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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 information, 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 generating 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 together 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 <2,6> 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