Basic Examples
Basic Examples
Here are the points on the first four sides of the spiral:
[4]
In[]:=
{{0,0},{0,1},{-1,1},{-1,0},{-1,-1},{0,-1},{1,-1}}
Out[]=
This shows the sequence of points in order on the first six sides:
Withs=
[6],Graphics[{{Pink,PointSize[.03],Point[s]},Arrow@Partition[s,2,1]}]
In[]:=
Out[]=
40 black sides with 20 red sides overlaid:
GraphicsLine@
[40],Red,Line@
[20]
In[]:=
Out[]=
The number of points in n sides:
Length@
@#&/@Range[0,10]
In[]:=
{1,2,3,5,7,10,13,17,21,26,31}
Out[]=
Table[Floor[(n+1)^2/4]+1,{n,0,10}]
In[]:=
{1,2,3,5,7,10,13,17,21,26,31}
Out[]=
Neat Examples
Neat Examples
This finds the coordinate pairs that are a prime distance along the square spiral:
UlamSquareSpiralPoints[n_]:=
[n][[Prime@Range@PrimePi[Floor[(n+1)^2/4]+1]]]
In[]:=
The larger points correspond to the primes 2, 3, 5, 7, 11, 13:
With{n=6},GraphicsLightGray,Line@
@n,Point@
@n,PointSize[.08],Point@UlamSquareSpiralPoints@n,ImageSize100
In[]:=
Out[]=
About 12% of the numbers up to 10101 are prime:
With[{n=200},Floor[(n+1)^2/4]+1]
In[]:=
10101
Out[]=
PrimePi@%
In[]:=
1240
Out[]=
Here are the first 1240 primes plotted along the square spiral:
Graphics@Point@UlamSquareSpiralPoints@250
In[]:=
About 11% of the numbers to 10101 are lucky:
This finds the coordinate pairs that are at lucky number distances along the square spiral:
Here are the first 1130 lucky numbers plotted along the square spiral: