Okay, please be gentle. I am not an geometry expert, just learned a
little in high school and enjoy seeing the relationships in lines,
geek that I am. My best attempt to illustrate what I found last night
is in the photo section. Since my scanner is seemingly unable to pick
up pencil lines, some of the lines had to be filled in via paint brush
and the pics are definitely off.
I constructed a Tetractys from a circle by extending the lines of the
hexagon formed by the division of the circumferance by the radius.
Labling the points as I have in the drawings, I then drew in line GE
and HA. Point L is located where line HA intersects the initial cirlce.
Placing a compas on point K and opening the distance to point L, I
drew an arc to the point intersecting the diagonal formed by line BC,
that point then being named Point M. Placing the point of the compass
then on L and opening to the distance of segment LM, I was then able
to divide the circle into five equal segments by defining points L, N,
O, P and Q (which again in my drawing is off because I used paintbrush).
So it would seem we have a direct relationship between two of the most
venerable Pythagorean symbols, namely the Tetraktys and the Pentalpha.
How does the golden ratio figure into this, since it is so involved
with the proportions of the pentagram? I so need one of you math
mavens to bail me out here.