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_n a_n/(a_{n −1}). In a note to the writer, Mr. Cossentino shared his wonder at discovering that (where the length of the spiral in the first quadrant, ₀∫^π/2 [r² + (dr/d)]^1/2 d, was determined to be 2.109962658) it was uniquely the case that the value φ for the base in K has the symmetric property that the length of the spiral from the y-axis in the first quadrant to - (a/[1 - r] = 5.523953954 [where a = 2.109962658, and r = φ⁻¹]) is identically the length of the spiral in the third quadrant (aφ² = 5.523953954). Upon examining these facts, the writer noticed that the ratio of the length of the φ-spiral from the x-axis in the fourth quadrant to - is to the length of the spiral through the first four quadrants of the coordinate plane precisely as 1.170820393 (= φ²/5^1/2) is to 1, and that it is, therefore, identically the constant ratio of the diameter of the base of each ‘golden bowl’ to the rim diameter of the bowl next smaller in the infinite pentagonal series of bowls,¹ and at once the ‘retrospective square value’ of *F₁ (= *F₁₀₀/φ⁹⁹).² He then superimposed the φ-spiral in the coordinate plane on the φ-spiral inscribed in the ‘golden rectangle’ the sides of which equal, respectively, 1 and φ, as here illustrated,

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_ec²/2h), 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)k⁴/h³c², 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).

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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 (φ² + 1)⁻¹ times the diagonal (1.175570505^-2 + [φ² + 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/φ² or 1/φ³ times the diagonal (2/φ² + 1/φ³ = 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 YBa₂Cu₃O_6+x.
⁴ 6.6261 multiplied by the length of the first four-quadrant turn of the spiral (8.640563268) equals, with a difference less than one in 1000, 1 radian, and multiplied by the length of the spiral from the fourth quadrant to - (10.11654769) equals, qua base factor, with a difference less than one in 1000, the inverse of the square of the total length of the perimeter of the ‘dead center’ cube (cf. Leahy, Foundation, Sections III.5 and IV.2), and, at once, the square-constant of the 100-square grid of the unum-founded Fibonacci series (cf. ibid., Section III.2).

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