Draft web text re vectors, quadrays, python, spatial geometry, crystallography, peer review solicited
Kirby Urner
urner at alumni.princeton.edu
Sat May 27 21:26:59 CEST 2000
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The essay below is on its way to the web, where it will be
supplemented with pictures and hyperlinks. I'm posting it
here with an invitation for feedback re how to improve it.
Please cc: pdx4d at teleport.com if you have pertinent remarks
and/or questions. Thanks in advance.
Kirby Urner
4D Solutions
May 27, 2000
=======================================
[NO TITLE]
INTRODUCTION
Quadrays may be considered a subclass of a more generic Vector
class. Quadrays define four basic directions, from the center
of a regular tetrahedron to its four corners, and map all other
points in space relative to these four vectors, labeled
(1,0,0,0)(0,1,0,0)(0,0,1,0)(0,0,0,1) -- with (0,0,0,0) their
common origin.
Are quadrays new? They certainly have some features in common
with the barycentrics and other homogenous coordinate systems.
This essay summarizes the work of various individuals without
advancing anyone's claims to priority. I invite readers to
provide me with feedback about prior and/or contemporaneous
partially overlapping research.
GEOMETRIC IMPLEMENTATION
Quadrays may be scaled and added in the usual fashion when
considered geometrically, i.e. a vector sum may be depicted as
a series of arrows placed tip to tail. The end result is a
vector from the origin to the last vector tip. Multiplication
by a floating point or real number k stretches a "Qvector" if
k>1 and shrinks it if 0=<k<1. If k<0, the negative sign
behaves as a "reversal operator," reorienting a Qvector
by 180 degrees, after which it is scaled by |k|. These
geometric cartoons are essentially equivalent to the Cartesian
ones.
ALGEBRAIC IMPLEMENTATION
Algebraically, the methods are slightly different. No vector
need involve all four quadrays in its construction, as any
point is either:
* in one of four quadrants
* in a plane between Qvectors
* on a basis Qvector
* is the origin itself
The corresponding 4-tuples will have 1,2,3 and 4 zeros
respectively, i.e. at least one coordinate will always be 0. We
may write this as {a,b,c,0} for a,b,c >=0, where {} means
"all permutations of the enclosed members". Note also that we
have no need for negative numbers in the normalized 4-tuple,
given all four basis vectors are positive (or unsigned).
COMPUTER LANGUAGE IMPLEMENTATION
My background as a math teacher, computer literarcy text
book contributor, and computer programmer, predisposes me
to blend more conventional mathematics notations with a
programming language.
In March, 1999, FoxPro Advisor, a trade magazine for Xbase
application developers, published my article about using
quadrays to drive ray tracings of colorful polyhedra
(see: http://advisor.com/Articles.nsf/ID/FA9903.URNEK01).
Since writing that article, I've become enamoured of Python
as a teaching language, because of the interactive command
line environment and spare, readable syntax. And unlike
Xbase, Python is free and cross-platform.
The section below is a transcript of an interactive Python
session, with >>> prompting user input. Computer output is
flush to the left.
Python 1.6a2 (#0, Apr 6 2000, 11:45:12) [MSC 32 bit (Intel)] on win32
Copyright 1991-1995 Stichting Mathematisch Centrum, Amsterdam
IDLE 0.6 -- press F1 for help
>>>
>>> from coords import Qvector
>>> v = Qvector((1,0,0,0)) # v = one of the basis quadrays
>>> -v # -v still has all positive coords
Qvector (0, 1, 1, 1)
>>> w = Qvector((2,3,1,1)) # all 4 coords > 0
>>> w # normalized form of same Qvector
Qvector (1, 2, 0, 0)
>>> x = Qvector((0,1,1,1)) # compare with -v
>>> y = v + x # y = v + (-v)
>>> y
Qvector (0.0, 0.0, 0.0, 0.0)
>>> z = Qvector((3,3,3,3)) # any (d,d,d,d) = (0,0,0,0)
>>> z
Qvector (0, 0, 0, 0)
>>> r = Qvector((-1,-2,0,4)) # use negatives when initializing
>>> r # normal form has no negatives
Qvector (1, 0, 2, 6)
NORMALIZATION
Clearly vector addition and negation are being handled according
to some different (non-XYZ) rules. Specifically, to obtain the
normal form of a quadray, we subtract its lowest coordinate from
the other 4, an operation with no net effect because (d,d,d,d)=
(0,0,0,0). Here's the algorithm:
def norm(self,plist):
# normalize such that 4-tuple all non-negative members
# with at least one zero member
return tuple( map(lambda x, y=min(plist): x-y, plist) )
The "lambda" operator is Python's way of letting us define a
function "on the fly", with x, y as parameters in this case.
plist is a passed 4-tuple, e.g. (-1,-2,0,4), with min(plist)
putting the lowest coordinate e.g. -2 in y by default. The
"map" operator then applies the rule x-min(plist) to each
member of plist, i.e. x becomes each member of plist in turn.
The resulting tuple is returned as the normalized form of
the quadray. This method is invoked whenever a quadray is
initialized, perhaps as a result of vector addition or
subtraction.
NOMENCLATURE
Whether we use the term "quadray" is not especially critical
and depends on the namespace of the author. David Chako
originally introduced his 4-tuple vector algebra on Synergetics-L
in December, 1996, without using this term, whereas D. Lloyd
Jarmusch mentioned using "quadrays" as early as 1981 for
essentially the same apparatus. Josef Hasslberger made a
similar beginning under the heading of "tetra space coordinates."
At one point I played with calling the six Cartesian vectors
(i,j,k,-i,-j,-k) "e-rays", because they go from (0,0,0) at the
center of a tetrahedron through its six mid-edges (hence "e"
for edge). By extension, quadrays could be called "v-rays",
because they go through the tetrahedron's four vertices, or,
equivalently, "f-rays" through the face-centers (Hasslberger's
depiction) -- "equivalently" because the tetrahedron is
self-dual, i.e. "through the faces" is "through the vertices"
of the dual, also a regular tetrahedron.
More important than what they're called is how they work, and
here again we have some divergence in their implementation.
QUADRAYS, TETRAYS & SIMPLICIAL COORDINATES
The most detailed elaboration of quadrays of which I am
personally aware occured on Synergetics-L, subsequent to
Chako's getting the ball rolling, and in this context the
research forked into quadrays and tetrays as distinct topics,
the difference being that tetrays use the identity (1,1,1,1)
to tick off an additional, fifth "temporal direction".
Possibly we'll end up classifying quadrays as a kind of
"simplicial coordinate system", where "simplicial" derives from
"simplex", another name for the tetrahedron. The differences
in implementation may reduce to conventions around how we
choose to normalize our 4-tuples.
ALTERNATIVE NORMALIZATIONS
Quadrays as originally defined by Chako used only non-negative
integers, meaning {a,b,c,0} did not sum to any specific
constant k = a+b+c+d. On the other hand, because (d,d,d,d)
may be added to (a,b,c,d) with no net effect -- (d,d,d,d)=
(0,0,0,0) -- we have the option to normalize to any integer
we like. The same rules apply if we allow (a,b,c,d) to be
real or floating point numbers.
Tom Ace used the k=0 normalization to derive a dot product,
and a more streamlined distance method, proving the importance
of remaining flexible about how we normalize.
def norm0(self):
# normalize such that sum of 4-tuple members = 0
a,b,c,d = self.quadray()
k = (a+b+c+d)/4.0
return tuple( map(lambda x,y=k: x-y, [a,b,c,d]) )
def dot(self,v1):
# return the dot product of self with another vector
# return a scalar
scalar = 0
a = self.norm0()
b = v1.norm0()
for i in range(4):
scalar = scalar + a[i] * b[i]
return 0.5*scalar
def length(self):
# return this vector's length
# same method as for XYZ vectors
return self.dot(self) ** 0.5
MORE QUADRAY METHODS
Ace also came up with a cross product method and 4x4 rotation
matrices, with (1,0,0,0)...(0,0,0,1) as the axes of rotation,
none of which depend on normalization to zero. Here's a cross
product method:
def cross(self,v1):
# return the cross product of self with another vector
# return a Qvector
A=Qvector((1,0,0,0))
B=Qvector((0,1,0,0))
C=Qvector((0,0,1,0))
D=Qvector((0,0,0,1))
a1,b1,c1,d1 = v1.quadray()
a2,b2,c2,d2 = self.quadray()
k= (2.0**0.5)/4.0
sum = (A*c1*d2 - A*d1*c2 - A*b1*d2 + A*b1*c2
+ A*b2*d1 - A*b2*c1 - B*c1*d2 + B*d1*c2
+ b1*C*d2 - b1*D*c2 - b2*C*d1 + b2*D*c1
+ a1*B*d2 - a1*B*c2 - a1*C*d2 + a1*D*c2
+ a1*b2*C - a1*b2*D - a2*B*d1 + a2*B*c1
+ a2*C*d1 - a2*D*c1 - a2*b1*C + a2*b1*D)
return k*sum
As Tom Ace explains at his website, the above is simply an
expansion of the following determinant:
| A B C D |
| k k k k |
| a1 b1 c1 d1 |
| a2 b2 c2 d2 |
VOLUME METHODS
Ace also developed a volume method consistent with the usual
equivalence in XYZ of a scalar triple product with a
determinant i.e.
| Ax Ay Ax |
A dot (B cross C) = | Bx By Bz | (A,B,C are vectors)
| Cx Cy Cz |
The scalar triple product likewise gives the volume of a
parallelepiped with edges ABC from a common vertex, or of
a corresponding tetrahedron, if we multiply the above
scalar by k=(1/6) -- plus take the absolute value if we
want to eliminate a possible negative sign.
The volume of tetrahedron ABCD using quadray coordinates
is likewise |k*det[M]| where AB, AC, AD = quadrays
(a1,b1,c1,d1),(a2,b2,c2,d2),(a3,b3,c3,d3) and M is
the matrix:
| 1 1 1 1 |
| a1 b1 c1 d1 |
| a2 b2 c2 d2 |
| a3 b3 c3 d3 |
If we set k=(1/4), then the volume of a regular tetrahedron
with all edges 1 would be 1 (see below). An regular
icosahedron consisting of 20 irregular tetrahedra, with radials
of ~0.951 and outer edges = 1, would have a volume of about
18.51.
>>> ico = Icosahedron() # create Icosahedron object
>>> ico.vertices["O1"].length() # a center-to-vertex vector
0.95105651629515353
>>> ico.volume # compute volume
18.512295868219159
The Icosahedron's constructor method sets the starting volume
using the expression: 20*vol(O1,P1,R1) where the vol method
returns (1./4.)*abs(det((O1,P1,R1))). O1,P1,R1 are vectors from
the icosa's center to the three vertices of one triangular facet.
Vectors define direction. Any 3 non-coplanar edges of a
tetrahedron, expressed as vectors, will provide sufficient
information to determine the other three edges. These
vectors may either share a common vertex, or define an
"open triangle" zig-zag. Either way, they provide
sufficient input for various vector-based volume methods.
In contrast, simple edge lengths, being scalar quantities,
do not define direction. To specify three non-coplanar
lengths does not uniquely determine the remaining three
edges. If we label a tetrahedron's six edges a,b,c,d,e,f
-- a,b,c connecting a common vertex to an opposite
triangle with edges d,e,f -- then we will need all
six values in order to compute a specific volume.
Leonhard Euler long ago solved the problem of how to express
a tetrahedron's volume solely in terms of its edge lengths.
Java programmer Gerald de Jong rederived the method and
expressed his result in the form of a Java applet.
The computation involves using "closed triangles" (edges
from the same face), "open triangles" (3-edge zig-zags),
and "opposite pairs" (edges with no endpoints in common).
Gerald's expression is also specifically designed to return
a volume of 1 when the six edges of the tetrahedron are 1.
Here's a Python version of the algorithm:
def vol2(edges):
a2,b2,c2,d2,e2,f2 = map(lambda x: x**2, edges)
open = ( f2 * a2 * b2 + d2 * a2 * c2
+ a2 * b2 * e2 + c2 * b2 * d2
+ e2 * c2 * a2 + f2 * c2 * b2
+ e2 * d2 * a2 + b2 * d2 * f2
+ b2 * e2 * f2 + d2 * e2 * c2
+ a2 * f2 * e2 + d2 * f2 * c2 )
closed = ( a2 * b2 * d2 + d2 * e2 * f2
+ b2 * c2 * e2 + a2 * c2 * f2 )
oppos = ( a2 * e2 * (a2 + e2)
+ b2 * f2 * (b2 + f2)
+ c2 * d2 * (c2 + d2))
return (abs(open - closed - oppos)/2.0)**0.5
This method has nothing specifically to do with quadrays, since
length is an attribute of the generic Vector class. I mention
it more because of its historical role in the quadray thread
on Synergetics-L.
David Chako conjectured that all tetrahedra with vertices at
the centers of fcc spheres might have whole number volumes,
provided the initial tetrahedron -- defined by four intertangent
fcc spheres -- were defined as volumetric unity.
Until Robert Gray developed an algebraic proof for this
conjecture, de Jong's Java applet and a MathCad worksheet
featuring Euler's original formula, provided an initial
source of empirical evidence for its validity (i.e. failed
to turn up any counter-examples). Other volume methods
would have served, but this was how events actually
unfolded.
In terms of quadrays, fcc sphere centers may be expressed
as sums of vectors with coordinates {2,1,1,0}. For example,
if s were an fcc sphere center, s + (2,1,1,0) + (1,1,2,0)
+ (1,2,0,1) = (4,4,3,1) = (3,3,2,0) would be another fcc
sphere center relative to s.
I will return to quadrays in a sphere packing context in
the notes on crystallography below.
CONVERTING TO/FROM XYZ (ONE POSSIBLE IMPLEMENTATION)
Differences in implementation may arise around how we choose to
define quadrays in relation to XYZ. Perhaps the most obvious
convention would be to define the length of each basis vector
{1,0,0,0} as unity.
However, in my own research, it's more useful and practical
to apply a different "control length", such that the six edges
of the "home base" tetrahedron are either 1 or 2 units. This
is because my home base tetrahedron is constructed by closest-
packing four equi-radius spheres, such that sphere centers
define the tetrahedron's four vertices. This means the
tetrahedron's edges are 2R (R=sphere radius) or 1D (1=sphere
diameter). Therefore I want my four quadrays to have a length
of root(6)/2 or root(6)/4 respectively.
>>> v = Qvector((1,0,0,0))
>>> v.length()
0.61237243569579447
>>> 6**0.5/4.0 # compare with root(6)/4
0.61237243569579447
With these preliminaries out of the way, we have sufficient
information to uniquely pair normalized (a,b,c,d) Qvectors
with Cartesian (x,y,z) Vectors and vice versa:
>From the Vector class:
def quadray(self):
# return (a,b,c,d) quadray based on current (x,y,z)
x=self.xyz[0]
y=self.xyz[1]
z=self.xyz[2]
a = (2/root2) * ((x>=0)*x + (y>=0)*y + (z>=0)*z)
b = (2/root2) * ((x<0)*(-x) + (y<0)*(-y) + (z>=0)*z)
c = (2/root2) * ((x<0)*(-x) + (y>=0)*y + (z<0)*(-z))
d = (2/root2) * ((x>=0)*x + (y<0)*(-y) + (z<0)*(-z))
return self.norm((a,b,c,d))
>From the Qvector subclass of Vector:
def __init__(self,arg,*flag):
# initialize a vector at an (a,b,c,d) tuple (= arg)
# NOTE: in accompanying essay, xyz units = sphere diameter
# i.e. Vector((1,0,0)).length() is 1 D, therefore quadray
# inputs must be scaled by 1/2 to fit this context, i.e.
# tetra edge defined by 2 basis quadrays = 1 D.
if len(arg)==3: arg = Vector(arg).quadray() # if 3-tuple passed
self.coords = self.norm(arg)
a,b,c,d = self.coords
self.xyz = ((0.5/root2) * (a - b - c + d),
(0.5/root2) * (a - b + c - d),
(0.5/root2) * (a + b - c - d))
Let's look at these methods in action:
>>> from coords import *
>>> i = Vector((1,0,0))
>>> i.xyz
(1, 0, 0)
>>> i.quadray()
(1.4142135623730949, 0.0, 0.0, 1.4142135623730949)
>>> j = Vector((-1,0,-3))
>>> q = Qvector(j.quadray())
>>> q
Qvector (0.0, 1.4142135623730949, 5.6568542494923797, 4.2426406871192848)
>>> q.xyz
(-0.99999999999999978, 0.0, -2.9999999999999996)
Note that floating point numbers lose a small amount of
information as we convert back and forth between Qvectors and
Cartesian Vectors (or "Cvectors").
USEFULNESS IN SPATIAL GEOMETRY
"Are quadrays useful?", is the final question I should address,
as I reach the close of this essay. I've found them to be so
in the context of conceptualizing the relationships among
polyhedra.
The initial home base tetrahedron with coordinates (1,0,0,0),
(0,1,0,0)(0,0,1,0)(0,0,0,1) will generate 22 other points of
interest by simple vector addition (negation is not required).
These 26 points A-Z are sufficient to define a tetrahedron,
inverse tetrahedron, cube, octahedron, rhombic dodecahedron
and cuboctahedron (David Chako supplied coordinates for the
first few of these in his initial post on the topic).
Furthermore, the size and orientation of these six shapes
matches that given by R. Buckminster Fuller in his 'Synergetics',
wherein these shapes all have whole number volumes, given
the tetrahedron is Fuller's geometric model of 3rd powering
(i.e. 1x1x1 = unit volume tetrahedron).
So with Qvectors, we bring whole number coordinates to an initial
set of shapes which already have whole number volumes. This
is an aesthetically pleasing construct, and may help streamline
the introduction to both spatial geometry and object-oriented
programming, as per my "numeracy + computer literacy" approach
(currently Python-focused).
"""
By Kirby Urner, Oregon Curriculum Network
Ver 0.1: May 24, 2000
"""
from coords import *
import math
#==================[ Points of Interest ]====================
"""
* 26 data points A-Z define six polyhedra in the concentric hierarchy
* the Jitterbug Transformation creates a basis for additional vertices
Labels of Numbers of
Shape Volume Vertices Vertices, Edges, Faces
---------------------------------------------------------
Tetrahedron 1 A-D 4 6 4
Inv Tetra 1 E-H 4 6 4
Duo-tet Cube 3 A-H 8 12 6
Octahedron 4 I-N 6 12 8
Rh Dodecahedron 6 A-N 14 24 12
Cuboctahedron 20 O-Z 12 24 14
See: http://www.inetarena.com/~pdx4d/ocn/graphics/povlabels.gif
Using the quadray apparatus (4 basis vectors to the corners
of a regular tetrahedron with edges = 1 sphere diameter)
See: http://www.teleport.com/~pdx4d/quadray/quadray.gif
"""
A = Qvector((1,0,0,0)) # center to corner of tetrahedron
B = Qvector((0,1,0,0)) # "
C = Qvector((0,0,1,0)) # "
D = Qvector((0,0,0,1)) # "
# tetrahedron's dual (also a tetrahedron i.e. inverted tet)
E,F,G,H = B+C+D, A+C+D, A+B+D, A+B+C
# tetrahedron + dual (inverted tet) = duo-tet cube
# octahedron vertices from pairs of tetrahedron radials
I,J,K,L,M,N = A+B, A+C, A+D, B+C, B+D, C+D
# octahedron + dual (cube) = rhombic dodecahedron
# cuboctahedron vertices from pairs of octahedron radials
O,P,Q,R,S,T = I+J, I+K, I+L, I+M, N+J, N+K
U,V,W,X,Y,Z = N+L, N+M, J+L, L+M, M+K, K+J
...
APPLICATIONS TO CRYSTALLOGRAPHY
Quadrays take to a sphere packing context like fish to water
and so might serve some streamlining role in crystallography.
Scott Childs has researched this possiblity. Russell Chu's
investigations are also relevant in this connection.
The 12 quadrays {2,1,1,0} define the vertices of a cuboctahedron
relative to the origin (0,0,0,0). This arrangement, with
12 nearest neighbors at the corners of a cuboctahedron, is
the face centered cubic lattice (fcc).
Quadrays with integer coordinates define the centers of
fcc spheres, as well as the centers of all tetrahedral and
octahedral voids between these spheres.
Given the volume expression (1/4)|k*det[M]|, we know that
any non-coplanar arrangement of any four such vertices (i.e.
with integer coordinates) will define a tetrahedral volume
evenly divisible by 1/4. Those tetrahedra with all vertices
in the same fcc lattice will have whole number volumes
(a sufficient, but not a necessary condition).
Selectively illuminated, these all-integer, quadray defined
vertices may present face centered, body centered, hexagonally
close packed, or simple cubic arrangements of points.
We may also sort all these vertices into four interpenetrating
fcc lattices, using the rhombic dodecahedron for guidance.
Rhombic dodecahedra fill space in an fcc pattern, with the
spheres inscribed and intertangent through the rhombic face
centers. The corners of the rhombic dodecahedron occur at the
centers of the voids surrounding each fcc sphere.
The rhombic dodecahedron has 8 points at the opposite ends of
short face diagonals, in a cubic arrangement -- this is the
cube of volume 3, relative to the dodeca's volume of 6. This
cube defines two interpenetrating unit-volume tetrahedra, A-D
and E-H, as per the above labeling scheme (see 26 points of
interest). To each tetrahedron, there corresponds a unique fcc
lattice.
In addition, the remaining 6 vertices of the rhombic
dodecahedron, which occur at opposite ends of long face
diagonals, define an octahedral arrangement. The corresponding
vectors, I-N, define yet another fcc lattice.
The sphere at the center of our rhombic dodecahedron, at
(0,0,0,0), has 12 nearest neighbors O-Z. This is the original
lattice. So we may identify the four lattices as follows:
{(1:'O-Z'),(2:'I-N'),(3:'A-D'),(4:'E-H')}.
Vertices from 1+2, or 3+4, will define a simple cubic lattice.
All 4 lattices taken together will define a body cubic centered
lattice. The hcp arrangement cannot be expressed as a sum of
whole fcc lattices, but its vertices will nevertheless have all
integer quadray coordinates.
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