Didn't actually pan out the way I thought it would, anyway:
Here's the example of reciprocal growth of polygons 3 4 5 8 10 and 12 using relationship of radius / side lenght. Notice the connecting points of radius and side of each preceding / succeeding polygon. Some interesting characteristics may be found. Value for hexagon is 1 for both radius & side, and serves as a turning point where radius gets proportionally longer - this can be seen in the growth pattern as well - shapes start pushing out of the perimeter.
radius of 1 and polygion with side length of 1 has been marked with thicker line.so taking simple example of sqrt(3)
radius = 1
side = sqrt(3)
radius = 1/sqrt(3)
side = 1
We also know that for decagon similar values are Phi and 1/Phi.
Growth follows powers of the polygon value when following reciprocal progression.
This happens also in fibonacci:
1 2 3 5 8 13 21 34 55 89 144 ...
144 / 89 = 1.6179775.. ≈ Phi
144 / 55 = 2.6181818.. ≈ Phi^2
144 / 34 = 4.2352941.. ≈ Phi^3
144 / 21 = 6.8571428.. ≈ Phi^4 etc..
If you divide any number with its reciprocal you get ^2.
For example:
5 / 0.2 = 25 = 5^2
5 / 0.04 = 125 = 5^3 (0.2/0.04=5)
probably not groundbraking information, however, I think it shows what I meant when I said that fibonacci / phi is numerically very closely tied to decagon shape.
Tema anterior
