Phi in Colonies of Honeybees

Honeybees: A honeybee colony consists of a queen, a few drones and lots of workers. The female bees (queens and workers) all have two parents, a drone and a queen. Drones, on the other hand, hatch from unfertilized eggs. This means they have only one parent. Therefore, Fibonacci numbers express a drone's family tree in that he has one parent, two grandparents, three great-grandparents and so forth.


(To read more on bees, click on the link within the word "honeybee". You will be redirected to HowStuffWorks.com)

Phi in Flowers

Flowers and branches: Some plants express the Fibonacci sequence in their growth points, the places where tree branches form or split. One trunk grows until it produces a branch, resulting in two growth points. The main trunk then produces another branch, resulting in three growth points. Then the trunk and the first branch produce two more growth points, bringing the total to five. This pattern continues, following the Fibonacci numbers. Additionally, if you count the number of petals on a flower, you'll often find the total to be one of the numbers in the Fibonacci sequence. For example, lilies and irises have three petals, buttercups and wild roses have five, delphiniums have eight petals and so on. (Thanks to HowStuffWorks for this info)

Phi in the Mona Lisa

Prevalent in the major works of Leonardo Da Vinci and underlying many of his design compositions, is the Golden Ratio ("Golden Mean"), a ratio of approximately 1:1.618, found in nature and creation, and inherent in the Fibonacci sequence. The Golden Rectangle, the Golden Triangle, and the Golden Pyramid, all based on the Golden Ratio are all appear prominent in the work of Leonardo Da Vinci.

"The ratio is always the same. 1 to 1.618, over and over and over again. The patterns, mathematical in design, are hidden in plain sight. You just have to know where to look. 7,080,360,000 people. And only a few of us can see the connections. Today we’ll send over 300 billion emails, 19 billion text messages. Yet we’ll still feel alone. The average person will say 2,250 words to 7.4 other individuals. Will these words be used to hurt or to heal? There’s an ancient Chinese myth about the Red Thread of Fate. It says that the gods have tied a red thread around every one of our ankles and attached it to all of the people whose lives we are destined to touch. This thread may stretch or tangle. But it will never break."

 

Jake Bohm, Touch

Calculation

Two quantities a and b are said to be in the golden ratio φ if:

One method for finding the value of φ is to start with the left fraction. Through simplifying the fraction and substituting in b/a = 1/φ,

it is shown that,

Multiplying by φ gives

which can be rearranged to

Using the quadratic formula gives the only positive solution as,

(Images from Wikipedia)

Phidias

Phidias, (490–430 BC) made the Parthenon statues that seem to embody the golden ratio.

Mark Barr was the one who decided to make the Golden Ratio's sign the Greek letter Phi, after Phidias.