"Nature's equations are always balanced"
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I would like to associate the lines from the book "The Da Vinci Code" by Dan Brown with this particular Pencil Sketching post...
"Nature's equations are always balanced"...

When I happened to read "The Da Vinci Code" by Dan Brown, I realised nature is a perfectionalist whose creations are perfect.
One part of the book speaks about this number 1.618...which is called PHI . Pronounced as "fee", this number has a great secret hidden behind it and PHI is considered the most beautiful number...a "Divine Proportion".
Mathematically, when you consider the Fibonacci series, 1 1 2 3 5 8 13 21..., the quotients of adjacent terms possessed the properly of approaching the number 1.618. This is just one simple example of PHI in Math...but there are numerous equations mentioned point blank with all flowing towards this number 1.618.
Not only mathematically, but also, when you study the relationship of females and males in a honeybee community, usually the number of female bees outnumber the number of male bees (ofcourse...a fact)...but have you ever wondered what would be the number when you divide the number of female bee by the number of male bee!!!!...maan...Dan Brown says, its PHI!!!!

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NO WAY!!!! will be one's answer..but he continues his argument with Nature...
Sunflower seeds grow in opposing spiral...and the ratio of each rotation's diameter to the next...PHI!!!!
Spiraled pinecone petals, leaf arrangement on plant stalks, insect segmentation...all display obedience to this Divine Proportion
But the most amazing fact that he pointed is...the human body is literally made of building blocks whose proportional ratios "always" equal to PHI!!!!
Now...if you are not ready to agree...he challenges us to take a tape measure...Measure the distance from the tip of your head to the floor....then divide that by the distance from your belly button to the floor...Guess what number you get...Oops...PHI!!!!!
Want another example....Measure the distance from your shoulder to your fingertips, then divide it by the distance from your elbow to your fingertips...PHI...Hip to floor divided by knee to floor...PHI...Finger Joins, Toes, Spinal Divisions...PHI PHI PHI...
And,here are the lines from the book that you'll enjoy as I did....
Da Vinci ... Fibonacci numbers ... the pentacle.
Incredibly, all of these things were connected by a single concept so fundamental
to art history that Langdon often spent several class periods on the topic.
PHI.He felt himself suddenly reeling back to Harvard, standing in front of his ”Symbolism
in Art” class, writing his favorite number on the chalkboard.1.618
Langdon turned to face his sea of eager students. ”Who can tell me what this
number is?”
A long-legged math major in back raised his hand. ”That’s the number PHI.” He
pronounced it fee ....
As Langdon loaded his slide projector, he explained that the number PHI was
derived from the Fibonacci sequence−a progression famous not only because the sum
of adjacent terms equaled the next term, but because the quotients of adjacent terms
possessed the astonishing property of approaching the number 1.618−PHI!
Despite PHI’s seemingly mystical mathematical origins, Langdon explained, the
truly mind-boggling aspect of PHI was its role as a fundamental building block in
nature. Plants, animals, and even human beings, all possessed dimensional properties
that adhered with eerie exactitude to the ratio of PHI to 1.
”PHI’s ubiquity in nature,” Langdon said, killing the lights, ”clearly exceeds coincidence,
and so the ancients assumed the number PHI must have been preordained
by the Creator of the universe. Early scientists heralded one-point-six-one-eight as
the Divine Proportion.”
”Hold on,” said a young woman in the front row. ”I’m a bio major and I’ve never
seen this Divine Proportion in nature.”
”No?” Langdon grinned. ”Ever study the relationship between female and males
in a honeybee community?”
”Sure. The female bees always outnumber the male bees.”
”Correct. And did you know that if you divide the number of female bees by the
number of male bees in any beehive in the world, you always get the same number?”
”You do?”
”Yup. PHI.”
The girl gaped. ”NO WAY!”
”Way!” Langdon fired back, smiling as he projected a slide of a spiral seashell.
”Recognize this?”
”It’s a nautilus,” the bio major said. ”A cephalopod mollusk that pumps gas into
its chambered shell to adjust its buoyancy.”
”Correct. And can you guess what the ratio is of each spiral’s diameter to the
next?”
The girl looked uncertain as she eyed the concentric arcs of the nautilus spiral.
Langdon nodded. ”PHI. The Divine Proportion. One-point-six-one-eight to one.”
The girl looked amazed.
Langdon advanced to the next slide—a close-up of a sunflower’s seed head. ”Sun-
flower seeds grow in opposing spirals. Can you guess the ratio of each rotation’s
diameter to the next?”
”PHI?” everyone said.
”Bingo.” Langdon began racing through slides now−spiraled pinecone petals, leaf
arrangement on plant stalks, insect segmentation−all displaying astonishing obedience
to the Divine Proportion.
”This is amazing!” someone cried out.


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”Yeah,” someone else said, ”but what does it have to do with art?”
”Aha!” Langdon said. ”Glad you asked.” He pulled up another slide−a pale
yellow parchment displaying Leonardo da Vincis famous male nude−The Vitruvian
Man−named for Marcus Vitruvius, the brilliant Roman architect who praised the
Divine Proportion in his text De Architectura.
”Nobody understood better than Da Vinci the divine structure of the human
body. Da Vinci actually exhumed corpses to measure the exact proportions of human
bone structure. He was the first to show that the human body is literally made of
building blocks whose proportional ratios always equal PHI.”
Everyone in class gave him a dubious look.
”Don’t believe me?” Langdon challenged. ”Next time you’re in the shower, take
a tape measure.”
A couple of football players snickered.
”Not just you insecure jocks,” Landgon promoted. ”All of you. Guys and girls.
Try it. Measure the distance from the top of your head to the floor. Then divide that
by the distance from your belly button to the floor. Guess what number you get.”
”Not PHI!” one of the jocks blurted out in disbelief.
”Yes, PHI,” Langdon replied. ”One-point-six-one-eight. Want another example?
Measure the distance from your shoulder to your fingertips, and then divide it by
the distance from your elbow to your fingertips. PHI again. Another? Hip to floor
divided by knee to floor. PHI again. Finger joints. Toes. Spinal divisions. PHI. PHI.
PHI. My friends, each of you is a walking tribute to the Divine Proportion.”
Even in the darkness, Langdon could see they were all astounded. He felt familiar
warmth inside. This is why he taught. ”My friends, as you can see, the chaos of the
world has an underlying order. When the ancients discovered PHI, they were certain
they had stumbled across God’s building block for the world, and they worshipped
Nature because of that. And one can understand why. God’s hand is evident in
Nature, and even to this day there exist pagan, Mother Earth−revering religions.
Many of us celebrate nature the way pagans did, and don’t even know it. May Day is
a perfect example, the celebration of spring the earth coming back to life to produce
her bounty. The mysterious magic inherent in the Divine Proportion was written at
the beginning of time. Man is simply playing by nature’s rules, and because art is
man’s attempt to imitate the beauty of the Creator’s hand, you can imagine we might
be seeing a lot of instances of the Divine Proportion in art this semester.”
Over the next half hour, Langdon showed them slides of artwork by Michelangelo,
Albrecht D¨urer, Da Vinci, and many others, demonstrating each artist’s intentional
and rigorous adherence to the Divine Proportion in the layout of his compositions.
Langdon unveiled PHI in the architectural dimensions of the Greek Parthenon, the
pyramids, of Egypt, and even the United Nations Building in New York. PHI appeared
in organizational structures of Mozarts sonatas, Beethoven’s Fifth Symphony,
as well as the works of Bart´ok, Debussy, and Schubert. The number PHI, Langdon
told them, was even used by Stradivarius to calculate the exact placement of the
f-holes in the construction of his famous violins.

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”In closing,” Langdon said, walking to the chalkboard, ”we return to symbols.”
He drew five intersecting lines that formed a five-pointed star. ”This symbol is one of
the most powerful images you will see this term. Formally known as a pentagram−or
pentacle, as the ancients called it−this symbol is considered both divine and magical
by many cultures. Can anyone tell me why that might be?”
Stettner, the math major, raised his hand. ”Because if you draw a pentagram,
the lines automatically divide themselves into segments according to the Divine Proportion.”

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Langdon gave the kid a proud nod. ”Nice job. Yes, the ratios of line segments in
a pentacle all equal PHI, making this symbol the ultimate expression of the Divine
Proportion. For this reason, the five-pointed star has always been the symbol of
beauty and perfection associated with the goddess and the sacred feminine.”
The girls in class beamed ...