With or without algebraic proof – I produced it here in 2009 – we will immediately find out some interesting aspects:
Once the four circles have been identified by their respective diameters in decreasing golden proportion, let's now move on to the first initiative, which consists in tracing from the vertical point
on the larger circumference, of diameter =
, a line tangent to the circumference Φ
, which will meet the primary at point
- drawing the symmetrical
VB line to
B we will have drawn a triangle whose base
AB is tangent to the circle Φ3;
therefore we would have obtained
B also following the reverse process.
- a mirror line at
VA halves the side at the point
X which is tangency of
VA to the circle Φ; and reaches exactly the point of tangency at Φ3 in the middle of
AB; in particular a line parallel to
AB from the point
X is tangent to Φ2, perhaps I should add 'obviously'; but the best is yet to come…
- The sum of
VB is equivalent to
AB×Φ, i.e. the base of this especial triangle is the Golden Section of the sum of the sides; the supreme balancing factor in the structure of the ordered physical universe; this had since long led me to define it as the Great Golden Triangle par excellence.
Driven by these premises, I forwarded to an in-depth study that proved to be decisive; and now the time has come to grapple with the numbers.
Given the diameter of the primary circle as a unit of measurement, therefore with a value of
1.0, the circumference will be
= π and the quadrant arc
VQ = π/4 (in a linear sense, as opposed to angular measurement in radians).
In this case the To measure
VD height of the
triangle will be
0,5 + Φ
0,1180339887498948482…+ 0,5], which in our construct turns out to be the golden section Φ of the diameter = 1:
VA it will be neessary according to the theorem of Pythagoras:
AD = √AC² – CD²