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Topic: Conway on Trilinear vs Barycentric coordinates
Replies: 13   Last Post: Feb 16, 1999 8:05 AM

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Kirby Urner

Posts: 4,713
Registered: 12/6/04
Article on Quadrays in March 1999 FoxPro Advisor
Posted: Feb 6, 1999 6:33 PM
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My article touching on quadrays in the context of 'Teaching
Object-Oriented 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:

-- 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

For those unfamiliar with quadrays, this Neo-Cartesian system
uses 4-tuples instead of the familiar xyz 3-tuples, 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

These four vectors span volume (given vector addition and
scalar multiplication of the usual sort), giving a unique,
simplest-terms 4-tuple (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 simplest-terms 4-tuple
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 unit-radius spheres packing context (fcc) and also defines
this tetrahedron to be the unit of volume.

The home base tetrahedron and its self-dual, 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 space-filler within which the
unit-radius fcc spheres nest and "kiss" through the face
centers (12 K-points where long and short diagonals of the
12 rhombic faces intersect, and where long diagonal is equal
in length to the inter-spheric center-to-center 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 co-exist 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 philosopher-engineer
Ludwig Wittgenstein.[5]


[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.


(0,1) (0,0) (1,0)



* (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 unit-volume tetrahedron comprised of 4 intertangent
unit-radius fcc spheres (i.e. the quadrays "home base" tetrahedron).
It follows that any tetrahedralizable volume with fcc vertices is
likewise whole-number volumed.

[3] '4D geometry' is embedded within Fuller's "explorations in
the geometry of thinking" or "synergetic-energetic 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, thanks to Robert
Gray et al.

[4] cite

[5] cite

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