Historical presentation · 2019

Unfolding Complex Trees with n-fold rotational symmetry

Bernat Espigulé ·
Illustrating Mathematics, ICERM
32 slides · Recovered static slides

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Slide 1: Unfolding Complex Trees with n-fold rotational symmetry Bernat Espigulé bernat@espigule.com Xavier Jarque & Núria Fagella Illustrating Mathematics, ICERM, Brown Universit
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Unfolding Complex Trees with n-fold rotational symmetry Bernat Espigulé bernat@espigule.com Xavier Jarque & Núria Fagella Illustrating Mathematics, ICERM, Brown University, October 2019 www.ComplexTrees.com

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Slide 2: \"Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire.\" Archiv för Matemat., Astron. och Fys. 1, 681-702, 1904. Helge von Koch\u2
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\"Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire.\" Archiv för Matemat., Astron. och Fys. 1, 681-702, 1904. Helge von Koch\u2028(1870 –1924) www.ComplexTrees.com 

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Slide 3: www.ComplexTrees.com  \"Les courbes planes ou gauches et les surfaces composées de parties semblales au tout. » J. l'École Polytech., 227-247 and 249-291, 1939. Paul Lév
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www.ComplexTrees.com  \"Les courbes planes ou gauches et les surfaces composées de parties semblales au tout. » J. l'École Polytech., 227-247 and 249-291, 1939. Paul Lévy \u2028(1886 –1971)

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Slide 4: r0 =  r3 Linear Extension of the Trunk LLL r1 = www.ComplexTrees.com r n
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r0 =  r3 Linear Extension of the Trunk LLL r1 = www.ComplexTrees.com r n

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Slide 5: www.ComplexTrees.com
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www.ComplexTrees.com

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Slide 6:  r = 0.61803… 45º 90º 60° 108° rSC = Classification of Fractal Trees 135º www.ComplexTrees.com 120° 144° 180º rmin = ½ = 0.5 rmax = 0.707…=
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 r = 0.61803… 45º 90º 60° 108° rSC = Classification of Fractal Trees 135º www.ComplexTrees.com 120° 144° 180º rmin = ½ = 0.5 rmax = 0.707…=

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Slide 7: 1/Φ = Φ - 1= (1+√5 )½ - 1 ≈ 0.618… 60° The Beauty of Fractals: Six Different Views By Dennis Gulick, Jon Scott MAA, Feb 28, 2011 “Excursions Through a Forest of Golden Fr
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1/Φ = Φ - 1= (1+√5 )½ - 1 ≈ 0.618… 60° The Beauty of Fractals: Six Different Views By Dennis Gulick, Jon Scott MAA, Feb 28, 2011 “Excursions Through a Forest of Golden Fractal Trees” Tara D. Taylor, St. Francis Xavier University 144° 108° “Finding Gold in the Forest” www.ComplexTrees.com 120°

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Slide 8: The Beauty of Fractals: Six Different Views By Dennis Gulick, Jon Scott MAA, Feb 28, 2011 “Excursions Through a Forest of Golden Fractal Trees” Tara D. Taylor, St. Franci
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The Beauty of Fractals: Six Different Views By Dennis Gulick, Jon Scott MAA, Feb 28, 2011 “Excursions Through a Forest of Golden Fractal Trees” Tara D. Taylor, St. Francis Xavier University Mathematical Details for the article of Frame & Mandelbrot Self-Contacting Fractal Trees by Prof. Donald C. West © (1999) Scaling Ratio r Self-Contacting Binary Fractal Trees Angle θ Tara D. Taylor “Computational Topology and Fractal Trees”, PhD Thesis, Dalhousie University, Canada (2005)  www.ComplexTrees.com http://faculty.plattsburgh.edu/don.west/trees/index.htm Michael Frame and Benôit B. Mandelbrot The Mathematical Intelligencer, 1999 Springer-Verlag

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Slide 9: www.ComplexTrees.com
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www.ComplexTrees.com

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Slide 10: www.ComplexTrees.com
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www.ComplexTrees.com

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Slide 11: π/5 = 36° 1/Φ ≈ 0.618… Nonlinear Dynamics and Chaos: Applications in Atmospheric Sciences A.M.Selvam Fig. 6. Quasicrystalline structure of the quasiperiodic Penrose tilin
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π/5 = 36° 1/Φ ≈ 0.618… Nonlinear Dynamics and Chaos: Applications in Atmospheric Sciences A.M.Selvam Fig. 6. Quasicrystalline structure of the quasiperiodic Penrose tiling pattern and Fibonacci sequence. A Hidden Golden Tree A Golden Cantor Set Roger L. Kraft The American Mathematical Monthly, Vol. 105, No. 8 (Oct., 1998), pp. 718-725  www.ComplexTrees.com

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Slide 12: Angle θ The Complete Tip to Tip Contact Plot 1/Φ = Φ - 1= (1+√5 )½ - 1 ≈ 0.618… 
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Angle θ The Complete Tip to Tip Contact Plot 1/Φ = Φ - 1= (1+√5 )½ - 1 ≈ 0.618… 

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Slide 14:  Angle θ The Complete Tip to Tip Contact Plot
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 Angle θ The Complete Tip to Tip Contact Plot

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Slide 15: www.ComplexTrees.com Harmonic Fractal Trees 
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www.ComplexTrees.com Harmonic Fractal Trees 

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Slide 18: www.ComplexTrees.com
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www.ComplexTrees.com

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Slide 19: www.ComplexTrees.com 
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www.ComplexTrees.com 

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Slide 20: www.ComplexTrees.com
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www.ComplexTrees.com

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Slide 21: “If you wish to advance into the infinite, explore the finite in all directions.” ⎯ Goethe. Holomorphic Dynamics Group Núria Fagella & Xavier Jarque www.ComplexTrees.com
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“If you wish to advance into the infinite, explore the finite in all directions.” ⎯ Goethe. Holomorphic Dynamics Group Núria Fagella & Xavier Jarque www.ComplexTrees.com @bernatree Thank you! Bernat Espigulé

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Slide 24: “Our soul is composed of harmony, and harmony is never bred save in moments when the proportions of objects are seen or heard.” ⎯ Leonardo da Vinci
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“Our soul is composed of harmony, and harmony is never bred save in moments when the proportions of objects are seen or heard.” ⎯ Leonardo da Vinci

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Slide 25: Harmonic Trees  1/Φ = Φ - 1= (1+√5 )½ - 1 ≈ 0.618…
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Harmonic Trees  1/Φ = Φ - 1= (1+√5 )½ - 1 ≈ 0.618…

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Slide 26:  Harmonic Trees Imagen 5
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 Harmonic Trees Imagen 5

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Slide 28: Symmetric Fractal Trees H. GÖTZE ⎯ Castel del Monte, Gestalt und Symbol der Architektur Friedrichs II. 3. Auflage Prestel, München 1991 1991, Dr. Susanne Krömker 
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Symmetric Fractal Trees H. GÖTZE ⎯ Castel del Monte, Gestalt und Symbol der Architektur Friedrichs II. 3. Auflage Prestel, München 1991 1991, Dr. Susanne Krömker 

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Slide 29:  Symmetric Fractal Trees Imagen 5
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 Symmetric Fractal Trees Imagen 5

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Slide 30: Imagen 2 Symmetric Fractal Trees 
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Imagen 2 Symmetric Fractal Trees 

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Slide 31: Imagen 7 Symmetric Fractal Trees  Imagen 10 Imagen 6 Imagen 11 Imagen 12 Imagen 5 Imagen 8 Imagen 9
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Imagen 7 Symmetric Fractal Trees  Imagen 10 Imagen 6 Imagen 11 Imagen 12 Imagen 5 Imagen 8 Imagen 9

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Slide 32: Imagen 1
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Imagen 1
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