The Golden Mean - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/the_golden_mean.pdfThe ﬁrst written deﬁnition of the Golden Ratio comes from Euclid’s elements.

The Golden Mean

Leonardo Pisano Bigolio

Who knows who he is?

c. 1170- c. 1250

aka Leonardo de Pisa

aka Fibonacci

Father was a trader and through travel, learned of Hindu-Arabicnumber systemWrote Liber Abaci (1202)

Book of Abacus or Book of Calculationpopularized Hindu–Arabic numerals in EuropeAdvocated using digits 0-9 and place valueShowed practical importance of this number system tobookkeepingIncluded a problem about the growing population of rabbits basedon idealized assumptions



c. 1170- c. 1250


aka Fibonacci





c. 1170- c. 1250


aka Fibonacci





c. 1170- c. 1250


aka Fibonacci





c. 1170- c. 1250


aka Fibonacci





c. 1170- c. 1250


aka Fibonacci


Book of Abacus or Book of Calculation

popularized Hindu–Arabic numerals in EuropeAdvocated using digits 0-9 and place valueShowed practical importance of this number system tobookkeepingIncluded a problem about the growing population of rabbits basedon idealized assumptions



c. 1170- c. 1250


aka Fibonacci


Book of Abacus or Book of Calculationpopularized Hindu–Arabic numerals in Europe

Advocated using digits 0-9 and place valueShowed practical importance of this number system tobookkeepingIncluded a problem about the growing population of rabbits basedon idealized assumptions



c. 1170- c. 1250


aka Fibonacci


Book of Abacus or Book of Calculationpopularized Hindu–Arabic numerals in EuropeAdvocated using digits 0-9 and place value

Showed practical importance of this number system tobookkeepingIncluded a problem about the growing population of rabbits basedon idealized assumptions



c. 1170- c. 1250


aka Fibonacci


Book of Abacus or Book of Calculationpopularized Hindu–Arabic numerals in EuropeAdvocated using digits 0-9 and place valueShowed practical importance of this number system tobookkeeping

Included a problem about the growing population of rabbits basedon idealized assumptions



c. 1170- c. 1250


aka Fibonacci



Fibonacci’s Rabbits


Fibonacci Sequence

fn = fn−1 + fn−2, f0 = f1 = 1

11

= 1

21

= 2

32

= 1.5

53

= 1.67

85

= 1.6

138

= 1.625

limn→∞

fn+1

fn≈ 1.61803

= Φ


Fibonacci Sequence

fn = fn−1 + fn−2, f0 = f1 = 1

11

= 1

21

= 2

32

= 1.5

53

= 1.67

85

= 1.6

138

= 1.625

limn→∞

fn+1

fn≈ 1.61803

= Φ


Fibonacci Sequence

fn = fn−1 + fn−2, f0 = f1 = 1

11

= 1

21

= 2

32

= 1.5

53

= 1.67

85

= 1.6

138

= 1.625

limn→∞

fn+1

fn≈ 1.61803

= Φ


Fibonacci Sequence

fn = fn−1 + fn−2, f0 = f1 = 1

11

= 1

21

= 2

32

= 1.5

53

= 1.67

85

= 1.6

138

= 1.625

limn→∞

fn+1

fn≈ 1.61803

= Φ

The Golden Mean

It was frequently used by the ancient Greeks.

The Golden Mean

There are golden ratios all over the regular pentagon.

Consequently, Φ is often attributed to Pythagaros since thePythagoreans used the regular pentagon as their symbol.

The Golden Mean

There are golden ratios all over the regular pentagon.

Consequently, Φ is often attributed to Pythagaros since thePythagoreans used the regular pentagon as their symbol.

The Ancient Greeks

The first written definition of the Golden Ratio comes from Euclid’selements.

DefinitionA straight line is said to have been cut in extreme and mean ratio,when as the whole line is to the greater segment, so the greater to theless.

• • •a b

a + ba

=ab

The Ancient Greeks



• • •a b

a + ba

=ab

The Ancient Greeks



• • •a b

a + ba

=ab

The Ancient Greeks



• • •a b

a + ba

=ab

Why Φ?

Mark Barr was the first to use Φ in honor of Pheidias.

Pheidias (490-430 BC) was a Greek sculpter of

Statue of Zeus atOlympia

Athena Promachos

Why Φ?




Athena Promachos

Why Φ?




Athena Promachos

Derivation of Φ

a + ba

=ab

= Φ

⇒ a = bΦbΦ + b

bΦ=

bΦ

bΦ + 1

Φ= Φ

Φ + 1 = Φ2

Φ2 − Φ− 1 = 0

Φ =1±√

52

Φ = 1.61803,−.61803

Derivation of Φ

a + ba

=ab

= Φ

⇒ a = bΦ

bΦ + bbΦ

=bΦ

bΦ + 1

Φ= Φ

Φ + 1 = Φ2

Φ2 − Φ− 1 = 0

Φ =1±√

52

Φ = 1.61803,−.61803

Derivation of Φ

a + ba

=ab

= Φ

⇒ a = bΦbΦ + b

bΦ=

bΦ

b

Φ + 1Φ

= Φ

Φ + 1 = Φ2

Φ2 − Φ− 1 = 0

Φ =1±√

52

Φ = 1.61803,−.61803

Derivation of Φ

a + ba

=ab

= Φ

⇒ a = bΦbΦ + b

bΦ=

bΦ

bΦ + 1

Φ= Φ

Φ + 1 = Φ2

Φ2 − Φ− 1 = 0

Φ =1±√

52

Φ = 1.61803,−.61803

Derivation of Φ

a + ba

=ab

= Φ

⇒ a = bΦbΦ + b

bΦ=

bΦ

bΦ + 1

Φ= Φ

Φ + 1 = Φ2

Φ2 − Φ− 1 = 0

Φ =1±√

52

Φ = 1.61803,−.61803

Derivation of Φ

a + ba

=ab

= Φ

⇒ a = bΦbΦ + b

bΦ=

bΦ

bΦ + 1

Φ= Φ

Φ + 1 = Φ2

Φ2 − Φ− 1 = 0

Φ =1±√

52

Φ = 1.61803,−.61803

Derivation of Φ

a + ba

=ab

= Φ

⇒ a = bΦbΦ + b

bΦ=

bΦ

bΦ + 1

Φ= Φ

Φ + 1 = Φ2

Φ2 − Φ− 1 = 0

Φ =1±√

52

Φ = 1.61803,−.61803

Derivation of Φ

a + ba

=ab

= Φ

⇒ a = bΦbΦ + b

bΦ=

bΦ

bΦ + 1

Φ= Φ

Φ + 1 = Φ2

Φ2 − Φ− 1 = 0

Φ =1±√

52

Φ = 1.61803,−.61803

A Couple Properties

Property 1

Φ2 = Φ + 1

Property 2

1Φ

= φ

= .61803

= Φ− 1

A Couple Properties

Property 1

Φ2 = Φ + 1

Property 2

1Φ

= φ

= .61803

= Φ− 1

A Couple Properties

Property 1

Φ2 = Φ + 1

Property 2

1Φ

= φ

= .61803

= Φ− 1

A Couple Properties

Property 1

Φ2 = Φ + 1

Property 2

1Φ

= φ

= .61803

= Φ− 1

The Golden Mean

It is believed that the Egyptians knew of it as well

Egyptians and the Golden Mean

The Egyptians lacked the ability to calculate the slant height forpyramids unless a 3 : 4 : 5 triangle since they did not have thePythagorean Theorem or anything similar to aid them.

Egyptians and the Golden Mean

The Egyptians lacked the ability to calculate the slant height forpyramids unless a 3 : 4 : 5 triangle since they did not have thePythagorean Theorem or anything similar to aid them.

Back to the Greeks

So how could the Greeks have constructed segments with the correctproportion for the Golden Mean?

Construct a 1 : 2 rectangleaffix a piece of string to the lower left corner, run it around apoint at the upper right corner and down to the right corner.How long is this piece of string?Since the string is

√5 + 1 units long, fold it in half.

1 +√

52

If we instead want 1Φ , we can get that too.

Beginning with same rectangle, affix string in same manner.Fold 1 unit back over the diagonal to get

√5− 1 units

From starting mark, fold in half.√

5− 12

Back to the Greeks


Construct a 1 : 2 rectangle

affix a piece of string to the lower left corner, run it around apoint at the upper right corner and down to the right corner.How long is this piece of string?Since the string is


1 +√

52



√5− 1 units


5− 12

Back to the Greeks


Construct a 1 : 2 rectangleaffix a piece of string to the lower left corner, run it around apoint at the upper right corner and down to the right corner.

How long is this piece of string?Since the string is


1 +√

52



√5− 1 units


5− 12

Back to the Greeks


Construct a 1 : 2 rectangleaffix a piece of string to the lower left corner, run it around apoint at the upper right corner and down to the right corner.How long is this piece of string?

Since the string is√

5 + 1 units long, fold it in half.

1 +√

52



√5− 1 units


5− 12

Back to the Greeks




1 +√

52



√5− 1 units


5− 12

Back to the Greeks




1 +√

52



√5− 1 units


5− 12

Back to the Greeks




1 +√

52


Beginning with same rectangle, affix string in same manner.

Fold 1 unit back over the diagonal to get√

5− 1 unitsFrom starting mark, fold in half.

√5− 12

Back to the Greeks




1 +√

52



√5− 1 units


5− 12

Back to the Greeks




1 +√

52



√5− 1 units


5− 12

Where We See In Geometry



1

Φ

θ

θ = 2sin−1

(12Φ

)

= 2sin−1(

12Φ

)= 36◦

So, the angles for the triangle are 36◦ for the top angle and 72◦ for thecongruent base angles.

And, this works whenever the base b to the slant height a is in therelation a

b = Φ.


1

Φ

θ

θ = 2sin−1

(12Φ

)

= 2sin−1(

12Φ

)= 36◦



b = Φ.


1

Φ

θ

θ = 2sin−1

(12Φ

)

= 2sin−1(

12Φ

)

= 36◦



b = Φ.


1

Φ

θ

θ = 2sin−1

(12Φ

)

= 2sin−1(

12Φ

)= 36◦



b = Φ.


1

Φ

θ

θ = 2sin−1

(12Φ

)

= 2sin−1(

12Φ

)= 36◦



b = Φ.


1

Φ

θ

θ = 2sin−1

(12Φ

)

= 2sin−1(

12Φ

)= 36◦



b = Φ.

Golden Pentagons

a + b

b

a

Golden Pentagons

a + b

b

a

Golden Pentagons

a + b

b

a

Golden Pentagons

a + b

b

a

Golden Rectangles

Golden Rectangles

Golden Rectangles

Golden Rectangles

Golden Rectangles

Constructing the Ancient Greek Way

Draw a simple square


Draw a line from the midpoint of one side to an opposite corner.


Use that line as the radius to draw an arc that defines the height of therectangle.





Logarithmic Curve

We can similarly create a logarithmic curve by starting with a goldenisosceles triangle and then taking the side corresponding to Φ as theshorter side in an adjacent triangle.

The Golden Mean - Dr. Travers Page of Mathbtravers.weebly.com/.../6/7/2/9/6729909/the_golden_mean.pdfThe ﬁrst written deﬁnition of the Golden Ratio comes from Euclid’s elements.

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