Basic Examples

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

○
TriangularSpiralPoints
[4]
In[]:=
{0,0},-
1
2
,
3
2
,{-1,0},-
3
2
,-
3
2
,-
1
2
,-
3
2
,
1
2
,-
3
2
,
3
2
,-
3
2
,{1,0},
1
2
,
3
2
,0,
3
,-
1
2
,
3
3
2
Out[]=

This shows the sequence of points in order for the first six sides:

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

40 black sides with 20 red sides overlaid:

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

The number of points in the first n sides are 1 more than the triangular numbers:

Length@
○
TriangularSpiralPoints
@#&/@Range[0,10]
In[]:=
{1,2,4,7,11,16,22,29,37,46,56}
Out[]=
Table[1+1/2n(n+1),{n,0,10}]
In[]:=
{1,2,4,7,11,16,22,29,37,46,56}
Out[]=

Neat Examples

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

UlamTriangularSpiralPoints[n_]:=
○
TriangularSpiralPoints
[n][[​​Select[Range[1+1/2n(n+1)],PrimeQ]​​]]
In[]:=

The larger points correspond to the primes 2, 3, 5, 7, 11:

With{n=4},GraphicsLightGray,Line@
○
TriangularSpiralPoints
@n,Point@
○
TriangularSpiralPoints
@n,PointSize[.08],Point@UlamTriangularSpiralPoints@n,ImageSize100
In[]:=
Out[]=

About 12% of the numbers up to 11326 are prime:

With[{n=150},1+1/2n(n+1)]
In[]:=
11326
Out[]=
PrimePi@%
In[]:=
1369
Out[]=

Here is a plot of the first 1369 primes:

Graphics@Point@UlamTriangularSpiralPoints@150
In[]:=

About 11% of the numbers up to 11326 are lucky:

This finds the coordinate pairs that are at lucky number distances along the triangular spiral:

Here are the first 1248 lucky numbers plotted along the triangular spiral: