Entity-oriented spatial coding and discrete topological spatial relations

9.7 MB·

by  Weining Zhu

ENTITY-ORIENTED SPATIAL CODING AND DISCRETE
TOPOLOGICAL SPATIAL RELATIONS
WEININGZHU
6
1 Abstract. Basedonanewlyproposedspatialdatamodel–spatialchromatic
0 model(SCM),wedevelopedaspatialcodingscheme,calledfull-codedordinary
2 arrangedchromaticdiagram(full-OACD).Full-OACDisatypeofspatialtes-
sellation,wherespaceispartitionedintoanumberofsubspacessuchascells,
n edges,andvertexes. Thesesubspacesarecalledspatialparticlesandassigned
a with unique codes – chromatic codes. The generation, structures, compu-
J
tations, and properties of full-OACD are introduced and relations between

5 chromaticcodesandparticlespatialtopologyareinvestigated,indicatingthat
1 chromatic codes provide a potential useful and meaningful tool not only for
spatialanalysisingeographicalinformationscience,butalsoforotherrelevant

] disciplinessuchasdiscretemathematics,topology,andcomputerscience.
G
C
.
s
c 1. Introduction
[
Coding the objects has been widely used in many scientific and technological

1 fields, such as telecommunications, bioinformatics, and computer cryptography,
v
in which information has been expressed, transferred, and interpreted by various
7
codes in numbers, strings, or symbols. In geographic information science (GIS),
1
8 there are also some relevant applications of coding. For example, a geographical
3 coordinatesystemsprovidesacodingschemeusingasingleoraseriesofcoordinates
0 to represent a spatial entity or region [1], [2]. Spatial index assigns codes (indexes)
.
1 to spatial objects so that they can be rapidly retrieved from spatial databases [3],
0 [4]. Ingeocodingsystems,landlotsandzipcodesallowspatiallocationsandpostal
6 addresses to be readily memorized and exclusively identified [5], [6].
1
The objective of this study is to do the similar work for coding the pure space
:
v itself. ActuallyaplannarCartesiancoordinatesystemisalsoacodingschemewhere
Xi apointinspaceiscodedbysuchasacoordinate(x,y). Basedonanewlyproposed
GIS data model – spatial chromatic model (SCM) [7], we suggest a spatial coding
r
a scheme, called full-coded ordinary arranged chromatic diagram (full-OACD). Full-
OACDcanbetakenasanextensionofOACD,whichisastandardpatternofSCM.
SCM has demonstrated its significant potentials for GIS theories and applications
in diverse aspects: the first law of geography, reasoning spatial topology, point
pattern recognition, and generalized Voronoi algorithms, etc. [8], [7].
Space in SCM is defined as the object-oriented space where the elementary unit
is a cell. A cell is characterized by its chromatic code, typically a string of nat-
ural numbers. One problem of OACDs is that only cells are coded, but cellular
boundariesandfeaturenodes,suchasedgesandvertexesgeneratedfromhalf-plane
partitions, have not been coded, and hence we may lose some particular spatial in-
formation, for example, the subspaces somewhere that are unable to be assigned
Key words and phrases. Spatialcoding,spatialtopology,spatialchromaticmodel.
1
2 WEININGZHU
to any cell. To solve this problem, we therefore extended OACD to full-OACD,
a full-space coding scheme. In full-OACD, all spatial components, including cells,
edges and vertexes, are coded in a spatially and mathematically consistent way.
The below sections will introduce, analyze, and discuss the procedures of gener-
ating full-OACDs, some important definitions, notations, properties, and theorems
(Section 2), topological relations among cells, edges, vertexes, and complexes (Sec-
tion 3), as well as their spatial implications, notes, and suggested future work
(Section 4).
2. Full-coded ordinary arranged chromatic diagram
Let P = {p ,p ,...,p } is a point set containing n points associated with an
1 2 n
index set I = {1,2,...,n}. The point set is also called the generator set and
points in P are generators, which can be treated as geographical entities or just
anygeneralobjects. ThesetQisafamilyofsubsetsofP consistingofallunordered
point-pairs in P, that is, Q={{p ,p }|p ,p ∈P, i(cid:54)=j, i,j ∈I}. The generation
i j i j
of a full-OACD follows the below steps.
Step (1): With respect to a point-pair q ={p ,p }∈Q, using their perpen-
i j
dicular bisector pb(cid:104)i,j(cid:105) to partition the space into two half-planes hp(i,j)
andhp(j,i),whereapointpinhp(i,j)iswithEuclideandistanced(p,p )<
i
d(p,p ), in hp(j,i) with d(p,p ) < d(p,p ), and in pb(cid:104)i,j(cid:105) with d(p,p ) =
j j i i
d(p,p ).
j
Step (2): Assign two half-planes hp(i,j) and hp(j,i) the codes (p0, p0, ...,
1 2
p1,...,p0)and(p0,p0,...,p1,...,p0),respectively,inwhichthesubscript
i n 1 2 j n
numbercorrespondstotheindexofeachpoint,andthesuperscriptnumber
is the assigned numerical variable t(q). In this way, only for points p or
i
p , t(q) = 1, but for the others, t(q) = 0. Similarly, assign the bisector
j
1 1
pb(cid:104)i,j(cid:105) with code (p0, p0, ..., p2, ..., p2, ..., p0), that is, for both p and
1 2 i j n i
p , t(q) = 1, but for the others, t(q) = 0. See the simplest full-OACD
j 2
generated from two entities in Fig.1.
Step (3): Repeatsteps(1)and(2)forallk = 1n(n−1)point-pairsinQ,and
2
then overlay the 2k half-planes so that they generate a spatial tessellation,
containing a number of faces, edges, and vertexes.
Step (4): The chromatic code of each face, edge, and vertex is the sum of
the values t(q) that are acquired from each half-plane partition, that is,
 (cid:80) (cid:80) (cid:80) (cid:80) 
t(q) t(q) t(q) t(q)
q∈Q q∈Q q∈Q q∈Q
p1 ,p2 ,...,pi ,...,pn  (2.1)
NotethatthepointsetP couldbeinanydimensionalspaceRm,andhenceeach
partitiondividesthespaceintotwohalf-spacesratherthanhalf-planes. Thisstudy
mainlyfocusesontheplanarfull-OACDsinspaceR2. Fig.2showstheprocedureof
generatingafull-OACDfrom3points(Fig.2a)inplane, denotedbyOACD(3,R2).
Through half-plane partitions, we get 6 half-planes in Fig.2b-2d, then we overlap
them together into a diagram such that in Fig.2e, and finally we sum the t(q)’s to
compute chromatic code for each subspace in the diagram Fig.2f.
In step (2), if we do not assign t(q) = 1 to any bisectors, then the obtained
2
diagram is OACD. Therefore edges and vertexes in OACDs are without codes.
ENTITY-ORIENTED SPATIAL CODING 3
Figure 1. A full-coded OACD generated from 2 entities.
This makes the important difference bewteen OACD and full-OACD, where edges
and vertexes are with codes.
Thesubspaces, i.e., faces, edges, andvertexesgeneratedinfull-OACDarecalled
spatial particles (denoted by Ω), and faces are particularly called cells (denoted by
ζ), whichhasbeenpreliminarilystudiedinOACD[8], [7]. Chromaticcodesofpar-
ticlesaren-tuplessuchasΩ(t ,t ,...,t ),inwhichthenumbert iscalledthechro-
1 2 n i
matic component of p in the code, or the component at location i. Easy to know
i
thatt willbeeitherintegerorhalf-integer. Sometimes,ifweareonlyinterestedin,
i
say, components of p and p , then a chromatic code Ω(t ,t ,...,t ,...t ,...,t )
i j 1 2 i j n
can be rewritten in a short form such as Ω(t ,t )∪(T ), or just Ω(t ,t ).
i j others i j
Fig.3showsanothertwoexamplesoffull-OACDs. Fig.3aisanoriginalfull-coded
OACD(4,R2) and Fig.3b is a homomorphic part of a full-coded OACD(6,R2),
where each spatial particles are coded in 6-tuples. Observing particle patterns and
codes in these full-OACDs we can find out many interesting properties.
Definition 1. Given a particle Ω(t ,t ,...,t ), the ascending order of its chro-
1 2 n
matic components is called the chromatic base of the particle, and denoted by
β(Ω)={t(cid:48),t(cid:48),...t(cid:48) }.
1 2 n
Forexample, cellsζ (0,2,3,1)andζ (2,1,3,0)bothhavethesamebaseβ(ζ )=
1 2 1
β(ζ ) = {0,1,2,3}. If two components are equal, their orders are in random. For
2
example, the base of edges (3,0,3,3) and (3,3,3,0) are both {0,3,3,3}. Chro-
2 2 2 2 2 2
matic codes are actually the permutations of different bases. In previous studies,
chromatic base was also called the primary code of a cell [8].
Definition 2. If two particles Ω (t , t , ..., t , ..., t ) and Ω (t , t , ...,
1 11 12 1i 1n 2 21 22
t , ..., t ) have the same chromatic codes, then they are called equi-color, and
2i 2n
denoted by Ω =Ω , that is,
1 2
∀i,t =t ⇔Ω =Ω (2.2)
1i 2i 1 2
otherwise, Ω (cid:54)=Ω .
1 2
If they have the same chromatic bases, then they are called equi-base, denoted by
Ω ∼=Ω , that is, if β(Ω )={t(cid:48) , t(cid:48) , ..., t(cid:48) , ..., t(cid:48) } and β(Ω )={t(cid:48) , t(cid:48) , ...,
1 2 1 11 12 1i 1n 2 21 22
t(cid:48) , ..., t(cid:48) }, then
2i 2n
∀i,t(cid:48) =t(cid:48) ⇔Ω ∼=Ω (2.3)
1i 2i 1 2
otherwise, Ω (cid:29)Ω .
1 2
4 WEININGZHU
R
(a)
G
B
0,1,1
22 1,1,0
22
1,0,0
1,0,0
0,1,0 0,0,1
1,0,1
2 2 0,0,1 0,1,0
(b) (c)
(d)
0 12 12
0 1 0 1 0 0
112 012 00 0 1 01 0 0 32,32,0 2,12,12
1 0 0
1 0 0 0 0 1 0 0 1 2,1,0
01 10 00 0 12 12 11 00 00 112 00 012 1,2,0 2,0,1 32,0,32
012 10 012 0 1 00 1 01212 012 012 001 000 110 12,2,12 1,1,1 1,0,2
0 1 0 0 0 1 0,2,1
0 1 0 0 0 1
0 0 1 0,1,2
0 1 0 00 00 11 12,12,2
(e) 00 012 112 12 12 0 (f) 0,32,32
0 1 0
Figure 2. The procedure of generating a full-OACD(3, R2). (a)
The generator set consists of three points markedwith color R, G,
and B; (b)-(d) Half-plane partitions and assignments of chromatic
codes with respect to perpendicular bisectors pb(cid:104)B,G(cid:105), pb(cid:104)G,R(cid:105),
and pb(cid:104)R,B(cid:105), respectively. (e) Overlapping all the six half-planes
in (b)-(d) together; and (f) Adding all chromatic components to-
gether to form the chromatic codes.
Property 1. Given two particles Ω and Ω ,
1 2
Ω =Ω ⇒Ω ∼=Ω (2.4)
1 2 1 2
and hence
Ω (cid:29)Ω ⇒Ω (cid:54)=Ω (2.5)
1 2 1 2
This property indicates that if two cells are equi-color, they must be equi-base,
and if they are not equi-base, they are impossible to be the equi-color.
The number of cells, edges, and vertexes in a full-coded OACD(n,R2) depends
on the point pattern of the generator set P. This study mainly focuses on the
ENTITY-ORIENTED SPATIAL CODING 5
2301
1302 Bisector
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2310 <1,2>
3210 <<13,,46>> 452013 345012
<<41,,65>> 352014 254013 245103
0312 <1,3> 253014
<4,5> 342015 245013 145203
3120 <<12,,63>> 243015
<3,5> 045213
0213 <<25,,65>> 234015 235014 145023
045123
3021 0132 Ver2te-Ix 324015 134025 135024
2031 3-I 314025 035124
0123 214035 124035 025134
125034
1032
125043
Note:
32--II vveerrtteexx 12125252 (a) ((12)) ACo =d 1e0s of edges and vertexes are doubled 124053 (b)
1023
Figure 3. Twoexamplesoffull-OACDs. (a)afull-OACD(4,R2);
(b) Homomorphic part of a full-OACD(6, R2).
general cases of P in a plane: (1) no more than two bisectors are parallel, and (2)
no more than three bisectors are concurrent, except that they are generated from
the three point-pairs which make a triangle.
Definition 3. In a general case of the point set P, any three point-pairs from three
different points generate a vertex, called 3-I vertex (i.e., the intersection of three
perpendicular bisectors of a triangle), denoted by ϕ3I; and any two point-pairs from
4 different points generate a vertex, called 2-I vertex (i.e., the intersection of two
perpendicular bisectors), denoted by ϕ2I.
Therefore vertexes ϕ in full-coded OACD(n,R2) are either 2-I or 3-I, see their
examples in Fig.3.
(cid:88)C2
Property 2. An OACD(n,R2) contains n i−C3+1 cells, (C2)2−3C3 edges,
n n n
i=1
C3 3-I vertexes, and 1C2C2 2-I vertexes.
n 2 n n−2
Proof. The proof of the cell number could be referred to [8]. Here we only prove
the edge number. Suppose in a plane there are n lines which intersect with each
other, then each line is divided into n edges by the other n−1 lines, therefore the
n lines will generate n2 edges. The total n point will generate C2 lines (bisectors)
n
and hence (C2)2 edges. But every three points generate a vertex which will reduce
n
3 edges, therefore the total edge number will be (C2)2−3C3. (cid:3)
n n
Property 3. In an OACD(n), the chromatic base of cells is
N={0,1,...,n−1} (2.6)
This property has been proved by [8]. It implies that all cells are equi-base, and
any two components of a cell are not equal. Below we use N[i,j] to denote the
integers between i and j, and also including i and j.
Property 4. In an OACD(n), the chromatic bases of edges are
{N\{z,z+1},z+ 1,z+ 1} (2.7)
2 2
6 WEININGZHU
for z = N[0,n−2], meaning for each z from 0 to n−2, we obtain a base which
removes z and z+1 from N and then add two z+ 1.
2
Particularly, an edge (denote by η) generated by bisector pb(cid:104)i,j(cid:105) bears a code
z+1 z+1
η(x 2,x 2) (2.8)
i j
for z =N[0,n−2].
Proof. Suppose η is the edge between two cells ζ and ζ , therefore before the
1 2
partition of pb(cid:104)i,j(cid:105), ζ and ζ should be merged into a larger cell ζ with code
1 2
(xz,xz), that is, point i and j have the same component z. After the partition,
i j
ζ and ζ ’s codes will be (xz+1,xz) and (xz,xz+1), see the proof of Lemma 2 in
1 2 i j i j
[8]. With respect to all other bisectors pb(cid:104)i,x(cid:105) or pb(cid:104)j,x(cid:105), x ∈ I\{i,j}, if ζ has
not gained any components, then minimum of z could be 0; if ζ always gained
one component for all the other n−2 bisectors, then the maximum of z could be
z+1 z+1
n−2. Therefore η’s chromatic code will be (x 2,x 2), and their bases will be
i j
{N\{z,z+1},z+ 1,z+ 1}, for z =N[0,n−2]. (cid:3)
2 2
Property 5. The chromatic bases of 2-I vertexes are
{N\{z ,z ,z +1,z +1},z + 1,z + 1,z + 1,z + 1} (2.9)
1 2 1 2 1 2 1 2 2 2 2 2
for z =N[0,n−4] and z =N[z +2,n−2]. Particularly, a vertex ϕ2I generated
1 2 1
by two bisectors pb(cid:104)i,j(cid:105) and pb(cid:104)u,v(cid:105) bears a code
ϕ2I(xzi1+12,xzj1+12,xzu2+12,xzv2+12) (2.10)
or
ϕ2I(xzi2+12,xzj2+12,xzu1+12,xzv1+12) (2.11)
for z =N[0,n−4] and z =N[z +2,n−2].
1 2 1
Proof. Supposepb(cid:104)i,j(cid:105)andpb(cid:104)u,v(cid:105)arethelasttwobisectorspartitioningamerged
cell,thenaccordingtotheLemma2in[8],beforethetwopartitions,thecellshould
be with a code such as (xzi1,xzj1,xzu2,xzv2). Let z1 is the smaller integer, and then
z = z +∆. After the two partitions by pb(cid:104)i,j(cid:105) and pb(cid:104)u,v(cid:105), four new cells will
2 1
be generated with codes
(xz1,xz1+1,xz1+∆+1,xz1+∆) (2.12)
i j u v
(xz1+1,xz1,xz1+∆+1,xz1+∆) (2.13)
i j u v
(xz1,xz1+1,xz1+∆,xz1+∆+1) (2.14)
i j u v
(xz1+1,xz1,xz1+∆,xz1+∆+1) (2.15)
i j u v
If ∆=0 or 1, then we can always find that in some codes of Eq.(2.12)-(2.15), two
components are equal. For example, if ∆ = 0, there are two z ’s and two z +1’s
1 1
in Eq.(2.12), and if ∆=1, there are two z +1’s in Eq.(2.13). But cellular base is
1
N, meaning any two components are not equal, therefore ∆ ≥ 2. Because pb(cid:104)i,j(cid:105)
and pb(cid:104)u,v(cid:105) involve 4 points, then the maximum of z should be n−4, and hence
1
z =N[0,n−4], z =N[z +2,n−2]. Theremainderoftheprooffollowsalongthe
1 2 1
line of the proof of Property 4. (cid:3)
ENTITY-ORIENTED SPATIAL CODING 7
Property 6. The chromatic bases of 3-I vertexes are
{N\{z,z+1,z+2},z+1,z+1,z+1} (2.16)
for z =N[0,n−3]. Particularly, a vertex ϕ3I generated by three bisectors pb(cid:104)i,j(cid:105),
pb(cid:104)j,k(cid:105), and pb(cid:104)k,i(cid:105) bears a code
ϕ3I(xz+1,xz+1,xz+1) (2.17)
i j k
for z =N[0,n−3].
Proof. Suppose before the partitions of pb(cid:104)i,j(cid:105), pb(cid:104)j,k(cid:105), and pb(cid:104)k,i(cid:105), the merged
cellhasacode(xz,xz+∆1,xz+∆2), where∆ ≥0and∆ ≥0. Afterthepartitions,
i j k 1 2
six new cells will be generated with codes
(xz+2,xz+∆1+1,xz+∆2)∪(X ) (2.18)
i j k others
(xz+2,xz+∆1,xz+∆2+1)∪(X ) (2.19)
i j k others
(xz+1,xz+∆1,xz+∆2+2)∪(X ) (2.20)
i j k others
(xz+1,xz+∆1+2,xz+∆2)∪(X ) (2.21)
i j k others
(xz,xz+∆1+1,xz+∆2+2)∪(X ) (2.22)
i j k others
(xz,xz+∆1+2,xz+∆2+1)∪(X ) (2.23)
i j k others
We examine the below possible values of ∆ and ∆ .
1 2
(1) ∆ =1 or ∆ =2, ∆ =1 or ∆ =2.
1 1 2 2
If ∆ = 1 or ∆ = 2, for example, in Eq.(2.20) and (2.19) there will be two
1 1
components equalling z+1 or z+2; similarly, if ∆ = 1 or ∆ = 2, in Eq.(2.21)
2 2
and (2.18) there will be two components equalling z+1 or z+2.
(2) ∆ ≥3, ∆ ≥3.
1 2
AccordingtoEq.(2.22)and(2.23),theremustbevaluesz+1andz+2inX ,
others
because they are not in locations x , x , or x . However, according to Eq.(2.18)-
i j k
(2.21), z+1 and z+2 are already in x , so that they cannot be in X .
i others
From the above two cases we know that the only allowed values of ∆ and ∆
1 2
are both 0, and the merged cell must bear a code
(xz,xz,xz) (2.24)
i j k
Thenattheintersectionofthethreebisectors,the3-Ivertexacquirescomponents
1 at x and 1 at x from pb(cid:104)i,j(cid:105), 1 at x and 1 at x from pb(cid:104)j,k(cid:105), 1 at x and
2 i 2 j 2 j 2 k 2 k
1 at x from pb(cid:104)k,i(cid:105), and therefore gain a code
2 i
(xz+12+12,xz+12+12,xz+12+12)=(xz+1,xz+1,xz+1) (2.25)
i j k i j k
From Eq.(2.18)-(2.23), we know that X do not contain components z, z+1,
others
and z+2, thus we know the base of the 3-I vertex is in form of Eq.(2.16). Because
therangeofz inacellisfrom0ton−1,theminimumz shouldbe0andmaximum
z should be z+2=n−1⇒ z =n−3. (cid:3)
This property indicates that the chromatic codes of 3-I vertexes contain three
identical integers which are different from the rest integers in codes. If cancel one
z+1, Eq.(2.16) can be rewritten as
{N\{z,z+2},z+1,z+1} (2.26)
for z =N[0,n−3].
8 WEININGZHU
Theorem 1. Different types of particles in a full-OACD are not equi-base, that is,
ζ (cid:29)η (cid:29)ϕ2I (cid:29)ϕ3I (2.27)
This theorem provides an approach to determine particle types. For example,
if we see a particle with chromatic components being all different integers, then it
must be a cell; if it contains 2 half-integers, it must be an edge; if it contains 3
equal integers, it must be a 3-I vertex; and if contains 4 half-integers, it must be a
2-I vertex.
Notation 1. The component-counting function H(Ω,m) is a function counting
the number of m in the chromatic code of Ω, that is, the function tells how many
components equal to m.
Definition4. ThedifferencetupleoftwoparticlesΩ (t ,t ,...,t )andΩ (t ,
1 11 12 1n 2 21
t , ..., t ) is defined by
22 2n
Ψ(Ω ,Ω )=(ψ ,ψ ,...,ψ ) (2.28)
1 2 1 2 n
=(|t −t |,|t −t |,...,|t −t |)
11 21 12 22 1n 2n
where ψ =|t −t |.
i 1i 2i
Then the chromatic distance between the two particles is defined by
n
(cid:88)
δ(Ω ,Ω )= ψ (2.29)
1 2 i
i=1
and each ψ is called the chromatic distance at the component i, and denoted by
i
δ(ψ ).
i
In addition, the code distance between two particles is defined by
γ(Ω ,Ω )=n−H(δ(Ω ,Ω ),0) (2.30)
1 2 1 2
The chromatic distance is also called transition number T between two cells in
ourpreviousstudy,anditisactuallytheManhattandistancebetweentwoparticles.
The code distance is actually the Hamming distance between two particles if we
treat their codes and components as strings rather than numbers.
Definition 5. The union of m particles Ω (t , t , ..., t ), Ω (t , t , ..., t ),
1 11 12 1n 2 21 22 2n
..., Ω (t , t , ..., t ) is called a complex or a m-complex, denoted by Θ, and
m m1 m2 mn
its code is given by
m
(cid:88)
Θ{Ω ,Ω ,...,Ω }= Ω (2.31)
1 2 m m
i=1
(cid:32) m m m (cid:33)
(cid:88) (cid:88) (cid:88)
= t , t ,..., t .
i1 i2 in
i=1 i=1 i=1
These m particles are called the elemental particles of the m-complex. If the m
particlesareallcells,thenthem-complexisalsocalledam-cellcluster. Oneparticle
could be taken as a 1-complex.
Theorem 2. If ζ and ζ are two cells in a full-OACD, then
1 2
ζ (cid:54)=ζ , (2.32)
1 2
ENTITY-ORIENTED SPATIAL CODING 9
Figure 4. Two basic structural units of full-OACD. (a) 2-I unit;
(b) 3-I unit.
Figure 5. Three types of particle relations in 2-I/3-I units: adja-
cent (Adj.), interval (Int.) and opposite (Opp.).
Thistheoremhasbeenprovedby[8]. Ittellsthatanytwocellsarenotequi-color
– their codes are unique.
Because any vertex in OACD is either 2-I or 3-I, therefore the a full-OACD
is tessellated by two types of structural units such as the two in Fig.4: the one
containing ϕ2I is called 2-I unit (Fig.4a), and the other containing ϕ3I is called
3-I unit (Fig.4b). According to the proofs of Property 5 and 6, particle codes in
2-I/3-I units should be those shown in Fig.4, and then it is easy to calculate and
prove the below four properties.
10 WEININGZHU
Property 7. A 2-I unit generated by pb(cid:104)i,j(cid:105) and pb(cid:104)u,v(cid:105) contains the following
9 particles.
(1) One 2-I vertex with code
ϕ2I (x + 1,x + 1,x + 1,x + 1) (2.33)
ijuv i 2 j 2 u 2 v 2
(2) Four edges with codes
η (x + 1,x + 1,x +1,x ),η (x + 1,x + 1,x ,x +1) (2.34)
iju i 2 j 2 u v ijv i 2 j 2 u v
η (x +1,x ,x + 1,x + 1),η (x ,x +1,x + 1,x + 1)
uvi i j u 2 v 2 uvj i j u 2 v 2
(3) Four cells with codes
ζ (x +1,x ,x +1,x ),ζ (x +1,x ,x ,x +1) (2.35)
iu i j u v iv i j u v
ζ (x ,x +1,x +1,x ),ζ (x ,x +1,x ,x +1)
ju i j u v jv i j u v
Property 8. A 3-I unit generated by pb(cid:104)i,j(cid:105), pb(cid:104)j,k(cid:105) and pb(cid:104)k,i(cid:105) contains the
following 13 particles.
(1) One 3-I vertex with code
ϕ3I (x +1,x +1,x +1) (2.36)
ijk i j k
(2) Six edges with codes
η (x ,x + 3,x + 3),η (x +2,x + 1,x + 1) (2.37)
jki i j 2 k 2 ijk i j 2 k 2
η (x + 3,x ,x + 3),η (x + 1,x +2,x + 1)
kij i 2 j k 2 jki i 2 j k 2
η (x + 3,x + 3,x ),η (x + 1,x + 1,x +2)
ijk i 2 j 2 k kij i 2 j 2 k
Note, in Eq.2.37, the underlined index indicates the perpendicular bisector which
makes the edge.
(3) Six cells with codes
ζ (x +2,x +1,x ),ζ (x +2,x ,x +1) (2.38)
ijk i j k ikj i j k
ζ (x +1,x +2,x ),ζ (x ,x +2,x +1)
jik i j k jki i j k
ζ (x +1,x ,x +2),ζ (x ,x +1,x +2)
kij i j k kji i j k
Property 9. In 2-I unit space:
(1) The codes of the vertex ϕ2I is the average of (I) two edges which are in the
same bisectors, (II) the two cells which are opposite to the vertex, (III) all the four
edges, and (IV) all the four cells, that is,
ϕ2I = 1(η +η )= 1(η +η ) (2.39)
2 iju ijv 2 uvi uvj
= 1(ζ +ζ )= 1(ζ +ζ )
2 iu jv 2 iv jv
= 1(η +η +η +η )
4 iju ijv uvi uvj
= 1(ζ +ζ +ζ +ζ )
4 iu iv ju jv
(2) The codes of an edge η is the half of the two cells ζ and ζ which are
1 2
respectively on the two sides of the edge. If ξ ={ζ ,ζ }, then,
1 2
η = 1(ζ +ζ )= 1ξ (2.40)
2 1 2 2