Pleating the tam?

My first thought about making the eight-sided tam was that you had to pleat the fabric to make the brim.  I tried out this hypothesis using graph paper.

First, I made an octagon using a compass and ruler:

figure 1

Next I folded inward along the outline of the octagon, making pleats and moving counter-clockwise, which worked well until I got to the last fold (one point too short to reach band and another reaching out to the edge):

figure 2

So I had to rethink the pleating pattern, and here’s what I decide on:

Pleating the Tam

First, crease along all eight sides of octagon, then open out again.  This creates a grid of lines for folding:

figure 4

Next, fold inwards from the 8 sides in the following symmetrical pattern:  

figure 5

The red dots in figure 5 are defined by the creases in the original folding lines (figure 4) and in turn define the circle of the seam of the hat band. 

Here's my first attempt at making the tam, along these lines.