Prime numbers on Sqrt[7] spirals
Sándor Kabai
Prime numbers up to 4057 are selected from the the integer range 1 - 10000.
“p1” contains the coordinates of integers from 1 to 10000 placed along a spiral at 2Pi/Sqrt[7] angular spacing.
“vv” represents the coordinates of the prime numbers selected form p1.
Well recognisable spirals are formed when nth numbers are connected, where n is 8, 45, 90, 127, 135, 172, or 180.
Prime numbers up to 4057 are selected from the the integer range 1 - 10000.
“p1” contains the coordinates of integers from 1 to 10000 placed along a spiral at 2Pi/Sqrt[7] angular spacing.
“vv” represents the coordinates of the prime numbers selected form p1.
Well recognisable spirals are formed when nth numbers are connected, where n is 8, 45, 90, 127, 135, 172, or 180.
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If n is 45, then n spirals include all the prime numbers, but not every spiral contains prime number. 25 spirals contain all the primes. The angles between subsequent spirals seem to lack regularity.
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n=45;sp45=Tube[p1[[1;;;;n]],80];sp45a=Rotate[sp45,2Pi/45,{0,0,1}];sp45b=Rotate[sp45,6Pi/45,{0,0,1}];sp45c=Rotate[sp45,8Pi/45,{0,0,1}];sp45d=Rotate[sp45,12Pi/45,{0,0,1}];sp45e=Rotate[sp45,18Pi/45,{0,0,1}];sp45f=Rotate[sp45,20Pi/45,{0,0,1}];sp45g=Rotate[sp45,26Pi/45,{0,0,1}];sp45h=Rotate[sp45,30Pi/45,{0,0,1}];sp45i=Rotate[sp45,32Pi/45,{0,0,1}];sp45j=Rotate[sp45,36Pi/45,{0,0,1}];sp45k=Rotate[sp45,38Pi/45,{0,0,1}];sp45l=Rotate[sp45,42Pi/45,{0,0,1}];sp45m=Rotate[sp45,48Pi/45,{0,0,1}];sp45n=Rotate[sp45,50Pi/45,{0,0,1}];sp45o=Rotate[sp45,56Pi/45,{0,0,1}];sp45p=Rotate[sp45,60Pi/45,{0,0,1}];sp45r=Rotate[sp45,62Pi/45,{0,0,1}];sp45s=Rotate[sp45,66Pi/45,{0,0,1}];sp45q=Rotate[sp45,68Pi/45,{0,0,1}];sp45t=Rotate[sp45,72Pi/45,{0,0,1}];sp45u=Rotate[sp45,78Pi/45,{0,0,1}];sp45v=Rotate[sp45,80Pi/45,{0,0,1}];sp45w=Rotate[sp45,80.2Pi/45,{0,0,1}];sp45z=Rotate[sp45,86Pi/45,{0,0,1}];Graphics3D[{Cyan,sp45z,{Purple,sp45w},Red,sp45u,Red,sp45u,Cyan,sp45t,Red,sp45q,Red,sp45s,Red,sp45r,Red,sp45p,Cyan,sp45o,Red,sp45n,Red,sp45m,Cyan,sp45l,Red,sp45k,Red,sp45j,Red,sp45i,Red,sp45h,Cyan,sp45g,Red,sp45f,Red,sp45e,Cyan,sp45d,Red,sp45c,Red,sp45b,Red,sp45a,Red,sp45,Yellow,primen},ViewPoint->{0,0,100},Boxed->False,SphericalRegion->True,ImageSize->500];
If n is 45, 90, 135 or 180, then n spirals include all the prime numbers, but not every spiral contains prime numbers. Use the following rotations: if n =45, then 2 Pi/n, if n = 90, then 4 Pi/n, if n = 135, then 6 Pi/n, if n = 180, then 8 Pi/n.
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sp90=Tube[p1[[1;;;;90]],80];sp90a=Table[Rotate[sp90,i4Pi/90,{0,0,1}],{i,15}];Graphics3D[{integers,Black,sp90,Cyan,sp90a,Yellow,primen},ViewPoint->{0,0,100},Boxed->False,SphericalRegion->True,ImageSize->400];
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sp135=Tube[p1[[1;;;;135]],80];sp135a=Table[Rotate[sp135,i6Pi/135,{0,0,1}],{i,15}];Graphics3D[{integers,Black,sp135,Green,sp135a,Yellow,primen},ViewPoint->{0,0,100},Boxed->False,SphericalRegion->True,ImageSize->400]
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sp180=Tube[p1[[1;;;;180]],80];sp180a=Table[Rotate[sp180,i8Pi/180,{0,0,1}],{i,10}];Graphics3D[{integers,Pink,sp180a,Yellow,primen},ViewPoint->{0,0,100},Boxed->False,SphericalRegion->True,ImageSize->400];
If n is 127, then n spirals include all the prime numbers, and all spirals contain prime numbers.
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sp127=Tube[p1[[1;;;;127]],60];sp127a=Table[Rotate[sp127,i2Pi/127,{0,0,1}],{i,127}];Graphics3D[{integers,Black,sp127,Green,sp127a,Red,sp45,Pink,primen},ViewPoint->{0,0,100},Boxed->False,SphericalRegion->True,ImageSize->400]
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Graphics3D[{integers,Black,sp127,Green,sp127a,Red,sp45},ViewPoint->{0,0,100},Boxed->False,SphericalRegion->True,ImageSize->400];
If n is 164, then n spirals include all the prime numbers, and all spirals contain prime numbers, except one.
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sp164=Tube[p1[[1;;;;164]],60];sp164a=Table[Rotate[sp164,i8Pi/164,{0,0,1}],{i,164}];sp164ex=Rotate[sp164,-40Pi/164,{0,0,1}];Graphics3D[{integers,Blue,sp164ex,Red,sp164a,Black,sp164,Yellow,primen},ViewPoint->{0,0,100},Boxed->False,SphericalRegion->True,ImageSize->400];
If n is 172, then 85 spirals include all the prime numbers, and all spirals contain prime numbers.
sp172=Tube[p1[[1;;;;172]],70];sp172a=Table[Rotate[sp172,i4Pi/172,{0,0,1}],{i,86}];sp172ex=Rotate[sp172,44Pi/172,{0,0,1}];Graphics3D[{Black,sp172ex,Cyan,sp172a,Yellow,primen},ViewPoint->{0,0,100},Boxed->False,SphericalRegion->True,ImageSize->400]
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Sets of spirals are partially shown together
The central part is show by choosing ViewAngle->0.001.
Spiral No.8 is added once.