FIBONACCI SQUARES

I thought of an interesting progression of the “Fibonacci Series” the other morning.  The Series is the sequence of integers, each numeral of which is the sum of the two preceding numerals in the sequence.  I discovered that the sum of the squares of adjacent numerals in the Series yields higher alternating numerals later in the Series!  I have no clue of any particular significance.

Thus, the initial part of the Series, beginning with zero.  Note that “1” repeats twice, since “1” is the succeeding total of “0” and the first “1,” which necessarily follows “0”:

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, etc.

SQUARES:

0, 1, 1, 4, 9, 25, 64, 169, 441, etc.

SUMS OF ADJACENT SQUARES:

1, 2, 5, 13, 34, 89, 233, 610, etc.

It should be noted that above about 13” in the Series, the quotient ratio of adjacent numerals in the Series is the so-called “Golden Ratio” of phi (Φ—“fee”—1.618) or 1/1.618 (= 0.618 = 1/Φ).  

The Fibonacci Series may be graphically projected as the addition of ever-larger squares forming ever-larger, outward-spiraling “Golden Rectangles."  Note that the quotient ratio of the Series increases or decreases by exactly “1” depending upon which “direction” one is going.  Consider that any square, therefore, is equal to the unit “1” since it is the same length on all sides.  That may account for the frequent manifestation of the Series in Nature as semi-liquid protoplasm expanding in all directions as it grows larger, such as the ever-larger chambered nautilus, the sprouting of leaves up a stem, etc.

So, employing the Pythagorean Theorem, the “hypotenusal” diagonals of the ever-larger graphic “Golden Rectangles” were also thus measured as the square roots of the aforesaid sums of the squares:

1, 1.414, 2.236, 3.605, 5.831, 9.434, 15.264, 24.698, etc.

AND, the quotient ratios of these higher adjacent “hypotenuses” more or less continue to manifest as the “Golden Ratio” of phi (Φ) = 1.618!!!  WOW!!

Just for grins, I also looked at the following, but I perceived no patterns or sequences:

ADJACENT SQUARE DIFFERENCES:

1, 0, 3, 5, 16, 39, 105, 271, etc.

DIFFERENCES, WITH ROUNDED SQUARE ROOTS:

⎷-1 (i), 1² + 2, 2 ²  + 1, 4² + 0, 6² + 3, 10² + 5, 16² + 15, etc.

I have speculated as to whether or not an exhaust pipe would scavenge “fluid” exhaust gases more efficiently from an internal-combustion cylinder if the diameter OR circumference OR cross-sectional area of the pipe were progressively enlarged by the Golden Ratio.  The overall length of the pipe would also inject an unknown variable, considering that most exhaust ports are at least an inch or larger in diameter at the start, and the pipe diameter would become very much larger very quickly!  And, consider whether the exhaust pipe should taper smoothly, like a  megaphone or trombone bell, or should it be stepped?  

As exhaust gases cool as they flow away from the cylinder(s), their requisite volume would likely decrease, so a non-flared exhaust pipe might be adequate.  Some drag-racing mechanics have cooled intake manifolds with dry ice to allow more unburned fuel-air mixture to flow into the combustion chamber(s) and have insulated the exhaust manifolds to RETAIN exhaust heat and improve exhaust-gas flows outward.