The Mathematics of Phi

By Geoff Byron, Tyler Galbraith, and Richard Kim

It’s a “phi-nomenon!”

The History of Phi

WHAT IS PHI?

Phi can sometimes be misunderstood because it is known by so many different names:

Ex: mean and extreme ratio, golden proportion, golden mean, golden section, golden number, divine proportion, φ, or sectio divina

Phi is most often known as the golden ratio

VALUES FOR PHI

Two quantities are said to be in the golden ratio, if “the whole is to the larger part as the larger part is to the smaller part.”

This can be demonstrated by:

Phi is equal to the following quadratic equation:

Therefore, we have Phi take on the values of 1.618 and .618, which are often written as Phi = 1.618 and phi = .618

€

a + b

a=

a

b

€

1± 5

2

THE GOLDEN MEAN

From the graphic above we can derive the following about Phi:

A is 1.618 times B and B is 1.618 times C.Alternatively, C is .618 of B and B is .618 of A.

WHO FOUND PHI?There is debate over when and by who Phi was actually discovered.

Egyptians: The ratio is found in the dimensions of the Egyptian’s pyramids, yet there is no mathematical or historical proof that the Egyptians knew about Phi.Euclid: Most often, the finding of Phi is associated with the Greek mathematician, Euclid, who wrote about Phi in his series of books, Elements, around 300 B.C.

Euclid is attributed with finding the golden ratio and many of its properties.

WHO FOUND PHI?

Fibonacci: Fibonacci is given credit for adding to the properties of Phi by establishing the Fibonacci Sequence, but it is uncertain if Fibonacci himself ever found the connection between his sequence and Phi.

WHERE DID THE NAME PHI COME FROM?

It was not until the 1900’s that the numerical value of 1.618 was given the name Phi.

Until then it was only referred to as the golden ratio, divine proportion, golden mean, and golden section.

American mathematician Mark Barr first used the Greek letter phi to designate the proportion

Reasons for choosing Phi:Phi is the first letter of Phidias, who used the golden ratio in his sculptures, as well as the Greek equivalent to the letter “F,” the first letter of Fibonacci. Phi is also the 21st letter of the Greek alphabet, and 21 is one of the numbers in the Fibonacci series.

WHERE WAS PHI FIRST SEEN?

Phi was first seen in the design of the Great Pyramids. (2560 B.C.)

It can also be seen used excessively in the design of the Parthenon. (447 B.C.)

So, how is Phi derived?

Jacques Philippe Marie Binet

Developed a formula that finds any Fibonacci number without having to start from 1, 1, 2, 3, 5, 8, etc….

What old mathematicians found out about Phi

...11111 +++++=x

...1111112 ++++++=x

Square both sides:

xx +=12

012 =−−xx

Apply quadratic equation:

251±==Φ x

...61803399.0

...61803399.1

−=Φ−=Φ

Notice that phi differs by sign:

What old mathematicians also found about

12 +=xx

Can you find the pattern?

813

58

35

23

12

1

7

6

5

4

3

2

+=

+=

+=

+=

+=

+=

xx

xx

xx

xx

xx

xx

????=nx

Binet’s Formula

)1()( −+= nfibxnfibxn

Solve for fib(n).

Φ±=x

)1())(()(:

)1()(:

−+Φ−=Φ−

−+Φ=Φ

nfibnfibB

nfibnfibAn

n

Subtract B from A:

)]1())(([)1()()( −+Φ−−−+Φ=Φ−−Φ nfibnfibnfibnfibnn

5

)()(

nn

nfibΦ−−Φ

= Finds any Fibonacci number, assuming at n=1, Fib(1)=1.

What does Binet have to do with Phi?

If we look at Binet’s formula as

it approaches infinity, it

converges to phi.

fib(n+1) fib(n) fib(n+1)/fib(n)1 1 12 1 23 2 1.55 3 1.6666678 5 1.613 8 1.62521 13 1.61538534 21 1.61904855 34 1.61764789 55 1.618182144 89 1.617978233 144 1.618056377 233 1.618026610 377 1.618037987 610 1.6180331597 987 1.6180342584 1597 1.6180344181 2584 1.6180346765 4181 1.618034

Looking at convergence from a calculus perspective, what test

should we use to test convergence???

The RATIO TEST!

)(

)(lim

5

)(5

)(

lim)(

)1(lim

11

11

nn

nn

nnn

nn

nn nfib

nfib

Φ−−ΦΦ−−Φ

=Φ−−Φ

Φ−−Φ

=+ ++

∞→

++

∞→∞→

Φ=ΦΦ +

∞→ n

n

n

1

lim

Applications of Phi

Phi in NatureThere is no other number that recurs throughout life more so than does phi. When looking at nature, we see Phi, often times without realizing it.

Phi in NatureThe golden spiral is created by making adjacent squares of Fibonacci dimensions and is based on the pattern of squares that can be constructed with the golden rectangle.

If you take one point, and then a second point one-quarter of a turn away from it, the second point is Phi times farther from the center than the first point. The spiral increases by a factor of Phi.

This shape can be found in many shells, especially in nautilus.

Phi in Nature

Phi in ManThe Phi proportion itself can be found in the very bones that form our body's skeleton. For example, the three bones of any finger are related to one another by 1.618. Also, the wrist joint cuts the length from fingertip to elbow at 0.618

Ratios equal to Phi

Φ=BrowtoHeadofTip

BrowtoChin

Φ=

Navelto

HeadofTip

FootofBottom

toNavel

Phi in Design

The appearance of phi in all we see and experience creates a sense of balance, harmony and beauty. Mankind uses this same proportion found in nature to achieve balance, harmony and beauty in its own creations of art, architecture, colors, design, composition, space and even music.

Phi in Design

Works CitedFreitag, Mark. "Phi: That Golden Number." Golden Ratio. 2006. 11 May 2006 <http://jwilson.coe.uga.edu/EMT669/Student.Folders/Frietag.Mark/Homepage/Goldenratio/goldenratio.html>. Obara, Samuel. "Golden Ratio in Art and Architecture." University of Georgia Dept. of Mathematics Education. 2003. 11 May 2006 <http://jwilson.coe.uga.edu/EMT668/EMAT6680.2000/Obara/Emat6690/Golden%20Ratio/golden.html>."Phi / Golden Proportion." Nature's Word | Musings on Sacred Geometry. 2006. 11 May 2006 <http://www.unitone.org/naturesword/sacred_geometry/phi/in_nature/>. Place, Robert M. "Leonardo on the Tarot." The Alchemical Egg. 2000. 11 May 2006 <http://thealchemicalegg.com/leotaroN.html>. "The Arts - Design and Composition." Phi the Golden Number. 2006. 11 May 2006 <http://goldennumber.net/design.htm>.