/~pdx4d/links.html, thanks to Robert Gray et al.
[4] cite http://www.teleport.com/~pdx4d/quadintro.html
[5] cite http://www.teleport.com/~pdx4d/quadphil.html

