THE GOLDEN BOWLS & THE LOGARITHMIC SPIRAL

The writer is indebted to Mr. Joseph Cossentino
who, after reading certain of the writer’s work, was led to examine the equation for
the logarithmic spiral in polar coordinates, *r* = *K* ,
where *K* = φ^{2/},
and, where *a** _{n}* = the Fibonacci series (0, 1, 1, 2, 3, 5, 8, 13 . . .),
φ = lim

and Mr. Cossentino was able to confirm the
writer’s judgment that the origin of the coordinate plane so superimposed was at
1.170820378 times the length of the side of the square at each stage of the infinitely
φ-diminishing square in the direction of the turn of the spiral by
proving that since = 1 +
φ^{-4 }+
φ^{-8}
+ φ^{-16} . . ., the *y*-axis
is *S* = *a*/(1 - *r*) = 1/(1 - φ^{-4}) = 1.170820393, and that since
= φ^{-1} +
φ^{-5} +
φ^{-9}
. . ., the *x*-axis is S = *a*/(1 - *r*) =
φ^{-1}/(1 -
φ^{-4})
= 1.170820393 · φ^{-1}.
With the origin of the coordinate plane thus located within the ‘golden
rectangle’ as defined, the total length, beginning with the *x*-axis in the
first quadrant, of the first four axis segments contained between the latter and the
points of intersection with the spiral in its first trip through the four quadrants

= 4.095126124, while the length of the spiral
through that first tour of the four quadrants = 8.640563268, so that the total of both
lengths = 12.73568939 (40^{-1}),
which, *qua* base factor, with a difference less than one in 1000, is the
circumference of the circle circumscribing the six-pointed star described
above on this web (the
diameter of which, in turn, with a difference less than one in 10,000,
is the power of
φ_{ }
equal to
the metric value of the velocity of light,^{3} as well as, again *qua* base factor, the metric
diameter of Earth. The length of the φ-spiral
from the *x*-axis of the fourth quadrant to - (10.11654769) is,
with a difference less than one in 100, the half-radius of the circle circumscribing the
six-pointed star, which latter, in turn, with a difference less than one in 1000,
100
^{-2}.

Finally, that
φ-proportionality is the ‘missing link’ among the physical
constants is further confirmed with wonderful precision by the following fact: if to the
total length of the first four axis segments, as defined above, is added the length of the
segment of the *x*-axis which extends (beyond the side of the original rectangle,
i.e., beyond *CB*) to the point of intersection with the spiral where the latter
completes its transit through the fourth quadrant, then, so construed, the total length of
the *y*-axis and the *x*-axis segments (4.095126124 + 2.530927155 = 6.626053279)
equals with a difference less than one in a quarter of a million the base factor of *h*
(6.6260755E-34 J·s), the Planck constant of action (energy × time).^{4} Nor is this the only instance of significant coincidence
of the physical constants and golden section values. For example, note that in the Rydberg
constant, *R*_{ }(*m*_{e}c^{2}/2*h*), the numerator
1E-25
times φ^{-4},
while the denominator 2E-34 times the total length of axis
segments defined by the first four-quadrant sweep of the
φ-spiral (as defined above), which is to say that both the
numerator and the denominator of *R*_{} are functions of
the pure numeric of the φ-spiral. So
too, the Stefan-Boltzmann constant, , (^{2}/60)*k*^{4}/*h*^{3}*c*^{2},
where the numerator 1E-92 times the constant ratio, divided by
φ, of the axis segments to the lengths from the
origin to the sides of the golden rectangle [.96672815/φ], while the denominator
1E-84 times [(*a*/[1
- *r*]) - *a*] where *a* = φ^{-1},
i.e., times the length from the origin to the top of the golden rectangle fourth smaller
than *ABCD*, created as the spiral winds toward - (φ^{-3}/5^{1/2}).

_____________________________________________________

The writer notes that when the diagonal of the golden rectangle is drawn, as here illustrated,

the center of the
spiral located horizontally at 1.170820393 times the length of the side of the
main square is at the point
on the diagonal 1.175570505^{-2} or
(φ^{2} + 1)^{-1}
times
the diagonal (1.175570505^{-2} +
[φ^{2} + 1]^{-1
}= 1). The corner point on the diagonal beyond the
point located horizontally at 1.170820393 times the length of the side of the
main square, the corner point of the first interior golden rectangle none of whose edges
coincides the main rectangle, is at 2/φ^{2}
or 1/φ^{3 }times the diagonal (2/φ^{2}
+ 1/φ^{3}^{ }= 1).

**
Notes
**

^{1} Cf., on this web, **
The Golden Bowl Structure: The Platonic Line . . ..**

2 Cf. D.G. Leahy, *Foundation*:
*Matter the Body Itself *(Albany, 1996), Sections III.2 and III.6.

^{3} Cf., on
this web,
**The Golden Bowl Structure: The Platonic Line . . .**
, and **
Measure
of Superconductive YBa**** _{2}Cu_{3}O_{6+x}**.