GOLDEN RATIO & SQUARES

(E-mail to a friend 6/17/17.  Updated 2/7/18.)

I have been reading a lot, lately, about the Golden Ratio (ϕ = 1.618) and the Fibonacci Series which, as you know, is manifest in nature in many ways.

My friend, Bill Atalay, has published (several years ago) Math And The Mona Lisa about the use of mathematical proportion by Leonardo da Vinci, etc.  Bill is a professor of physics at Mary Washington, U.Va. and Princeton.  I may have already mentioned him.  I recall that I did so.  He goes into extensive detail about da Vinci’s use of the Golden Ratio and related factors.  He discusses the Great Pyramid as a manifestation of the Golden Ratio, which may be why I mentioned it to you earlier.

He references a stained-glass window (with photo) at the UN by Marc Chagall, which has 5 square panes x 8 square panes, a manifestation of two numbers in the Fibonacci Series, which mathematically invokes the Golden Ratio.  I have taken it upon myself to graph the expanding projection of the Series in a “logarithmic spiral” on graph paper, proceeding in successive squares and starting at 1-1-2-3-5-8-13-21-34-55-89-144, etc., making a right-angle turn at each break.  I ran out of space at “55” but it continues, of course, into infinity.

ANYWAY, as each number increments, it does so by adding an exact SQUARE of the long side of the preceding rectangle, such that the two side-by-side ones add a 2 x 2 square adjacent, then a 3 x 3 square is added to the long side of the (2 + 1 =) 3 x 2 rectangle, then a 5 x 5 square is added to the long side of the (3 + 2 =) 5 x 3 rectangle, then a 8 x 8 square is added to the long side of the (5 + 3 +) 8 x 5 rectangle, etc.  If one assigns an arbitrary value of “1” (1 x 1) to any square added, then the preceding rectangle that itself already measures as a Golden Rectangle is 0.618 of the square, 1/ϕ, being the mathematical inverse of ϕ!  Then that prior Golden Rectangle added to the new square becomes it’s own Golden Rectangle which is then 0.618 of the next larger square added, etc.!!  Therefore, each manifest Golden Rectangle is 1.618 of the preceding rectangle (1.0 + 0.618)!

THIS IS SIGNIFICANT and explains why (I believe) the Golden Ratio is manifest in nature!  Consider that a plant, a chambered nautilus, whatever, is going to grow LARGER by adding more tissue to its existing size, and it will do so more “efficiently” if added to the larger "side" of the existing tissue.  It might also be confined by the allowed physical space (like the nautilus shell), so it is forced to "spiral" out to the side again and again as it grows.  Now, the new growth produces a “SQUARE” of sorts because the new tissue has spread out more or less equally in all directions, and that is basically what a “square” is—it is spread out from side to side and end to end equally, and that "spread" will arguably be defined in scope by the size of the existing tissue from which it emanates.  Perhaps the existing long side of the prior rectangle serves as sort of a “brake” to tell the organism to stop growing for that particular sequence once it reaches the full-spread “square” equivalency.  I don’t know if this explains WHY it grows that way, but that is what I imagined.

I took my graph of the squared-off “spiral” and colored the succeeding squares added in different colors to readily show how it accretes size in such a maneuver.