# # Maple spreadsheet used to find coefficient for # a block pattern coloring for monochromatic # constellations. A supplement to the paper # "Finding patterns avoiding many monochromatic # constellations" by Steve Butler, Kevin Costello # and Ron Graham # # Written by Steve Butler # last updated on 10 June 2009 # # load the following procedures # (examples on how to use this are given below) # # LinearAlgebra package is only used to solve # linear system of equations, everything else # is built into the below procedures with(LinearAlgebra): # # computes coefficient given constellation and # subdivision pattern Constellation:=proc(Cpat,Spat) # compute the coefficient global CCoeff,preCCoeff,perturb,M,Q,Spatnew,blocks; local i,j; CCoeff:=0; if (perturb=true) then # if perturbing we need to load some matrices M:=Matrix(nops(Spat),nops(Spat)); Q:=Vector(nops(Spat)); Q[nops(Spat)]:=1; end if; for i from 1 by 2 while i<nops(Spat) do # start the recursions for red for j from 1 by 2 while j<nops(Spat) do RECURSEP([[Spat[i],Spat[j],1,j],[Spat[i+1],Spat[j],nops(Cpat),i+1],[Spat[i+1],Spat[j+1],1,j+1],[Spat[i],Spat[j+1],nops(Cpat),i]],Cpat,Spat,2,0); end do; end do; for i from 2 by 2 while i<nops(Spat) do for j from 2 by 2 while j<nops(Spat) do # start the recursions for blue RECURSEP([[Spat[i],Spat[j],1,j],[Spat[i+1],Spat[j],nops(Cpat),i+1],[Spat[i+1],Spat[j+1],1,j+1],[Spat[i],Spat[j+1],nops(Cpat),i]],Cpat,Spat,2,1); end do; end do; i:=true; # test for symmetry for j from 1 to nops(Cpat) do if (Cpat[j]+Cpat[nops(Cpat)+1-j]<>1) then i:=false; end if; end do; if (i=true) then CCoeff:=CCoeff/2; end if; i:=1; # next get common denominator for j from 2 to (nops(Cpat)-1) do i:=lcm(i,denom(Cpat[j])); end do; if (perturb=true) then preCCoeff:=CCoeff/i; printf("The pattern before perturbing gives coefficient:"); print(preCCoeff); perturb:=false; Spatnew:=Spat; for i from 1 to nops(Spat) do M[1,i]:=0; M[nops(Spat),i]:=0; end do; # finish editing M M[1,1]:=1; M[nops(Spat),nops(Spat)]:=1; Spatnew:=convert(LinearSolve(M,Q),list); # now solve the linear system of equations i:=true; for j from 1 to (nops(Spatnew)-1) do # check to see if answer is reasonable if (Spatnew[j]>Spatnew[j+1]) then i:=false; end if; end do; if (i=false) then printf("The pattern found by perturbing is inconsistent."); else i:=1; blocks:=Vector(nops(Spatnew)-1); for j from 1 to (nops(Spatnew)-1) do # find new block structure blocks[j]:=Spatnew[j+1]-Spatnew[j]; i:=lcm(i,denom(blocks[j])); end do; for j from 1 to (nops(Spatnew)-1) do blocks[j]:=i*blocks[j]; end do; blocks:=convert(blocks,list); printf("Perturbing suggests the following pattern:"); print(blocks); printf("The pattern after perturbing gives coefficient:"); Constellation(Cpat,Spatnew); perturb:=true; printf("This gives a difference (original)-(perturbed)"); preCCoeff-CCoeff; end if; else CCoeff:=CCoeff/i; print(CCoeff); end if; end proc: # # find the area of a triangle with points A, B, C AREAT:=proc(A,B,C) local AA, BB, CC; if (A[1]<=B[1] and B[1]<=C[1]) then AA:=A; BB:=B; CC:=C; elif (A[1]<=C[1] and C[1]<=B[1]) then AA:=A; BB:=C; CC:=B; elif (B[1]<=A[1] and A[1]<=C[1]) then AA:=B; BB:=A; CC:=C; elif (B[1]<=C[1] and C[1]<=A[1]) then AA:=B; BB:=C; CC:=A; elif (C[1]<=A[1] and A[1]<=B[1]) then AA:=C; BB:=A; CC:=B; else AA:=C; BB:=B; CC:=A; end if; if (AA[2]<=BB[2] and BB[2]<=CC[2]) then if ((BB[2]-AA[2])*(CC[1]-AA[1])>(CC[2]-AA[2])*(BB[1]-AA[1])) then (CC[1]-AA[1])*(CC[2]-AA[2])/2-(BB[1]-AA[1])*(CC[2]-BB[2])-(CC[1]-BB[1])*(CC[2]-BB[2])/2-(BB[1]-AA[1])*(BB[2]-AA[2])/2; else (CC[1]-AA[1])*(CC[2]-AA[2])/2-(CC[1]-BB[1])*(BB[2]-AA[2])-(CC[1]-BB[1])*(CC[2]-BB[2])/2-(BB[1]-AA[1])*(BB[2]-AA[2])/2; end if; elif (AA[2]<=CC[2] and CC[2]<=BB[2]) then (CC[1]-AA[1])*(BB[2]-AA[2])-(BB[1]-AA[1])*(BB[2]-AA[2])/2-(CC[1]-BB[1])*(BB[2]-CC[2])/2-(CC[1]-AA[1])*(CC[2]-AA[2])/2; elif (BB[2]<=AA[2] and AA[2]<=CC[2]) then (CC[1]-AA[1])*(CC[2]-BB[2])-(BB[1]-AA[1])*(AA[2]-BB[2])/2-(CC[1]-BB[1])*(CC[2]-BB[2])/2-(CC[1]-AA[1])*(CC[2]-AA[2])/2; elif (BB[2]<=CC[2] and CC[2]<=AA[2]) then (CC[1]-AA[1])*(AA[2]-BB[2])-(BB[1]-AA[1])*(AA[2]-BB[2])/2-(CC[1]-BB[1])*(CC[2]-BB[2])/2-(CC[1]-AA[1])*(AA[2]-CC[2])/2; elif (CC[2]<=AA[2] and AA[2]<=BB[2]) then (CC[1]-AA[1])*(BB[2]-CC[2])-(BB[1]-AA[1])*(BB[2]-AA[2])/2-(CC[1]-BB[1])*(BB[2]-CC[2])/2-(CC[1]-AA[1])*(AA[2]-CC[2])/2; else if ((AA[2]-BB[2])*(AA[1]-CC[1])>(AA[2]-CC[2])*(AA[1]-BB[1])) then (CC[1]-AA[1])*(AA[2]-CC[2])/2-(CC[1]-BB[1])*(AA[2]-BB[2])-(CC[1]-BB[1])*(BB[2]-CC[2])/2-(BB[1]-AA[1])*(AA[2]-BB[2])/2; else (CC[1]-AA[1])*(AA[2]-CC[2])/2-(BB[1]-AA[1])*(BB[2]-CC[2])-(CC[1]-BB[1])*(BB[2]-CC[2])/2-(BB[1]-AA[1])*(AA[2]-BB[2])/2; end if; end if; end proc: # # the area of (convex) polygon P AREAP:=proc(P) local i,n,area; n:=nops(P); area:=0; for i from 2 to n-1 do area:=area+AREAT(P[1],P[i],P[i+1]); end do; area; end proc: # # find the intersection of line between A and B and line y=mx+b INTERSECT:=proc(A,B,m,b) local mm,bb,U; if (A[1]=B[1]) then U[1]:=A[1]; U[2]:=m*A[1]+b; else mm:=(B[2]-A[2])/(B[1]-A[1]); bb:=A[2]-mm*A[1]; U[1]:=(bb-b)/(m-mm); U[2]:=m*U[1]+b; end if; convert(U,list); end proc: # # The heart of the worksheet. The idea is to take P # a polygon and then we slice it off according to the # current set of slopes that we are dealing with # note that each point is a quadruple, the first # two entries is the location of the point and the second # two give slope and intercept (indirectly) of line # on incident edge. RECURSEP:=proc(P,Cpat,Spat,iterate,MOD) global CCoeff; local i,j,k,Q,R,S,INTER,numABOVE,numBELOW,DONE,ptA,ptAloc,ptAfound,ptB,ptBloc,cntr; if (iterate=nops(Cpat)) then CCoeff:=CCoeff+AREAP(P); if (perturb=true) then loadM(P,Cpat,Spat,MOD); end if; # load the matrix if perturbing else Q:=P; DONE:=false; for i from 2 by 1 while (i<=nops(Spat) and DONE=false) do # see whether points of P are above or below in current line numABOVE:=0; numBELOW:=0; for j from 1 to nops(Q) do if (Q[j][2]<(Spat[i]-Cpat[iterate]*Q[j][1])/(1-Cpat[iterate])) then numBELOW:=numBELOW+1; INTER[j]:=-1; else numABOVE:=numABOVE+1; INTER[j]:=1; end if; end do; if (numBELOW>0) then # we can do something if (numABOVE=0) then #everything is between previous and current line DONE:=true; if ((i-MOD) mod 2=0) then RECURSEP(Q,Cpat,Spat,iterate+1,MOD); end if; else # have to slice part of Q off ptAfound:=false; # we now find how it slices in half for k from 1 to (nops(Q)-1) do if (INTER[k]*INTER[k+1]=-1) then if (ptAfound=false) then ptAfound:=true; ptAloc:=k; ptA:=INTERSECT(Q[k],Q[k+1],Cpat[iterate]/(Cpat[iterate]-1),Spat[i]/(1-Cpat[iterate])); else ptBloc:=k; ptB:=INTERSECT(Q[k],Q[k+1],Cpat[iterate]/(Cpat[iterate]-1),Spat[i]/(1-Cpat[iterate])); end if; end if; end do; if (INTER[nops(Q)]*INTER[1]=-1) then ptBloc:=nops(Q); ptB:=INTERSECT(Q[1],Q[nops(Q)],Cpat[iterate]/(Cpat[iterate]-1),Spat[i]/(1-Cpat[iterate])); end if; unassign('R'); unassign('S'); cntr:=1; for k from 1 to ptAloc do R[cntr]:=Q[k]; cntr:=cntr+1; end do; R[cntr][1]:=ptA[1]; R[cntr][2]:=ptA[2]; R[cntr][3]:=iterate; R[cntr][4]:=i; R[cntr]:=convert(R[cntr],list); cntr:=cntr+1; R[cntr][1]:=ptB[1]; R[cntr][2]:=ptB[2]; R[cntr][3]:=Q[ptBloc][3]; R[cntr][4]:=Q[ptBloc][4]; R[cntr]:=convert(R[cntr],list); cntr:=cntr+1; for k from (ptBloc+1) to nops(Q) do R[cntr]:=Q[k]; cntr:=cntr+1; end do; cntr:=1; for k from (ptAloc+1) to ptBloc do S[cntr]:=Q[k]; cntr:=cntr+1; end do; S[cntr][1]:=ptB[1]; S[cntr][2]:=ptB[2]; S[cntr][3]:=iterate; S[cntr][4]:=i; S[cntr]:=convert(S[cntr],list); cntr:=cntr+1; S[cntr][1]:=ptA[1]; S[cntr][2]:=ptA[2]; S[cntr][3]:=Q[ptAloc][3]; S[cntr][4]:=Q[ptAloc][4]; S[cntr]:=convert(S[cntr],list); if (INTER[ptAloc+1]=1) then # S is above line keep it, recurse on R Q:=convert(S,list); if ((i-MOD) mod 2=0) then RECURSEP(convert(R,list),Cpat,Spat,iterate+1,MOD); end if; else # S is below recurse it, keep R Q:=convert(R,list); if ((i-MOD) mod 2=0) then RECURSEP(convert(S,list),Cpat,Spat,iterate+1,MOD); end if; end if; end if; end if; end do; end if; end proc: # # the following loads the entries of M according to the polygon Pin loadM:=proc(Pin,Cpat,Spat,MOD) global M; local i,n,P; n:=nops(Pin); for i from 1 to n do P[i]:=Pin[i]; end do; P[n+1]:=P[1]; P[n+2]:=P[2]; P:=convert(P,list); for i from 2 to (nops(P)-1) do if (P[i][3]=nops(Cpat)) then # a vertical line if ((P[i][2]<P[i+1][2] and MOD=0) or (P[i][2]>P[i+1][2] and MOD=1)) then M[P[i][4],P[i-1][4]]:=M[P[i][4],P[i-1][4]]-Cpat[P[i][3]]/(Cpat[P[i-1][3]]-Cpat[P[i][3]]); M[P[i][4],P[i][4]]:=M[P[i][4],P[i][4]]+Cpat[P[i-1][3]]/(Cpat[P[i-1][3]]-Cpat[P[i][3]])+Cpat[P[i+1][3]]/(Cpat[P[i][3]]-Cpat[P[i+1][3]]); M[P[i][4],P[i+1][4]]:=M[P[i][4],P[i+1][4]]-Cpat[P[i][3]]/(Cpat[P[i][3]]-Cpat[P[i+1][3]]); else M[P[i][4],P[i-1][4]]:=M[P[i][4],P[i-1][4]]+Cpat[P[i][3]]/(Cpat[P[i-1][3]]-Cpat[P[i][3]]); M[P[i][4],P[i][4]]:=M[P[i][4],P[i][4]]-Cpat[P[i-1][3]]/(Cpat[P[i-1][3]]-Cpat[P[i][3]])-Cpat[P[i+1][3]]/(Cpat[P[i][3]]-Cpat[P[i+1][3]]); M[P[i][4],P[i+1][4]]:=M[P[i][4],P[i+1][4]]+Cpat[P[i][3]]/(Cpat[P[i][3]]-Cpat[P[i+1][3]]); end if; else if ((P[i][1]>P[i+1][1] and MOD=0) or (P[i][1]<P[i+1][1] and MOD=1)) then M[P[i][4],P[i-1][4]]:=M[P[i][4],P[i-1][4]]+1/(Cpat[P[i][3]]-Cpat[P[i-1][3]]); M[P[i][4],P[i][4]]:=M[P[i][4],P[i][4]]+((1-Cpat[P[i+1][3]])/(Cpat[P[i][3]]-Cpat[P[i+1][3]])+(1-Cpat[P[i-1][3]])/(Cpat[P[i-1][3]]-Cpat[P[i][3]]))/(1-Cpat[P[i][3]]); M[P[i][4],P[i+1][4]]:=M[P[i][4],P[i+1][4]]+1/(Cpat[P[i+1][3]]-Cpat[P[i][3]]); else M[P[i][4],P[i-1][4]]:=M[P[i][4],P[i-1][4]]-1/(Cpat[P[i][3]]-Cpat[P[i-1][3]]); M[P[i][4],P[i][4]]:=M[P[i][4],P[i][4]]-((1-Cpat[P[i+1][3]])/(Cpat[P[i][3]]-Cpat[P[i+1][3]])+(1-Cpat[P[i-1][3]])/(Cpat[P[i-1][3]]-Cpat[P[i][3]]))/(1-Cpat[P[i][3]]); M[P[i][4],P[i+1][4]]:=M[P[i][4],P[i+1][4]]-1/(Cpat[P[i+1][3]]-Cpat[P[i][3]]); end if; end if; end do; end proc: # # The following is a simplified version of using # Constellation, A is any realization of the constellation # and B is the block structure. blockConstellation:=proc(A,B) local i,AA,BB,nA,nB,total; nA:=nops(A); nB:=nops(B); for i from 1 to nA do AA[i]:=(A[i]-A[1])/(A[nA]-A[1]); end do; total:=0; for i from 1 to nB do total:=total+B[i]; end do; BB[1]:=0; for i from 1 to nB do BB[i+1]:=BB[i]+B[i]/total; end do; Constellation(convert(AA,list),convert(BB,list)); end proc:

# # the main function is blockConstellation. There are two # inputs, one a constellation pattern (any nontrivial # realization of the constellation can be used) and the # other a block pattern (the only important thing is the # relative block sizes). For instance to compute the # coefficient for the three term AP for the currently # best known block configuration we use the following. # # Note that it stores the number as the variable CCoeff # blockConstellation([1,2,3],[28,6,28,37,59,116,116,59,37,28,6,28]);

IyIkPCIiJSM+Iw==

# # we can also use the function to take an approximately # optimal block structure and perturb it to find the # best optimal block structure. For instance, # if experimentally we found a good block structure for # the constellation [0,1/3,1] was # 1101,193,577,583,989,1434,1115,2833,3680,3681,2830,1113,1434,988,582,575,194,1098 # then we run the same function but first set the # variable perturb to be true. # # There are several variables stored, namely: # preCCoeff = coefficient before perturbing # M = matrix set up to do the perturbation # CCoeff = coefficient after perturbing # blocks = block structure after perturbing # Spatnew = the perturbed new subdivision pattern # perturb:=true: blockConstellation([0,1/3,1],[1101,193,577,583,989,1434,1115,2833,3680,3681,2830,1113,1434,988,582,575,194,1098]);

The pattern before perturbing gives coefficient:

IyIpeG4ibyYiKisrK10o

Perturbing suggests the following pattern:

NzQiKDhBYiIiJzpDRiInXksiKSInUUIjKSIoW1hSIiIob10tIyIoVEdkIiIoNWYqUiIodmc+JkYrRipGKUYoRidGJkYlRiRGIw==

The pattern after perturbing gives coefficient:

IyIpIj5TZyIiKjNmdDYj

This gives a difference (original)-(perturbed)

IyIqVlM5PSgiMisrK11VXExLIg==

# # Below are some locally optimal patterns that have been found # and the corresponding coefficients. # perturb:=false;

SSZmYWxzZUclKnByb3RlY3RlZEc=

blockConstellation([1,2,3,4],[566124189415440472939626834822903743300467940483, 115903533761943477398551818347715476722877927241, 568011813340950665677009286694526323061781532322, 472083073090028493914605548954507028673457587863, 174690683867336844297305424871758992360029965453, 98464537567500111285074159909918993309405119848, 737681146409933099806596775369238915383890216793, 881071132072892536672404740128385947619685842609, 387204684955306822603896642766296540832568256888, 340852889156784985628080980878507258595675472221, 1398355239284691808801098670395696996980804292522, 2015438904391090234472652819593929714355629836078, 354924006068259988552316716495705798216298952575, 917029329994691011286378833655488533756529343857, 1246774229265384930907724953794401373314144038191, 543203071437439856124749368271693956037186323582, 2179716742907087903057122171392866104311765026441, 2172387005301046067153961748343296914044366107569, 546203621232713973260465876924982631234637232779, 1296607046453245562932414262768745367411919249702, 848633230480614872785768439513578746631939174778, 332362434790023921274974476865878572983006589230, 2079963873190082657423397539748746717308015584742, 1352139932444260494597496699603210730948918467199, 339606780510267312616862401149984633870549046619, 373718051493152648659948917556716014372715915533, 786614601718483336599780069288734659594156639775, 660138925526725209837882202781409057453701881412, 51505717888223458966003645016452574647172214751, 208563593370975774208121482104389596165334050300, 660659764939424259451601477074423264167731648445, 458220356230536594713018106829384942175966684332, 25106402444485772567927008361180285102508619312, 396852300398188165681981456085190506968094545183, 59492524857712314099066167971046557078820629622, 398049321798723913182570904641775780071858799086]);

IyJSdnpWbi8uUWZPOFNfP3FDIjQ1PUFJSCdSeiIiVFc4LVZXRCcqcG4hKmVeYVB2OCMpZkEjbyg9dTwvIg==

blockConstellation([0,3/10,1],[119228718024148022932189931580657635982754931074482856129, 18691552986407448127542596353996435011556813685407206923, 43414903502641894623053470110989379079145782700060829808, 39335006358699214018961309525073803600682059946044319294, 59568404324564399543354923445220036812752501950991192466, 65218435561732440851968505083027381648775941120746388044, 88590868826691039949503466845572967616486658230291521266, 133220283108409953564517789111130323060969637088249838121, 93226465812202717574621719255372174676037739002087565099, 237742481910002497917786014966420299021574301840059533420, 117462424188412208724660283562648928698367801161008307060, 325064923303547112477390127249934550921677394556516982920, 350974183591413827257151066712082235822065583756451433700, 387760155760299918352739149724606153513617491211563699550, 627598447210535105260699733564705298620649290117421860240, 494678955494794864755505378553879841909194617150330397260, 758995170811694095710028678036297985828955464551103628910, 1188831866344672077905397526879492832391147147437492377030, 1379682616857357911626832061556359777325347960248046326220, 1244330000419354963590626986370219029026974957186141025670, 706806102048667019900237484230774944079176586943410697556, 452533195818487800557168618902788427937652672716957516074, 569518765450023779286104769366344306140780241524258661490, 214358710307067643343612358754470693506750870525952742204, 411867052603214883184778637764209726625649720760646919066, 245963301825292715732177634858674510598997547862014580514, 171828053921238204296887175883383855209157047288088289546, 185934513011657238284789722120833274067958706617883216640, 78876763235405987948620241531340317805380396218276796734, 129927411688462573166743384337166076875574876059963420546, 47335397809623388304728341551067814718064129989968232180, 330448015483674712098060265915839560855397213756745144680]);

IyJmbihvYEF4JD12KGZhJipSLiIzLkMjZl5jWDwpekxRLlNOJD0iZ24rX1dYa10qKmU9XCNIMGs/JGZadiF6Z2kkeS1LLiM0TCN6

blockConstellation([0,1/4,1],[631190316306400325972, 23321256453714458087, 217557022114231121737, 100141050911243035573, 307999572837249273651, 195186988505399737119, 352380984908243720777, 371083801031004301427, 327740778575536748393, 731673351086130522702, 276829804229005434026, 1312646866569184356082, 655180137912135774030, 1385049712230510886616, 1325075842514182632280, 1115950741162114185650, 2270594446412925419934, 1819386224749383672352, 2886668317651138077410, 4528656516934616095468, 5233099613551747457920, 5233099613551747457920, 4528656516934616095468, 2886668317651138077410, 1819386224749383672352, 2270594446412925419934, 1115950741162114185650, 1325075842514182632280, 1385049712230510886616, 655180137912135774030, 1312646866569184356082, 276829804229005434026, 731673351086130522702, 327740778575536748393, 371083801031004301427, 352380984908243720777, 195186988505399737119, 307999572837249273651, 100141050911243035573, 217557022114231121737, 23321256453714458087, 631190316306400325972]);

IyI4WnpAWnN1UiFcISo0PSI5c2slbztgKGYsJyozR0o=

blockConstellation([0,3/7,1],[3747226371, 1294186425, 4688162203, 6106634656, 10656339794, 17684720639, 3238826362, 3238826362, 17684720639, 10656339794, 6106634656, 4688162203, 1294186425, 3747226371]);

IyItUDEpKlxDXyIvK3NTMz0kZiI=

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