The Golden Mean - Dr. Travers Page of...
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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
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
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
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
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
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 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
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 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
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 value
Showed practical importance of this number system tobookkeepingIncluded a problem about the growing population of rabbits basedon idealized assumptions
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 tobookkeeping
Included a problem about the growing population of rabbits basedon idealized assumptions
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
Fibonacci’s Rabbits
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’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’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’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
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
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
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
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
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?
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?
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
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
So how could the Greeks have constructed segments with the correctproportion for the Golden Mean?
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
√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
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
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
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
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
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 unitsFrom starting mark, fold in half.
√5− 12
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
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
Where We See In Geometry
Where We See In Geometry
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.
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.
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.
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.
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.
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.
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
Constructing the Ancient Greek Way
Draw a line from the midpoint of one side to an opposite corner.
Constructing the Ancient Greek Way
Use that line as the radius to draw an arc that defines the height of therectangle.
Constructing the Ancient Greek Way
Use that line as the radius to draw an arc that defines the height of therectangle.
Constructing the Ancient Greek Way
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.