[figure in the last page]
A more adequate approach to the so widespread spiral figuration could reinterpret it under the aspect of a lattice that crystallizes the golden proportion in the isomorphic partition of simple 90° arcs, each of which in fact represents one of our scaled circles, in its module unitary governed by the π.
Instead of an illusory tracing, the merit of the sequence would be rendered perhaps with less immediate success, but with greater aesthetic and mathematical coherence.
I find it interesting at this point to draw attention to the seasonal phases oriented to the four Elements – of which the DENSITY would be a more expressly appropriate parameter – which are polarized as hinges of matter at the appropriate vertices of the tetrahedron, rotating their faces as zodiacal triplicities on the sequenced sides and from there to the formation of living substances, that is all: where and how the 3 [the Intent – energy] combines with 4 [the Object – mass].
And the π, as we are titling it, cannot fail to be the keystone of this order.
For the many enthusiasts of the pi day, this pic of the four quarters could well figure as the ideal trait-d'union between the 'pi' and the slices of ‘pie’…
Why 'to split the hair in four'
Far from wanting to alienate the syntax of secular conventions, this perspective is aimed at a gnoseological approach that better frames the unexplored relationship that connects the square (the first container of the Golden Section) to the circular entities, from the aforementioned cross-section of the golden progression to the formulas of circle and circumference.
The surrogate notations would translate for a diameter = mt.1 more or less like this:
|Given that π is the arc = ¼ of circmference, its
square root is the side of the square having the area of the circle
|√π mt: 0,88665
|The )² of that side, or area of the circle, hence it returns to ||π mq.
|From )² of π the area of the square with a perimeter equal to the circumference will be obtained, a datum of little interest, were it not that its linear value is equivalent to the gold number, also symbolically confirming the intrinsic relationship between square and golden section.
So the family is complete, in thought and words.
0,78615² = 0,618034 = Φ
As for the formulas, if the one for the area of the circle becomes:
the one for calculating the circumference would be like on 4 sides:
4π × r²|
4π × 2r
If this should be |
π, in the light of these assumptions I will try to illustrate geometrically why π is the square root of Φ, multiplied by four.
Why root and why square? of this universal term, after several researches I have not found a single satisfactory description, that is, one that does not limit itself to listing the numerical results among the operands.
|Geometrically speaking, one could then agree that the square root of a number is thinkable as, or measurable with, the side [base, i.e. root] of a square whose area is equivalent to that number: just to give it plausible form and content.
The ideographic relationship thus established will allow, by directly using the value 2√Φ to conceive in a synchronous quadratic way both the calculation of the fixed circumference and the area of the circle, and the dynamic development in progressive stages of the spiral process.