Figure 2

Define to be the angle from
the axis of the cone to its surface.
I call this the apex angle.
I discovered by accident that certain apex angles
give interesting patterns in the crossline directions
while most do not.
I found these
patterns to be insensitive to the choice of *N*
and show them here for about *N*=1400.
Another value of apex angle
with an interesting pattern is .It gives the charming pattern in Figure 3.
(I've seen this pattern before on a party hat.
I attributed it to an ingenious artist.
Now I realize that like all mathematics,
this art existed before the big bang.)

Figure 3

The radius *r* of the cone divided by its altitude *z*
has a ratio
where is the angle from
the axis of the cone to its surface.
The area of a circle is
.The surface area of a cone is
.Dividing the area into *N* cells,
the surface area per cell is
.Taking any cell area to be square,
the length of a side is

(3) |

(4) |

(5) |

(6) |

4/20/1999