spare ridge
This first graph is based on trigonomic values of the pentagram. Particularly the value of 36° which is the outer angle of the star. Since the unit circle is 360°, 36° divides the circle into ten pieces! The divisions fall on some interesting values as well. Of course starting on the x-axis, 0° and 180°. Also sacred numbers 72°(value of the shem ha-mephorash), 108° (considered a holy number) 144° (144,000 in book of revelation) and 216°. 216=6×6×6.
I also overlayed the pentagram on top of the 10 divisions. Starting at 18° the pentagram touches the circle at 72° intervals because 360/5 is 72. Two interesting values, the left leg touches at 234° which I found interesting because 234 is a sequence. And, what confounds me the most is the right leg touches at 306°. Just a weird coincidence that it's a permutation of 36 and 360? Doesn't seem to be much explanation for it.
I also wanted to mention a right angle triangle with angles 36°/54°/90°. And, by extension, a 18°/72°/90° triangle.
The sines and cosines of these with a radius or hypotenuse of 1 gives the ratios:
√(10-2√5)/4: (√5+1)/4:1
And
(√5-1)/4: √(10+2√5)/4:1
Multiplying these ratios by 2, we have the values for φ and 1/φ for legs a and b!
φ= (1+√5)/2
1/φ= (1-√5)/2
φ is a clockwise spiral and 1/φ is a counterclockwise spiral.
light pewter