Basic Examples 
(4)
 

Here are the points on the first four legs of the spiral:
In[96]:=
[◼]
TriangularSpiralPoints
[4]
Out[96]=
{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

———
This shows the sequence of points in order for the first six sides:
In[97]:=
Withs=
[◼]
TriangularSpiralPoints
[6],Graphics[{{Pink,PointSize[.03],Point[s]},Arrow@Partition[s,2,1]}]
Out[97]=
———
40 black sides with 20 red sides overlaid:
In[98]:=
GraphicsLine@
[◼]
TriangularSpiralPoints
[40],Red,Line@
[◼]
TriangularSpiralPoints
[20]
Out[98]=
———
The number of points in the first
n
sides are one more than the triangular numbers:
In[67]:=
Length@
[◼]
TriangularSpiralPoints
@#&/@Range[0,10]
Out[67]=
{1,2,4,7,11,16,22,29,37,46,56}
In[68]:=
Table[1+1/2n(n+1),{n,0,10}]
Out[68]=
{1,2,4,7,11,16,22,29,37,46,56}

Neat Examples 
(2)
 
