Cette page fait suite à une demande sur la liste AmiTeX. Le nombre de départ est paramétrable à la ligne
k:=41
En changeant le 15 de la ligne
for l=1 upto 15:
on augmente ou on diminue la longueur de la spirale.
spiraleulam.mp [ source brut ]
prologues:=2; input TEX;
beginfig(1); k:=41; label(TEX(""&decimal(k)&""),(0,0)); pair ptd; ptd:=(0,0); u:=8mm; for l=1 upto 15: if (l mod 2)=1: for j=1 upto l: k:=k+1; label(TEX(""&decimal(k)&""),ptd+u*(0,j)); endfor; ptd:=ptd+u*(0,l); for j=1 upto l: k:=k+1; label(TEX(""&decimal(k)&""),ptd+u*(j,0)); endfor; ptd:=ptd+u*(l,0); fi; if (l mod 2)=0: for j=1 upto l: k:=k+1; label(TEX(""&decimal(k)&""),ptd+u*(0,-j)); endfor; ptd:=ptd+u*(0,-l); for j=1 upto l: k:=k+1; label(TEX(""&decimal(k)&""),ptd+u*(-j,0)); endfor; ptd:=ptd+u*(-l,0); fi; endfor; endfig; end
Cette spirale peut être également construite en indiquant que les nombres premiers.
spiulam2.mp [ source brut ]
prologues:=2; vardef pgcd(expr A,B) = save a,b,r; numeric a,b,r; a := A; b := B; forever: r := a mod b; a := b; b := r; exitunless r > 0; endfor; a enddef; vardef prem(expr P)= boolean reponse; reponse=false; numeric div; div=0; if P=1: div:=1; else: for g=2 upto P-1: if (pgcd(P,g)<>1): div:=div+1; fi; endfor; fi; if div=0: reponse:=true; fi; reponse enddef; vardef ulam(expr nb,pas)= save $; picture $; $=image( if prem(nb)=true: label(TEX(""&decimal(nb)&""),(0,0)); else: label(btex $\star$ etex,(0,0)); fi; pair ptd; ptd=(0,0); u:=8mm; k:=nb; for l=1 upto pas: if (l mod 2)=1: for j=1 upto l: k:=k+1; if prem(k)=true: label(TEX(""&decimal(k)&""),ptd+u*(0,j)); else: label(btex $\star$ etex,ptd+u*(0,j)); fi; endfor; ptd:=ptd+u*(0,l); for j=1 upto l: k:=k+1; if prem(k)=true: label(TEX(""&decimal(k)&""),ptd+u*(j,0)); else: label(btex $\star$ etex,ptd+u*(j,0)); fi; endfor; ptd:=ptd+u*(l,0); fi; if (l mod 2)=0: for j=1 upto l: k:=k+1; if prem(k)=true: label(TEX(""&decimal(k)&""),ptd+u*(0,-j)); else: label(btex $\star$ etex,ptd+u*(0,-j)); fi; endfor; ptd:=ptd+u*(0,-l); for j=1 upto l: k:=k+1; if prem(k)=true: label(TEX(""&decimal(k)&""),ptd+u*(-j,0)); else: label(btex $\star$ etex,ptd+u*(-j,0)); fi; endfor; ptd:=ptd+u*(-l,0); fi; endfor; ); $ enddef; input TEX;
beginfig(1); draw ulam(1,10); endfig;
beginfig(2); draw ulam(41,15); endfig; end
Une légère amélioration : la coloration des nombres premiers et du nombre de départ. J'ai ajouté également, en indice, le nombre de diviseurs autre que 1 et le nombre lui-même.
spiulam3.mp [ source brut ]
vardef pgcd(expr A,B) = save a,b,r; numeric a,b,r; a := A; b := B; forever: r := a mod b; a := b; b := r; exitunless r > 0; endfor; a enddef; vardef prem(expr P)= boolean reponse; reponse=false; numeric Div,DIV;%DIV pour compter le nombre de diviseurs Div=0;DIV=0; if P=1: Div:=1; else: for g=2 upto P-1: if (pgcd(P,g)<>1): Div:=Div+1; if (P mod g)=0: DIV:=DIV+1; fi; fi; endfor; fi; if Div=0: reponse:=true; fi; reponse enddef; vardef ulam(expr nb,pas)= save $; picture $; $=image( if prem(nb)=true: fill fullcircle scaled 7mm withcolor 0.9green; label(TEX(""&decimal(nb)&""),(0,0)); else: label(TEX(""&decimal(nb)&""),(0,0)) withcolor 0.9green; fi; pair ptd; ptd=(0,0); u:=8mm; k:=nb; for l=1 upto pas: if (l mod 2)=1: for j=1 upto l: k:=k+1; if prem(k)=true: fill fullcircle scaled 7mm shifted (ptd+u*(0,j)) withcolor 0.9white; label(TEX(""&decimal(k)&""),ptd+u*(0,j)); else: label(TEX(""&decimal(k)&"$_{"&decimal(DIV)&"}$"),ptd+u*(0,j)); fi; endfor; ptd:=ptd+u*(0,l); for j=1 upto l: k:=k+1; if prem(k)=true: fill fullcircle scaled 7mm shifted (ptd+u*(j,0)) withcolor 0.9white; label(TEX(""&decimal(k)&""),ptd+u*(j,0)); else: label(TEX(""&decimal(k)&"$_{"&decimal(DIV)&"}$"),ptd+u*(j,0)); fi; endfor; ptd:=ptd+u*(l,0); fi; if (l mod 2)=0: for j=1 upto l: k:=k+1; if prem(k)=true: fill fullcircle scaled 7mm shifted (ptd+u*(0,-j)) withcolor 0.9white; label(TEX(""&decimal(k)&""),ptd+u*(0,-j)); else: label(TEX(""&decimal(k)&"$_{"&decimal(DIV)&"}$"),ptd+u*(0,-j)); fi; endfor; ptd:=ptd+u*(0,-l); for j=1 upto l: k:=k+1; if prem(k)=true: fill fullcircle scaled 7mm shifted (ptd+u*(-j,0)) withcolor 0.9white; label(TEX(""&decimal(k)&""),ptd+u*(-j,0)); else: label(TEX(""&decimal(k)&"$_{"&decimal(DIV)&"}$"),ptd+u*(-j,0)); fi; endfor; ptd:=ptd+u*(-l,0); fi; endfor; ); $ enddef; input TEX;
beginfig(1); draw ulam(1,6); endfig;
beginfig(2); draw ulam(41,15); endfig; end
Suite à un exemple de Guillaume Conan, une représentation en 3D
spiulam4.mp [ source brut ]
input geometriesyr16; input TEX; vardef pgcd(expr A,B) = save a,b,r; numeric a,b,r; a := A; b := B; forever: r := a mod b; a := b; b := r; exitunless r > 0; endfor; a enddef; vardef prem(expr P)= boolean reponse; reponse=false; numeric Div,DIV;%DIV pour compter le nombre de diviseurs Div=0;DIV=0; if P=1: Div:=1; else: for g=2 upto P-1: if (pgcd(P,g)<>1): Div:=Div+1; if (P mod g)=0: DIV:=DIV+1; fi; fi; endfor; fi; if Div=0: reponse:=true; fi; reponse enddef; vardef ulam(expr nb,pas)= save spiulam; picture spiulam; spiulam=image( if prem(nb)=true: trace cercles((0,0,0),(0.1,0,0),(0,0,0),(1,0,0),(0,1,0)) dashed evenly; label(TEX(""&decimal(nb)&""),Projette((0,0,0))); else: label(TEX(""&decimal(nb)&""),Projette((0,0,0))) withcolor 0.9green; fi; color ptd; ptd=(0,0,0); coef:=0.25; k:=nb; for l=1 upto pas: if (l mod 2)=1: for j=1 upto l: k:=k+1; if prem(k)=true: trace cercles(ptd+coef*(0,j,0),ptd+coef*(0,j,0)+(0.1,0,0),(0,0,0),(1,0,0),(0,1,0)) dashed evenly;% withcolor 0.9green; label(TEX(""&decimal(k)&""),Projette(ptd+coef*(0,j,0))); else: label(TEX(""&decimal(k)&"$_{"&decimal(DIV)&"}$"),Projette(ptd+coef*(0,j,0))); fi; endfor; ptd:=ptd+coef*(0,l,0); for j=1 upto l: k:=k+1; if prem(k)=true: trace cercles(ptd+coef*(j,0,0),ptd+coef*(j,0,0)+(0.1,0,0),(0,0,0),(1,0,0),(0,1,0)) dashed evenly;%withcolor 0.9green; label(TEX(""&decimal(k)&""),Projette(ptd+coef*(j,0,0))); else: label(TEX(""&decimal(k)&"$_{"&decimal(DIV)&"}$"),Projette(ptd+coef*(j,0,0))); fi; endfor; ptd:=ptd+coef*(l,0,0); fi; if (l mod 2)=0: for j=1 upto l: k:=k+1; if prem(k)=true: trace cercles(ptd+coef*(0,-j,0),ptd+coef*(0,-j,0)+(0.1,0,0),(0,0,0),(1,0,0),(0,1,0)) dashed evenly;%withcolor 0.9green; label(TEX(""&decimal(k)&""),Projette(ptd+coef*(0,-j,0))); else: label(TEX(""&decimal(k)&"$_{"&decimal(DIV)&"}$"),Projette(ptd+coef*(0,-j,0))); fi; endfor; ptd:=ptd+coef*(0,-l,0); for j=1 upto l: k:=k+1; if prem(k)=true: trace cercles(ptd+coef*(-j,0,0),ptd+coef*(-j,0,0)+(0.1,0,0),(0,0,0),(1,0,0),(0,1,0)) dashed evenly;%withcolor 0.9green; label(TEX(""&decimal(k)&""),Projette(ptd+coef*(-j,0,0))); else: label(TEX(""&decimal(k)&"$_{"&decimal(DIV)&"}$"),Projette(ptd+coef*(-j,0,0))); fi; endfor; ptd:=ptd+coef*(-l,0,0); fi; endfor; ); spiulam enddef;
figureespace(-11u,-11u,11u,11u); Initialisation(5,30,20,1000); draw ulam(1,9); finespace;
figureespace(-10u,-10u,10u,10u); Initialisation(5,30,20,1000); draw ulam(41,7); finespace; end