Golden Ratio

Suppose I ask you, for example: What do the delightful petal arrangement in a red rose, Salvador Dali's famous painting "Sacrament of the Last Supper," the magnificent spiral shells of mollusks, and the breeding of rabbits all have in common? Hard to believe, but these very disparate examples do have in common a certain number or geometrical proportion known since antiquity, a number that in the nineteenth century was given the honorifics "Golden Number","Golden Ratio," and "Golden Section." A book published in Italy at the beginning of the sixteenth century went so far as to call this ratio the "Divine Proportion."

The first clear definition of what has later become known as the Golden Ratio was given around 300 B.C. by the founder of geometry as a formalized deductive system, Euclid of Alexandria.
In Euclid's words:
"A straight line is said to have been cut in extreme and mean ratio
when, as the whole line is to the greater segment, so is the greater
to the lesser."

The modern history of the golden ratio starts with Luca Pacioli's Divina Proportione of 1509, which captured the imagination of artists, architects, scientists, and mystics with the properties, mathematical and otherwise, of the golden ratio.
Here are listed for the first time in writing the five properties that make this great proportion worthy of the epithet "divine":

• 1. "That it is one only and not more." Pacioli compares the unique

value of the Golden Ratio to the fact that unity "is the supreme
epithet of God himself."

• 2. Pacioli finds a similarity between the fact that the definition of

the Golden Ratio involves precisely three lengths (AC, CB, and
AB in Figure 24) and the existence of a Holy Trinity, of Father,
Son, and Holy Ghost.

• 3. To Pacioli, the incomprehensibility of God and the fact that the

Golden Ratio is an irrational number are equivalent. In his own
words: "Just like God cannot be properly defined, nor can be understood
through words, likewise our proportion cannot be ever
designated by intelligible numbers, nor can it be expressed by
any rational quantity, but always remains concealed and secret,
and is called irrational by the mathematicians."

• 4. Pacioli compares the omnipresence and invariability of God to

the self-similarity associated with the Golden Ratio—that its
value is always the same and does not depend on the length of the
line being divided or the size of the pentagon in which ratios of
lengths are calculated.

• 5. The fifth reason reveals an even more Platonic view of existence

than Plato himself expressed. Pacioli states that just as God conferred
being to the entire cosmos through the fifth essence, represented
by the dodecahedron, so does the Golden Ratio confer
being to the dodecahedron, since one cannot construct the dodecahedron
without the Golden Ratio. He adds that it is impossible to
compare the other four Platonic solids (representing earth, water,
air, and fire) to each other without the Golden Ratio.