Phi square root series: (seed ratio 1.061997) (reciprocal 0.941622)

Besides the (Tor-Abbey) Lucas unit, a Mile unit is possible using the (Tor-Avebury) 61.8 axis. The harmonics remain the same.

Golden Triangles ( Mile ) analysis

At this scale Lucas numbers are identified in the yardages.

Using the algorithm's underlying equation:

unit is 1364/2 = 682

sqrt5 (2.2361) x 682 - 682 = 843

checking [1364 x phi = 843], [843 x phi = 521], [1364 - 843 = 521]

Lucas Series: [ 2, 1, 3, 4, 7, 11, 18 etc ] resolving to Phi as the effective ratio.

Golden Triangles ( Lucas ) analysis

Lucas golden right angled triangle developed:

distance Tor to Edmund's Mound (the fourth site related by the phi algorithm) resolved as 1364 x sqrt phi 0.786151 (a sub-division of the phi series)

Classic relationship to Phi of Fibonacci & Lucas series illustrating unsettled Phi convergence / resolution.

Mapping alternate values resolves Phi vacillation to two converging strands ( A & B ) with ratios of Phi squared & reciprocal phi squared.

Using the phi algorithm:

Two random yet resonant sites related (Lady Chapel of Glastonbury Abbey & Glastonbury Tor)

The phi harmonic relates a third site (Chalice Well).

If one maps Phi itself as a fibonacci-type series one is guaranteed a constant Phi ratio since it has the quality of being both additive and exponential. One can now introduce the square / square root function to generate sub-divisions to any resolution desired.

Below I have settled upon 8 divisions as an acceptable resolution.

Much simpler to translate all the above computations as radii of circles with each radius conforming to the sqrt phi series. (harmonic radiation / gravitation)

Notice the array of harmonic triangles conforming to the harmonics of the Great Pyramid of Giza.

(Any right angled triangle with a 38.2 angle will do this)

Vesica:- A bi-directional (reciprocal) relational device for any two points, and

container for an elegant procedural algorithm for the physical computation of the phi section.

Since the whole point of this geometric algorithm is to efficiently embody the underlying math equations and theorems, it can be taken on trust that it will produce the desired phi section [solid circles].

The Mile unit is an important feature of the Tor-Avebury axis such that it cannot be ignored. However the Lucas unit allows for a more direct construction of the triangle (38.2 angle from Tor as origin).

Mile & Lucas analyses offer different configurations relative to scale & chosen resolution of sqrt Phi subdivisions plus, of course, relative to the chosen number of phi ratio digits.

[there are over a million digits 'and counting' in the phi ratio - indeterminable - to infinity?]

And, further, as a bi-directional device, the reciprocal computation applies simultaneously.

And there are the reflections of these two computations which complete the list of bi-directional permutations.

The green radii below are irrational (i.e. indeterminable in essence)

Since my agenda is purely aesthetic (pattern seeking) I am comfortable with these ground-level offsets. Both Lucas & Mile patterns are in the vicinity of the resonant sites which themselves are in their own vicinity. Dealing with irrationals, by their very nature, is always about an acceptable 'vicinity'.

Like Fibonacci the Lucas Series is one of a myriad of possible additive series. However, Fibonacci & Lucas are unique in that they are the direct expressions of Phi and reciprocal phi (1.618034 & 0.618034).

Fibonacci:

consecutive subtraction

Lucas:

alternate addition & subtraction

Notice that whether PHI is developed as an additive series or an exponential series the resulting values are the same. Not true of its reciprocal phi.

Alternates

Above I highlighted the occurrence of alternate computation in resolving the vacillation of the Phi ratio in an additive series.

I also highlight the alternate treatment of Phi exponential values in processing the Lucas Series.

Further, alternate computation is involved in the development of the Fibonacci series to the Lucas series.

I have dealt with this development elsewhere (see Lucas page) but this diagram illustrates another method.

Here I am using the Fibonacci series as root for every level of computation plus increasing the spread of the alternate values. [spread 3, 5, 7]

This exposes the 3-6-9 series as the next level up from the Lucas series.

Elsewhere I have illustrated the framework nature of the 3-6-9 series within an additive series and it persists as such in this array

This is the complete pattern transposed to Modulo 9:

Detail: Isolated Vesica showing alternate addition processing (expanding 'petals'.

Oddly, this cycled array requires data from previous cycles (i.e whole numbers prior to '0'). Without this, the Lucas series would not acquire its initial digit '2'. However, that is assuming that the array is a 'closed' cycle. It probably is an advancing helical array seen in 'plan' view. ( 0 - 8 infinity)

But this perceived 'order', based upon truncating the unknown magnitude of irrationals, will probably disintegrate at more resolved levels.

A: Phi Rising

B: Phi Falling

As illustrated above Phi vacillation can be resolved if we split the array into two alternating groups A & B:

This arrangement seems nice and tidy - Two interlocking processes

(A: Reflective values & B: Reciprocal values).

The circular construction, passing through alternate vertical values, aligns the behaviour of Phi (falling or rising) for each series level.

Fibonacci A Phi Falling : Lucas A Phi Rising

Fibonacci B Phi Rising : Lucas B Phi Falling

Taking the Golden Ratio phi as a bearing : 61.8034 degrees.

Applying it to that extant idea of an East - West axis. Investigating that extant Glastonbury, Avebury, Stonehenge connection.

Vesica & phi algorithm:

Imagine a data-rich 1 yd radius circle at the summit of Glastonbury Tor radiating harmonically.

Square Root Phi series relative to a 30unit radius - Colour coded & Greyscale coded.