Evaluate

Basic Examples

Here are the points on the first four sides of the spiral:

○
SquareSpiralPoints
[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=
○
SquareSpiralPoints
[6],Graphics[{{Pink,PointSize[.03],Point[s]},Arrow@Partition[s,2,1]}]
In[]:=
Out[]=

40 black sides with 20 red sides overlaid:

GraphicsLine@
○
SquareSpiralPoints
[40],Red,Line@
○
SquareSpiralPoints
[20]
In[]:=
Out[]=

The number of points in n sides:

Length@
○
SquareSpiralPoints
@#&/@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

This finds the coordinate pairs that are a prime distance along the square spiral:

UlamSquareSpiralPoints[n_]:=
○
SquareSpiralPoints
[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@
○
SquareSpiralPoints
@n,Point@
○
SquareSpiralPoints
@n,PointSize[.08],Point@UlamSquareSpiralPoints@n,ImageSize100
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: