\\ VORONOI GRAPHS in DIMENSIONS 2 to 6: \\ \\ Contiguous Forms to the 7 perfect 6-Dimensional Forms \\ ============================================== \\ \\ The edges of the Voronoi graph are of the form \\ A+t*(B-A) \\ where A is a p6[i] and B a convenient V6[j] put into matrix form \\ by the program vec2mat, e.g \\ p6[1]+t*(vec2mat(V6[1])-p6[1]) \\ for the first one. One must have \\ i=1 and 1<=j<=11, or i=2 and j=12,13, or i=3 and 14<=j<=17, or i=5 and j=18 \\ AUXILLIARY PARI-GP PROGRAMS: \\ Tranformations symmetrix matrix <--> vector and converse {mat2vec(a,n,n1,v,k,i1,j)=; n=length(a);n1=n*(n+1)/2;v=vector(n1,k,0); k=0;for(i1=1,n,for(j=i1,n,k=k+1;v[k]=a[i1,j])); v} \\ {vec2mat(v,n,a,k)= n=floor(sqrt(2*length(v)));a=matrix(n,n,k,l,1); k=0;for(l=1,n,for(j=l,n,k=k+1;a[l,j]=v[k];a[j,l]=v[k])); a} \\ \\ ============================================== \\ VORONOI GRAPH in DIMENSION 2: \\ Contiguous Form to the perfect 2-Dimensional Form \\ --> 1 Class with corank 1 \\ ============================================== {p2=vector(1,k,0); p2[1]=[2,-1;-1,2]; } V2=vector(1,k,1); \\ 1 orbit of facets for A2=P_2^1 : graph=[1] V2[1]=[2,1,2]; {VR2=vector(1,i,0); VR2[1]=p2[1]+t*(vec2mat(V2[1])-p2[1]); } \\ \\ ============================================== \\ VORONOI GRAPH in DIMENSION 3: \\ Contiguous Form to the perfect 3-Dimensional Form \\ --> 1 Classes with corank 1 \\ ============================================== {p3=vector(1,k,0); p3[1]=[2,0,1;0,2,1;1,1,2]; } V3=vector(1,k,1); \\ 1 orbit of facets for A3=P_3^1 : graph=[1] V3[1]=[2,1,1,2,1,2]; {VR3=vector(1,i,0); VR3[1]=p3[1]+t*(vec2mat(V3[1])-p3[1]); } \\ \\ ============================================== \\ VORONOI GRAPH in DIMENSION 4: \\ Contiguous Forms to the 2 perfect 4-Dimensional Forms \\ --> 2 Classes with corank 1 \\ ============================================== {p4=vector(2,k,0); p4[1]=[2,0,1,1;0,2,1,1;1,1,2,1;1,1,1,2]; p4[2]=[2,1,1,1;1,2,1,1;1,1,2,1;1,1,1,2]; } V4=vector(2,k,1); \\ 2 orbits of facets for D4=P_4^1 : graph=[1,1] V4[1]=[4,0,2,2,2,0,1,2,1,2]; V4[2]=[2,1,1,1,2,1,1,2,1,2]; \\ 1 orbit of facets for A4=P_4^2 : graph=[1,0] {VR4=vector(2,i,0); VR4[1]=p4[1]+t*(vec2mat(V4[1])-p4[1]); VR4[2]=p4[1]+t*(vec2mat(V4[2])-p4[1]); } \\ \\ ============================================== \\ VORONOI GRAPH in DIMENSION 5: \\ Contiguous Forms to the 3 perfect 5-Dimensional Forms \\ --> 4 Classes with corank 1 \\ ============================================== {p5=vector(3,k,0); p5[1]=[2,0,1,1,1;0,2,1,1,1;1,1,2,1,1;1,1,1,2,1;1,1,1,1,2]; p5[2]=[4,1,2,2,2;1,4,2,2,2;2,2,4,1,1;2,2,1,4,1;2,2,1,1,4]; p5[3]=[2,1,1,1,1;1,2,1,1,1;1,1,2,1,1;1,1,1,2,1;1,1,1,1,2]; } V5=vector(4,k,1); \\ 4 orbits of facets for D5=P_5^1 : graph=[2,1,1] V5[1]=[2,0,0,1,1,2,1,1,1,2,1,1,2,1,2]; V5[2]=[2,1,1,1,1,2,1,1,1,2,1,1,2,0,2]; V5[3]=[4,1,2,2,2,4,2,2,2,4,1,1,4,1,4]/2; V5[4]=[2,1,1,1,1,2,1,1,1,2,1,1,2,1,2]; \\ 1 orbit of facets for A_5^3=P_5^2: graph=[1,0,0] \\ 1 orbit of facets for A_5 =P_5^3: graph=[1,0,0] {VR5=vector(4,i,0); VR5[1]=p5[1]+t*(vec2mat(V5[1])-p5[1]); VR5[2]=p5[1]+t*(vec2mat(V5[2])-p5[1]); VR5[3]=p5[1]+t*(vec2mat(V5[3])-p5[1]); VR5[4]=p5[1]+t*(vec2mat(V5[4])-p5[1]); } \\ \\ ============================================== \\ VORONOI GRAPH in DIMENSION 6: \\ Contiguous Forms to the 7 perfect 6-Dimensional Forms \\ --> 18 Classes with corank 1 \\ ============================================== \\ The seven 6-dimensional perfect latrtices {p6=vector(7,k,0); p6[1]=[2,1,1,1,1,0;1,2,1,1,1,0;1,1,2,1,1,1;1,1,1,2,1,1;1,1,1,1,2,1;0,0,1,1,1,2]; p6[2]=[4,1,1,1,2,2;1,4,1,1,2,2;1,1,4,1,2,2;1,1,1,4,2,2;2,2,2,2,4,1;2,2,2,2,1,4]; p6[3]=[2,1,1,1,1,2;1,2,1,1,1,2;1,1,2,1,1,2;1,1,1,2,1,2;1,1,1,1,2,2;2,2,2,2,2,4]; p6[4]=[4,1,2,2,2,2;1,4,2,2,2,2;2,2,4,1,1,2;2,2,1,4,1,2;2,2,1,1,4,2;2,2,2,2,2,4]; p6[5]=[4,-2,1,-1,-1,1;-2,4,-2,1,-1,-1;1,-2,4,-2,1,-1;-1,1,-2,4,-2,1;-1,-1,1,-2,4,-2;1,-1,-1,1,-2,4]; p6[6]=[4,1,2,2,2,2;1,4,2,2,2,2;2,2,4,1,2,2;2,2,1,4,2,2;2,2,2,2,4,1;2,2,2,2,1,4]; p6[7]=[2,1,1,1,1,1;1,2,1,1,1,1;1,1,2,1,1,1;1,1,1,2,1,1;1,1,1,1,2,1;1,1,1,1,1,2]; } V6=vector(18,k,1); \\ 11 orbits of facets for E6=P6_1 : graph=[3,1,3,2,1,1,0] V6[1]=[2,1,1,1,1,0,4,2,2,3,1,2,1,2,1,2,2,1,4,2,2]; V6[2]=[2,2,2,1,1,0,4,3,2,3,1,4,1,2,1,2,2,1,4,2,2]; V6[3]=[2,2,2,1,2,0,4,3,2,4,1,4,2,4,2,2,3,2,6,3,4]; V6[4]=[4,3,3,3,2,1,6,3,3,3,0,6,3,3,3,6,3,3,4,2,4]/2; V6[5]=[4,3,3,4,3,1,4,3,4,3,1,4,4,3,2,6,4,2,4,2,2]; V6[6]=[4,3,3,2,2,0,4,3,2,3,0,4,3,3,2,4,3,3,4,2,4]; V6[7]=[4,0,2,2,1,1,4,1,2,2,0,2,2,1,1,4,2,2,2,1,2]; V6[8]=[6,3,2,3,3,0,6,2,3,3,0,4,2,2,2,4,2,1,4,2,4]/2; V6[9]=[4,2,4,2,0,0,4,3,2,1,0,8,4,1,3,4,2,3,4,3,6]/2; V6[10]=[8,5,5,4,5,1,8,4,5,5,0,6,4,4,3,6,4,2,6,2,4]/2; V6[11]=[4,3,4,2,3,1,6,5,3,5,2,8,3,6,4,4,4,3,8,5,6]/2; \\------------------------------------------------------- \\ 3 orbits of facets for E6*=P6_2 : graph=[1,0,1,1,0,0,0]; V6[12]=[8,2,-2,-2,0,4,4,0,0,2,2,4,2,2,0,4,2,0,4,0,4]; V6[13]=[4,2,1,2,2,3,4,1,2,2,3,4,1,2,2,6,2,4,4,1,6]; \\------------------------------------------------------- \\ 8 orbits of facets for D6=P6_3 : graph=[3,1,1,1,0,1,1]; V6[14]=[2,2,1,1,2,3,4,1,1,2,4,2,1,2,2,2,2,2,4,4,6]; V6[15]=[4,3,1,1,2,4,6,2,3,3,6,4,1,2,4,4,2,4,4,5,10]/2; V6[16]=[4,3,2,3,3,5,6,2,4,3,6,4,3,3,4,6,3,6,6,6,10]/2; V6[17]=[2,2,1,2,2,3,4,1,3,3,4,2,1,1,2,4,3,4,4,4,6]; \\------------------------------------------------------- \\ 5 orbits of facets for P6_4 (semi-eutactic) : graph=[2,1,1,0,0,1,0]; V6[18]=[4,1,2,2,2,1,4,2,2,2,2,4,1,1,1,4,1,2,4,2,4]; \\------------------------------------------------------- \\ 1 orbit of facets for A_6^({2})=P6_5 : graph=[1,0,0,0,0,0,0]; \\ 3 orbits of facets for P6_6 : graph=[1,0,1,1,0,0,0]; \\ 1 orbit of facets for A6=P6_7 : graphe=[0,0,1,0,0,0,0]; \\ {VR6=vector(18,i,0); for(i=1,11,VR6[i]=p6[1]+t*(vec2mat(V6[i])-p6[1])); for(i=12,13,VR6[i]=p6[2]+t*(vec2mat(V6[i])-p6[2])); for(i=14,17,VR6[i]=p6[3]+t*(vec2mat(V6[i])-p6[3])); VR6[18]=p6[4]+t*(vec2mat(V6[18])-p6[4]); } \\ \\============================================================== \\ END OF FILE \\==============================================================