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Finding Gold -- the Golden Mean in Mathematics, Architecture, Arts & Life

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Finding Gold -- the Golden Mean in Mathematics, Architecture, Arts & Life Finding Gold { The Golden Mean in Mathematics, Architecture, Arts & Life Reimer K¨uhn Disordered Systems Group Department of Mathematics King's College London Cumberland Lodge Weekend, Feb 17{19, 2017 1 / 42 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 2 / 42 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 3 / 42 Golden Ratio - Definition & Numerical Value Euklid (' 300 BC): cutting a line in extreme and mean ratio & golden rectangle Numerical value p a + b a 1 ± 5 = ≡ ' , '2 − ' − 1 = 0 ) ' = a b ± 2 Golden Ratio p 1 + 5 ' = ' = = 1:618033988749894848204586833 ::: + 2 4 / 42 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 5 / 42 History Ancient Greece Discovery of the concept attributed to Phythagoras (∼ 569-475 BC) Description of 5 regular polyedra, the geometry of some involving ' by Plato (427-347 BC) Architecture of the Parthenon in Athens, completed 438 BC under 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" (some illustrations by Da Vinci1 (1452-1519)), on the mathematics of the golden ratio, its appearance in art, architecture, and in the Platonic solids, defines golden ratio as divine proportion, attributes divine and aesthetically pleasing properties to it. Johannes Kepler (1571-1630) proves that ratio of two successive Fibonacci numbers approaches the golden mean, Kepler \golden triangle", describes radii of planetary motion in terms 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 Map Physics: Roger Penrose (b.1931) Aperiodic Tilings 6 / 42 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 7 / 42 Golden Ratio { Construction Using compass and ruler 8 / 42 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 9 / 42 Q: Convergence? Independence of initial value '0? Representations | Iterative Solutions General idea: Use definition in the form ' = f(') and solve by iteration '0 ;'n+1 = f('n) ; n ≥ 0 10 / 42 Representations | Iterative Solutions General idea: Use definition in the form ' = f(') and solve by iteration '0 ;'n+1 = f('n) ; n ≥ 0 Q: Convergence? Independence of initial value '0? 11 / 42 1 'n = 1 + 'n−1 1 = 1 + 1 + 1 'n−2 1 = 1 + 1 + 1 1+ 1 'n−3 . 1 = 1 + 1 ! '? 1 + 1 1+ 1 1+ 1 1+ 1+::: Continued Fraction Representation 1 Continued fraction: Use ' = f(') = 1 + ' 12 / 42 1 = 1 + 1 + 1 'n−2 1 = 1 + 1 + 1 1+ 1 'n−3 . 1 = 1 + 1 ! '? 1 + 1 1+ 1 1+ 1 1+ 1+::: Continued Fraction Representation 1 Continued fraction: Use ' = f(') = 1 + ' 1 'n = 1 + 'n−1 13 / 42 1 = 1 + 1 + 1 1+ 1 'n−3 . 1 = 1 + 1 ! '? 1 + 1 1+ 1 1+ 1 1+ 1+::: Continued Fraction Representation 1 Continued fraction: Use ' = f(') = 1 + ' 1 'n = 1 + 'n−1 1 = 1 + 1 + 1 'n−2 14 / 42 . 1 = 1 + 1 ! '? 1 + 1 1+ 1 1+ 1 1+ 1+::: Continued Fraction Representation 1 Continued fraction: Use ' = f(') = 1 + ' 1 'n = 1 + 'n−1 1 = 1 + 1 + 1 'n−2 1 = 1 + 1 + 1 1+ 1 'n−3 15 / 42 Continued Fraction Representation 1 Continued fraction: Use ' = f(') = 1 + ' 1 'n = 1 + 'n−1 1 = 1 + 1 + 1 'n−2 1 = 1 + 1 + 1 1+ 1 'n−3 . 1 = 1 + 1 ! '? 1 + 1 1+ 1 1+ 1 1+ 1+::: 16 / 42 p 'n = 1 + 'n−1 q p = 1 + 1 + 'n−2 r q p = 1 + 1 + 1 + 'n−3 . v u s r u q p = t1 + 1 + 1 + 1 + 1 + ::: ! '? Nested Root Representation p Nested Root: Use '2 − ' − 1 = 0 $ ' = f(') = 1 + ' 17 / 42 q p = 1 + 1 + 'n−2 r q p = 1 + 1 + 1 + 'n−3 . v u s r u q p = t1 + 1 + 1 + 1 + 1 + ::: ! '? Nested Root Representation p Nested Root: Use '2 − ' − 1 = 0 $ ' = f(') = 1 + ' p 'n = 1 + 'n−1 18 / 42 r q p = 1 + 1 + 1 + 'n−3 . v u s r u q p = t1 + 1 + 1 + 1 + 1 + ::: ! '? Nested Root Representation p Nested Root: Use '2 − ' − 1 = 0 $ ' = f(') = 1 + ' p 'n = 1 + 'n−1 q p = 1 + 1 + 'n−2 19 / 42 . v u s r u q p = t1 + 1 + 1 + 1 + 1 + ::: ! '? Nested Root Representation p Nested Root: Use '2 − ' − 1 = 0 $ ' = f(') = 1 + ' p 'n = 1 + 'n−1 q p = 1 + 1 + 'n−2 r q p = 1 + 1 + 1 + 'n−3 20 / 42 Nested Root Representation p Nested Root: Use '2 − ' − 1 = 0 $ ' = f(') = 1 + ' p 'n = 1 + 'n−1 q p = 1 + 1 + 'n−2 r q p = 1 + 1 + 1 + 'n−3 . v u s r u q p = t1 + 1 + 1 + 1 + 1 + ::: ! '? 21 / 42 Iteration & Convergence 22 / 42 Iteration & Convergence 23 / 42 Iteration & Convergence 24 / 42 Iteration & Convergence 25 / 42 Iteration & Convergence 26 / 42 Iteration & Convergence (Local) convergence for jf 0(')j < 1 Here global convergence irrespective of initial condition. 27 / 42 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 28 / 42 Golden Ratio in Geometry Kepler triangle, pentagon (' = b=a), pentagram Golden rhombus, rhombic triacontahedron, dodecahedron, icosahedron p For a = 2 dodecahedron has (r; ρ, R) = ('2/ξ; '2; 3'), and p qp icosahedron has (r; ρ, R) = ('2= 3; '; ξ'), where ξ = 5=' 29 / 42 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 30 / 42 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 divine proportion, attributes divine and aesthetically pleasing properties 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 of arts and architecture. 31 / 42 Architecture Pyramids Cheops and Chephren, within an arcmin, and half a p degree respectively within golden inclination, ' ' 4/π, Parthenon, United Nations building (Oscar Niemeier, 1950). 32 / 42 Visual Arts Leonardo Da Vinci(1452-1519), Heinrich Agrippa (1486-1535), Georges Seurat (1859-1891), Salvador Dali (1904-1989), Piet Mondrian (1872-1944) 33 / 42 But look and ye shall find Golden Ratio in Design Proportions in iconic products close to golden ratio Credit cards, National Geographic logo, KitKat logo, but interestingly not smart phones (i-phone, samsung galaxy, blackberry) or tablets (i-pad, samsung galaxy). Are they missing opportunities? 34 / 42 Golden Ratio in Design Proportions in iconic products close to golden ratio Credit cards, National Geographic logo, KitKat logo, but interestingly not smart phones (i-phone, samsung galaxy, blackberry) or tablets (i-pad, samsung galaxy). Are they missing opportunities? But look and ye shall find 35 / 42 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 36 / 42 Fibonacci Sequence and the Golden Ratio Leonardo of Pisa (Fibonacci, c. 1170- c. 1250); introduced Hindu-Arabic numeral system to Europe; \Liber Abaci" (1202) Fibonacci Sequence, describing growth of a population of rabbits. 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!1 Fn+1=Fn = ' exists, then from (*) Fn+1=Fn = 1 + Fn−1=Fn by taking limits, get p 1 + 5 ' = 1 + 1=' , ' = 2 37 / 42 Fibonacci - Convergence Rate Fibonacci sequence fromh 2nd order linear recursion Fn+1 = Fn + Fn−1 ;F0 = 0 ;F1 = 1 : (∗) n 2 Solve using ansatz Fn = Aλ ; solves iff λ − λ − 1 = 0! Two solutions λ1;2 = '±. (recall '+ = '; '− = 1 − ') n n Fn = A'+ + B'− Initial conditions ! A = −B = p1 5 ! n 1 n 1 − ' Fn = p ' 1 − 5 ' 38 / 42 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 x 1 1 x x 1 n+1 = = n ; 1 = yn+1 1 0 yn y1 0 writing this as x x 1 n+1 = B n = ··· = Bn yn+1 yn 0 The matrix B has eigenvalues '±! ) detB = −1 Find F F Bn = n+1 n Fn Fn−1 B and thus B2 are chaotic maps; B2 is chaotic and area preserving (discrete analogue of a Hamiltonian system) 39 / 42 Phase Space View | Arnol'd's Cat Map 2 Look at xn+1 = B xnmod 1, i.e. dynamics restricted to a torus. Defines Arnol'd's cat map (V.I. Arnol'd 1937-2010). Paradigm of chaotic volume preserving map, played key role in theory of ergodic systems Eigenvalues irrationally related: high iterate of cat image will cover the torus uniformly 40 / 42 Fibonacci Sequence in Nature Fibonacci spiral, phyllotaxis, shells, hands,.
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