Finding Gold – The Golden Mean inMathematics, Architecture, Arts & Life

Reimer Kuhn

Disordered Systems GroupDepartment of Mathematics

King’s College London

Cumberland Lodge Weekend, Feb 17–19, 2017

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Outline

1 Golden Ratio – Definition & Numerical Value

2 History

3 Construction

4 Representations

5 Geometry

6 Architecture and the Arts

7 The Fibonacci Sequence and the Golden Ratio

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Outline

1 Golden Ratio – Definition & Numerical Value

2 History

3 Construction

4 Representations

5 Geometry

6 Architecture and the Arts

7 The Fibonacci Sequence and the Golden Ratio

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Golden Ratio - Definition & Numerical Value

Euklid (' 300 BC): cutting a line in extreme and mean ratio& golden rectangle

Numerical value

a+ b

a=a

b≡ ϕ ⇔ ϕ2 − ϕ− 1 = 0 ⇒ ϕ± =

1±√

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Golden Ratio

ϕ = ϕ+ =1 +√

52

= 1.618033988749894848204586833 . . .

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Outline

1 Golden Ratio – Definition & Numerical Value

2 History

3 Construction

4 Representations

5 Geometry

6 Architecture and the Arts

7 The Fibonacci Sequence and the Golden Ratio

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History

Ancient Greece

Discovery of the concept attributed to Phythagoras(∼ 569-475 BC)Description of 5 regular polyedra, the geometry of someinvolving ϕ by Plato (427-347 BC)Architecture of the Parthenon in Athens, completed 438 BCunder Phidias (∼ 480-43- BC)First known written account in Euclid (∼ 325-265 BC),“Elements”, including proof of irrationality.

Renaissance

Luca Pacioli (1445-1517), “De Divina Proportione” (someillustrations by Da Vinci1 (1452-1519)), on the mathematics ofthe golden ratio, its appearance in art, architecture, and in thePlatonic solids, defines golden ratio as divine proportion,attributes divine and aesthetically pleasing properties to it.Johannes Kepler (1571-1630) proves that ratio of twosuccessive Fibonacci numbers approaches the golden mean,Kepler “golden triangle”, describes radii of planetary motion interms of Platonic solids in “Mysterium Cosmographicum”.

19th & 20th Century

Arts: Georges Seurat (1859-1891) The Bathers, Salvador Dali(1904-1989) Last Supper, Piet Mondrian (1872-1944)Compositions . . .Mathematics: Vladimir Arnol’d (1931 -2010) Cat MapPhysics: Roger Penrose (b.1931) Aperiodic Tilings

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Outline

1 Golden Ratio – Definition & Numerical Value

2 History

3 Construction

4 Representations

5 Geometry

6 Architecture and the Arts

7 The Fibonacci Sequence and the Golden Ratio

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Golden Ratio – Construction

Using compass and ruler

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Outline

1 Golden Ratio – Definition & Numerical Value

2 History

3 Construction

4 Representations

5 Geometry

6 Architecture and the Arts

7 The Fibonacci Sequence and the Golden Ratio

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Representations — Iterative Solutions

General idea: Use definition in the form

ϕ = f(ϕ)

and solve by iteration

ϕ0 , ϕn+1 = f(ϕn) , n ≥ 0

Q: Convergence? Independence of initial value ϕ0?

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Representations — Iterative Solutions

General idea: Use definition in the form

ϕ = f(ϕ)

and solve by iteration

ϕ0 , ϕn+1 = f(ϕn) , n ≥ 0

Q: Convergence? Independence of initial value ϕ0?

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Continued Fraction Representation

Continued fraction: Use ϕ = f(ϕ) = 1 + 1ϕ

ϕn = 1 +1

ϕn−1

= 1 +1

1 + 1ϕn−2

= 1 +1

1 + 11+ 1

ϕn−3

...

= 1 +1

1 + 11+ 1

1+ 1

1+ 11+...

→ ϕ?

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Continued Fraction Representation

Continued fraction: Use ϕ = f(ϕ) = 1 + 1ϕ

ϕn = 1 +1

ϕn−1

= 1 +1

1 + 1ϕn−2

= 1 +1

1 + 11+ 1

ϕn−3

...

= 1 +1

1 + 11+ 1

1+ 1

1+ 11+...

→ ϕ?

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Continued Fraction Representation

Continued fraction: Use ϕ = f(ϕ) = 1 + 1ϕ

ϕn = 1 +1

ϕn−1

= 1 +1

1 + 1ϕn−2

= 1 +1

1 + 11+ 1

ϕn−3

...

= 1 +1

1 + 11+ 1

1+ 1

1+ 11+...

→ ϕ?

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Continued Fraction Representation

Continued fraction: Use ϕ = f(ϕ) = 1 + 1ϕ

ϕn = 1 +1

ϕn−1

= 1 +1

1 + 1ϕn−2

= 1 +1

1 + 11+ 1

ϕn−3

...

= 1 +1

1 + 11+ 1

1+ 1

1+ 11+...

→ ϕ?

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Continued Fraction Representation

Continued fraction: Use ϕ = f(ϕ) = 1 + 1ϕ

ϕn = 1 +1

ϕn−1

= 1 +1

1 + 1ϕn−2

= 1 +1

1 + 11+ 1

ϕn−3

...

= 1 +1

1 + 11+ 1

1+ 1

1+ 11+...

→ ϕ?

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Nested Root Representation

Nested Root: Use ϕ2−ϕ− 1 = 0 ↔ ϕ = f(ϕ) =√

1 + ϕ

ϕn =√

1 + ϕn−1

=√

1 +√

1 + ϕn−2

=

√1 +

√1 +

√1 + ϕn−3

...

=

√√√√1 +

√1 +

√1 +

√1 +√

1 + . . .→ ϕ?

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Nested Root Representation

Nested Root: Use ϕ2−ϕ− 1 = 0 ↔ ϕ = f(ϕ) =√

1 + ϕ

ϕn =√

1 + ϕn−1

=√

1 +√

1 + ϕn−2

=

√1 +

√1 +

√1 + ϕn−3

...

=

√√√√1 +

√1 +

√1 +

√1 +√

1 + . . .→ ϕ?

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Nested Root Representation

Nested Root: Use ϕ2−ϕ− 1 = 0 ↔ ϕ = f(ϕ) =√

1 + ϕ

ϕn =√

1 + ϕn−1

=√

1 +√

1 + ϕn−2

=

√1 +

√1 +

√1 + ϕn−3

...

=

√√√√1 +

√1 +

√1 +

√1 +√

1 + . . .→ ϕ?

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Nested Root Representation

Nested Root: Use ϕ2−ϕ− 1 = 0 ↔ ϕ = f(ϕ) =√

1 + ϕ

ϕn =√

1 + ϕn−1

=√

1 +√

1 + ϕn−2

=

√1 +

√1 +

√1 + ϕn−3

...

=

√√√√1 +

√1 +

√1 +

√1 +√

1 + . . .→ ϕ?

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Nested Root Representation

Nested Root: Use ϕ2−ϕ− 1 = 0 ↔ ϕ = f(ϕ) =√

1 + ϕ

ϕn =√

1 + ϕn−1

=√

1 +√

1 + ϕn−2

=

√1 +

√1 +

√1 + ϕn−3

...

=

√√√√1 +

√1 +

√1 +

√1 +√

1 + . . .→ ϕ?

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Iteration & Convergence

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Iteration & Convergence

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Iteration & Convergence

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Iteration & Convergence

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Iteration & Convergence

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Iteration & Convergence

(Local) convergence for |f ′(ϕ)| < 1Here global convergence irrespective of initial condition.

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Outline

1 Golden Ratio – Definition & Numerical Value

2 History

3 Construction

4 Representations

5 Geometry

6 Architecture and the Arts

7 The Fibonacci Sequence and the Golden Ratio

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Golden Ratio in Geometry

Kepler triangle, pentagon (ϕ = b/a), pentagram

Golden rhombus, rhombic triacontahedron, dodecahedron, icosahedron

For a = 2 dodecahedron has (r, ρ,R) = (ϕ2/ξ, ϕ2,√

3ϕ), and

icosahedron has (r, ρ,R) = (ϕ2/√

3, ϕ, ξϕ), where ξ =q√

5/ϕ

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Outline

1 Golden Ratio – Definition & Numerical Value

2 History

3 Construction

4 Representations

5 Geometry

6 Architecture and the Arts

7 The Fibonacci Sequence and the Golden Ratio

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Luca Pacioli, Da Vinci

Luca Pacioli (1445-1517), “De Divina proportione”, on the(i)mathematics of the golden ratio, (ii) its appearance in art,(Vitruvian) architecture, and (iii)in the Platonic solids(illustrations by Da Vinci); defines golden ratio as divineproportion, attributes divine and aesthetically pleasingproperties to it. In part believed to be plagiarized from Piero della

Francesca (translated into Italian without acknowledgement)

Pacioli’s work has received considerable attention in world ofarts and architecture.

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Architecture

Pyramids Cheops and Chephren, within an arcmin, and half adegree respectively within golden inclination,

√ϕ ' 4/π,

Parthenon, United Nations building (Oscar Niemeier, 1950).

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Visual Arts

Leonardo Da Vinci(1452-1519), Heinrich Agrippa(1486-1535), Georges Seurat (1859-1891), Salvador Dali(1904-1989), Piet Mondrian (1872-1944)

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Golden Ratio in Design

Proportions in iconic products close to golden ratio

Credit cards, National Geographic logo, KitKat logo, butinterestingly not smart phones (i-phone, samsung galaxy,blackberry) or tablets (i-pad, samsung galaxy). Are theymissing opportunities?

But look and ye shall find

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Golden Ratio in Design

Proportions in iconic products close to golden ratio

Credit cards, National Geographic logo, KitKat logo, butinterestingly not smart phones (i-phone, samsung galaxy,blackberry) or tablets (i-pad, samsung galaxy). Are theymissing opportunities?

But look and ye shall find

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Outline

1 Golden Ratio – Definition & Numerical Value

2 History

3 Construction

4 Representations

5 Geometry

6 Architecture and the Arts

7 The Fibonacci Sequence and the Golden Ratio

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Fibonacci Sequence and the Golden Ratio

Leonardo of Pisa (Fibonacci, c. 1170- c. 1250); introducedHindu-Arabic numeral system to Europe; “Liber Abaci”(1202)

Fibonacci Sequence, describing growth of a population ofrabbits.

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987...

Fn+1 = Fn + Fn−1 , F0 = 0 , F1 = 1 . (∗)

If limn→∞ Fn+1/Fn = ϕ exists, then from (*)

Fn+1/Fn = 1 + Fn−1/Fn

by taking limits, get

ϕ = 1 + 1/ϕ ⇔ ϕ =1 +√

52

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Fibonacci - Convergence Rate

Fibonacci sequence fromh 2nd order linear recursion

Fn+1 = Fn + Fn−1 , F0 = 0 , F1 = 1 . (∗)

Solve using ansatz Fn = Aλn; solves iff λ2 − λ− 1 = 0!

Two solutions λ1,2 = ϕ±. (recall ϕ+ = ϕ, ϕ− = 1− ϕ)

Fn = Aϕn+ +Bϕn

−

Initial conditions → A = −B = 1√5

Fn =1√5ϕn

(1−

(1− ϕϕ

)n)

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Phase Space View

2nd order recursion ⇔ coupled pair of 1st order recursions

Fn+1 = Fn + Fn−1 , F0 = 0 , F1 = 1 . (∗)

xn ≡ Fn , yn ≡ Fn−1

Get(xn+1

yn+1

)=(

1 11 0

)=(xn

yn

),

(x1

y1

)=(

10

)writing this as(

xn+1

yn+1

)= B

(xn

yn

)= · · · = Bn

(10

)The matrix B has eigenvalues ϕ±! ⇒ detB = −1 Find

Bn =(Fn+1 Fn

Fn Fn−1

)B and thus B2 are chaotic maps; B2 is chaotic and areapreserving (discrete analogue of a Hamiltonian system)

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Phase Space View — Arnol’d’s Cat Map

Look at xn+1 = B2xnmod 1, i.e. dynamics restricted to atorus.

Defines Arnol’d’s cat map (V.I. Arnol’d 1937-2010). Paradigmof chaotic volume preserving map, played key role in theory ofergodic systems

Eigenvalues irrationally related: high iterate of cat image willcover the torus uniformly

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Fibonacci Sequence in Nature

Fibonacci spiral, phyllotaxis, shells, hands,. . .

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An analysis ends, when the patient realizes

it could go on forever.

THANK YOU!

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