The Golden Ratio (Phi)

Bud Kroll

image:: jwilson.coe.uga.edu

F

What is the Golden Ratio?

1

A C

Construct a line segment AC

What is the Golden Ratio?

2

A B C

Divide the line segment at point B, such that the ratio of the

longer portion of the segment (AB) to the shorter portion of

the segment (BC) is the same as the ratio of the entire

segment (AC) to the longer portion (AB).

𝑨𝑩

𝑩𝑪=

𝑨𝑪

𝑨𝑩

Deriving the value of the Golden Ratio

3

A B C

If we give the full segment AC a unit value of 1, and give the

larger segment (AB) a value of x, the smaller segment has a

value of 1-x.

1

1-X X

We know the ratios of the segments to each other, so we can now set

up a proportional equation

4

A B C

1

1-X X

𝒙

𝟏 − 𝒙=

𝟏

𝒙

We can cross multiply and set up a quadratic equation in standard

form …

5

A B C

1

1-X X

𝒙𝟐 = 𝟏 − 𝒙

𝒙𝟐 + 𝒙 − 𝟏 =0

𝒙

𝟏 − 𝒙=

𝟏

𝒙

…and we can use the Quadratic Formula to solve for the roots

6

𝒙𝟐 + 𝒙 − 𝟏 =0 𝑎 = 1 𝑏 = 1 𝑐 = −1

𝑥 =−𝑏 ± 𝑏2 − 4𝑎𝑐

2𝑎

𝑥 =−(1) ± (1)2−4(1)(−1)

2(1)

𝑥 =−1 ± 1 − (−4)

2

Continued …

…and we can use the Quadratic Formula to solve for the roots

7

𝑥 =−1 ± 1 − (−4)

2

𝑥 =−1 + 5

2 𝑥 =

−1 − 5

2

(A negative, or

extraneous root,

which we can ignore)

𝑥 = 0.618034 (𝑟𝑜𝑢𝑛𝑑𝑒𝑑)

… allowing us to compute the Golden Ratio

8

A B C

1

1-X X

𝑥 = 0.618034

1

𝑥= 1.618034

𝑥

1 − 𝑥= 1.618034

Some golden things about the Golden Ratio

9

How is it that the ratio of the full width to the larger piece is exactly 1

greater than the ratio of the larger piece to the full width?

𝑥 = 0.618034

1

𝑥= 1.618034

A B C

1

1-X X

Some golden things about the Golden Ratio

10

How is it that the ratio of the full width to the larger piece is exactly 1

greater than the ratio of the larger piece to the full width?

This doesn’t happen normally.

If the ratio were 2:1, 1

0.5− 1 ≠

0.5

1

If the ratio were 1.5:1, 1

.66666− 1 ≠

.66666

1

But 1

0.618033− 1 =

1.618033

1

Some golden things about the Golden Ratio

11

If 1

𝑥=

𝑥

1−𝑥 , why is 𝑥 =

1

𝑥− 1 when 1/x is the Golden Ratio?

Going back to our cross-multiplication:

𝑥2 = 1 − 𝑥

Divide through by x:

𝑥2

𝑥=

1

𝑥−

𝑥

𝑥

which simplifies to

𝑥 =1

𝑥− 1

Instead of using a line segment, let’s use a rectangle whose

dimensions are 1.66 units wide and 1 unit high

12

A E D

Create a rectangle ABCD. Divide the rectangle with segment EF, such

that the ratio of AE to ED is the same as the ratio of the width of AD to

AE. (You have created a square AEFB.)

C B F

And plot the upper right hand corner of the resulting square

13

A E D

C B F

Now let’s do it again with the smaller rectangle, rotating clockwise to

remain aligned with the width…

14

A E D

C B F

And again …

15

A E D

C B F

And again …

16

A E D

C B F

And again …

17

A E D

C B F

And then connect the dots …

The result is a logarithmic spiral.

image:: jwilson.coe.uga.edu

The angles in a logarithmic spiral remain constant as you move

around the spiral, even though the spiral is getting tighter and tighter

19

We frequently observe logarithmic spirals in objects created by

human beings …

20

… and nature

21

The Golden Ratio is thought to be the most “pleasing” to people – the

Egyptians used Phi in designing the Great Pyramids c. 2500 BCE

22 www.goldennumber.net/architecture

The walls of the Great Pyramids are at an angle of 51.83 degrees.

Cos(51.83°) = 0.618, or 𝒑𝒉𝒊 .

The Greeks used it to design the Parthenon c. 448 BCE

23

The Parthenon (Athens) www.goldennumber.net/architecture

As well as Notre Dame (1163) among others

24 www.goldennumber.net/architecture

Phi can be estimated using an infinite series of fractions

25

1 +1

1 +1

1 +1

1 +1

1 +1

1 +1

𝑒𝑡𝑐

By Fractions

1 + 1/Previous Answer = New Answer

1 + 1/1 2

1 + 1/2 1.5

1 + 1 + 1/1.5 1.666666667

1 + 1 + 1/1.66666666666667 1.6

1 + 1 + 1/1.6 1.625

1 + 1 + 1/1.625 1.615384615

1 + 1 + 1/1.61538461538462 1.619047619

1 + 1 + 1/1.61904761904762 1.617647059

1 + 1 + 1/1.61764705882353 1.618181818

1 + 1 + 1/1.61818181818182 1.617977528

1 + 1 + 1/1.61797752808989 1.618055556

1 + 1 + 1/1.61805555555556 1.618025751

1 + 1 + 1/1.61802575107296 1.618037135

1 + 1 + 1/1.61803713527851 1.618032787

1 + 1 + 1/1.61803278688525 1.618034448

1 + 1 + 1/1.61803444782168 1.618033813

1 + 1 + 1/1.61803381340013 1.618034056

1 + 1 + 1/1.61803405572755 1.618033963

1 + 1 + 1/1.61803396316671 1.618033999

1 + 1 + 1/1.6180339985218 1.618033985

In 20 iterations we can converge very

close to Phi

Insights:: Mark Freitag

http://jwilson.coe.uga.edu/emt669/student.folders/frietag.

mark/homepage/goldenratio/goldenratio.html

It can also be approximated using an infinite series of the

square root of 1

26

By Square Root

Square Root of (1+Previous Answer) = New Answer

0

Sqrt(1+0) 1

Sqrt(1+1) 1.414213562

Sqrt(1+1.4142135623731) 1.553773974

Sqrt(1+1.55377397403004) 1.598053182

Sqrt(1+1.59805318247862) 1.611847754

Sqrt(1+1.61184775412525) 1.616121207

Sqrt(1+1.61612120650812) 1.617442799

Sqrt(1+1.61744279852739) 1.617851291

Sqrt(1+1.61785129060967) 1.617977531

Sqrt(1+1.61797753093474) 1.618016542

Sqrt(1+1.61801654223149) 1.618028597

Sqrt(1+1.61802859747023) 1.618032323

Sqrt(1+1.618032322752) 1.618033474

Sqrt(1+1.61803347392815) 1.61803383

Sqrt(1+1.61803382966122) 1.61803394

Sqrt(1+1.61803393958879) 1.618033974

Sqrt(1+1.61803397355828) 1.618033984

Sqrt(1+1.61803398405543) 1.618033987

Sqrt(1+1.61803398729922) 1.618033988

Sqrt(1+1.61803398830161) 1.618033989

Likewise, in 20 iterations we can

converge very close to Phi

𝑥 = 1 + 1 + 1 + 1 + 1 + ⋯

Insights:: Mark Freitag

http://jwilson.coe.uga.edu/emt669/student.folders/frietag.

mark/homepage/goldenratio/goldenratio.html