There is an object called the Spiral of Life which has some very fascinating properties in relation to Pi and phi. The object suggests that there exists some approximations to pi and phi which play a very important role in describing the object itself while not detracting in any way the significance of pi and phi themselves.
A pi approximation using key measurements along the Lewin orbit
The Lewin orbit is a location where the ever-expanding path and shifted orbital path conjoin with a high degree of synchronicity between them. The four vertices of the expanding path are intervals 77, 88, 99 and 112. The vertices of the Lewin orbit in mid-shift are 77, 89, 101, and 113. The key intervals forming an approximation of pi come from the intersection of expanding and orbiting paths.. An easy way to visualize these intervals is to imagine a bow and arrow at the vertices of the Lewin orbit (77, 89, 101, 113) with an additional interval - 88 - representing the grip location and interval 89 representing the arrow rest where the arrow is launched. In this object 113 represents the serving where the arrow drawn in preparation for launch. 77 and 101 are where string is attached to bow at bottom and top respectively. In this scenario, the sum of the four intervals along the bow ( 77, 88, 89, and 101) divided by the serving (113) leaves us with 355/113 - a very close approximation to pi.
Other Pi coincidences
The full shift of the Lewin orbit represents a shift in every vertex. In other words at mid-shift we have as vertices intervals 77, 89, 101 and 113 while full-shift is simply one more at each vertex resulting in 78, 90, 102 and 114. The expanding intervals of the model are centered on intervals 1, 2, 3 and 4. This simply suggests the first four intervals are tightly woven such that all are considered centered intervals of the object. The product of the first interval of the Lewin orbit 77 and the centered core interval 4 and the fully-shifted interval on opposite side (north) 102 results in 31416.
In this model a new generation expansion is highly suggested beginning where the Lewin orbit comes full circle with the overlap of intervals 77 and 125. Thus the first four intervals of the Spiral of Life correspond not only with the timing of intervals 125, 126, 127 and 128 but also 77, 78, 79 and 80 the sum of which is 314. Dividing this result by the first Passover interval - 100 - and we reach the pi approximation 3.14 (Interval 100 is so-named because the new vertices of the orbital path "passes over" interval 100 and 101 before arriving at 102 to complete the full shift corresponding with the arrival of interval two.)
Finally a coincidence surrounding pi and phi
This is what prompted me reply to this forum. The location where a new generation expansion begins is where intervals 77 and 125 conjoin. Set as a decimal fraction that gives us .616. No, it's not phi but it's very close. But what's more, in exploring this fraction uniformly inward I happened along a very close approximation to Pi which I hope you will find as interesting and eye-opening as I did.
The spiral's four arms expand according to the Golden Egg sequence. It is a recursive sequence like the Fibonacci sequence only it begins with 0, 0 and requires adding one to each recursion. This results in a largely prime column (save 9) whose intervals north are 3, 9, 13, 23, 37, 61... and whose intervals south are (2+3), 5, 11, 17, 29, 47, 77...
If we examine the south column where 77 and 125 conjoin to form the Lewin orbit and explore the resulting fraction inward we are met with these results:
125 x .616 = 77
77 x .616 = 47.43
47.43 x .616 = 29.22
29.22 x .616 = 18.00
18.00 x .616 = 11.09
11.09 x .616 = 6,83
6.83 x .616 = 4.21
4.21 x .616 = 2,59
2.59 x .616 = 1.60
1.60 x .616 =.9833
.9833 x .616 = .6058
.6058 x .616 = .3731
.3731 x .616 = .2298
,2298 x .616 =.1415924131
A very nice approximation of Pi's decimal fraction!
