Length & Width
The value of this recursion becomes much more clear in two dimensions. Again, we begin with the simplest case: a rectangle.
Make a rectangle ABCD with sides in PHI ratio:
Then, inside this rectangle, draw a square BCFE (yellow). Just as in Extreme & Mean Proportion, the long side (AB) is divided by the short side (AD) into PHI ratio segments. There is a smaller rectangle AEFD remaining, and its sides are in PHI ratio.
Now, repeat this operation. Make another square DFHG (red) inside the second, smaller rectangle ADFE. Again, a new, smaller rectangle AEHG remains, and its sides are also in PHI ratio.
Perform the operation again. Draw another square AGJI (blue) inside the remaining rectangle AEHG, and another rectangle AGJI remains. And the sides of this new rectangle are in PHI ratio.
Once more, draw a new square AILM (green) inside the remaining rectangle AGJI to create a new rectangle AIML whose sides are in the same PHI ratio.
Each time we repeat this operation of squaring the rectangle, the PHI ratio reappears. So, PHI recurs in two dimensions.
By a similar, but opposite process, we could create larger rectangles instead of smaller ones, and the PHI ratio would recur at each expanding scale.
Thus, in this two dimensional, rectilinear "flatland," you see that the PHI ratio is a natural harmonic proportion that recreates itself at each successive scale.
PHI is a Greek letter, and the ancient Greeks adopted the Divine Ratio extensively as sacred measure and aesthetic proportion in their art and architecture. In Athens, the great Parthenon temple was composed of many PHI ratio rectangles. The Greeks sought divine guidance at the Oracle of DelPHI, whose symbol was a serpent coiled around an egg. Later, you will understand this symbol is an archetype of profound biological significance.
In art, PHI is considered the most pleasing aesthetic proportion. But few modern architects or artists use this Golden Mean in their designs. And while our engineering makes profuse use of the circle, triangle and square, rarely has the Golden Mean been used to create form, shape or function.
Four & Five
Embedded Fractal Nest
But in Nature, (PHI)ve-sided symmetry is a 5-sided pentagon, not a 4-sided rectangle. The right angle ideas embedded in our four-square mindset are challenged to think in five-sided figures.
In a Pentacle, draw diagonals on the inner, inverted pentagon to create a new inverted star. Inside this star, is a smaller pentagon, now in point-up position.
Adding new diagonals can continue endlessly. Pentagons and pentangles recur in ever smaller scales. Each is in PHI ratio to the previous, larger scale.
So, recursion in PHI-sided symmetry forms nests of stars and pentagons. Not only ratio, but shape recurs at each smaller scale. Repetition of pattern through an endless series of scales forms an embedded nest. This doesn't happen in 3- or 4-sided symmetry, and depicts PHI's unique ability to pass pattern between scales—a quality needed for communication and memory.
In math, this is a basis for fractals—how order emerges from chaos—how pattern exists at every scale.
In medieval times, astronomer and mathmetician Johannes Kepler used PHI to calculate and accurately predict the orbital diameters and speed of the planets
Add & Multiply
Standing Wave Form
Nesting reveals another unique property of PHI. In Extreme & Mean Proportion, if the long segment is 1, the ratio becomes:
1.618... = 1 / .618...
Or, in symbolic equation:
PHI = 1 / (PHI - 1) = 1.618...
And, by simple algebra:
PHI - 1 = 1 / PHI = .618...
PHI + 1 = PHI2 = 2.618...
These extraordinary equations combine addition left of the equal sign with multiplication on the right. PHI provides a unique interaction between addition and multiplication—a singular point where to add is to multiply, and to subtract is to divide. No other real number has these qualities.
Our physical world is more than matter. Since Einstein's Relativity Theory, we now know that physical matter is also waves of energy: sound, electromagnetism, and perhaps others, such as gravity waves. To a wave, there are no lines—only circles.
When waves of different frequencies meet, they interact—they add and multiply. The PHI ratio permits waves to interact in orderly, harmonic patterns. One simple portrayal of this interaction loks like this:
Thus, waves of different length form a stable nest. PHI is how varied vibrations condense to create a standing wave form.
In electronics, this is "heterodyne," and is how radio and TV signals are modulated onto a carrier wave.
How does the U.S. Pentagon communicate with underground bases and undersea submarines?
by Pentagram coded into ELF waves
In telecommunications, PHI allows information to be conveyed over a full spectrum of frequencies.