Fractals & Statistical Models I Nonlinear Computational Science - PowerPoint PPT Presentation

Fractals & Statistical Models I Nonlinear Computational Science in Action by Example Rubin H Landau Sally Haerer, Producer-Director Based on A Survey of Computational Physics by Landau, Páez, & Bordeianu with Support from the National Science Foundation Course: Computational Physics II 1 / 1

Fractal = Fractional Dimension (Math) Dimension def = ? � = n “fractals:” Mandelbrot, IBM Geometric fractals: 1 d f Statistical: random, local d f Agree: line, triangle, cube Hausdorff–Besicovitch d f Uniform density, side L M ( L ) ∝ L d f Works for 1-D, 2-D, ∝ L d f ρ = M ( L ) ∝ L d f − 2 3-D! A L 2 2 / 1

Our 1st Fractal: Sierpi´ nski Gasket Game of Chance (Randomness) Sierpin.py Draw equilateral triangle. 1 Place dot at P = ( x 0 , y 0 ) within 2 Random 1 2, or 3: 3 If 1, � P = ( � P + � 1)/2 4 If 2, � P = ( � P + � 2)/2 5 If 3, � P = ( � P + � 3)/2 6 Repeat 10,000 × new P 7 300 10,000 points ( x k + 1 , y k + 1 ) =( x k , y k ) + � n 200 2 100 n = int ( 1 + 3 r i ) 0 0 100 200 300 3 / 1

Determine d f via ρ = CL d f − 2 Geometric Sierpi´ nski Gasket (Midpoints) Self-similar: part ∼ whole Dots, m dot = 1, or ln ρ ∝ ( d f − 2 ) log L ρ new = 3 4 ρ old (fills space less) d f − 2 = ∆ log ρ ∆ log L d f = 2 + ∆ log ρ ( L ) ∆ log L = 2 + log 1 − log 3 4 log 1 − log 2 ≃ 1 . 585 4 / 1

Example 2: Beautiful Plants Nature + chance ⇒ high regularity & symmetry? EG fern or tree = beautiful, graceful & random? Key: self-similar, fractal Simple random algorithm ⇒ beauty? If algorithm → ferns ⇒ algorithm ∃ fern? 5 / 1

Self-Affine Connection (Theory) Point–Point Relation ⇒ Self Similarity Recall Sierpi´ nski gasket: ( x k + 1 , y k + 1 ) = ( x k , y k ) / 2 + ( a n , b n ) / 2 s = scaling factor = 1 ( a n , b n ) = translation 2 , s > 0 = amplification, s < 0 = reduction ( x ′ , y ′ ) = s ( x , y ) (General Scaling) ( x ′ , y ′ ) = ( x , y ) + ( a x , a y ) (General Translation) x ′ = x cos θ − y sin θ, (Rotation) y ′ = x sin θ + y cos θ (Rotation) Affine Connection = point–point scale + rotate + translate 6 / 1

Barnsley’s Fern (Fern3D.py) Affine Connection + Random (2D)  ( 0 . 5 , 0 . 27 y n ) , with 2% probability ,      ( − 0 . 139 x n + 0 . 263 y n + 0 . 57     0 . 246 x n + 0 . 224 y n − 0 . 036 ) , with 15% probability ,      ( x , y ) n + 1 = ( 0 . 17 x n − 0 . 215 y n + 0 . 408   0 . 222 x n + 0 . 176 y n + 0 . 0893 ) , with 13% probability ,       ( 0 . 781 x n + 0 . 034 y n + 0 . 1075      − 0 . 032 x n + 0 . 739 y n + 0 . 27 ) , with 70% probability.  Start ( x 1 , y 1 ) = ( 0 . 5 , 0 . 0 ) ∆ d f different parts Repeat iterations Stem = compress fronds Not completely self-similar Nonlinear indirectly 7 / 1

Barnsley’s Fern (Fern3D.py) Affine Connection + Random  2 % , r < 0 . 02 ,      15 % , 0 . 02 ≤ r ≤ 0 . 17 ,   Select with probability P = 13 % , 0 . 17 < r ≤ 0 . 3 ,       70 % , 0 . 3 < r < 1 .  Combined rules (program):  ( 0 . 5 , 0 . 27 y n ) , r < 0 . 02 ,    ( − 0 . 139 x n + 0 . 263 y n + 0 . 57      0 . 246 x n + 0 . 224 y n − 0 . 036 ) , 0 . 02 ≤ r ≤ 0 . 17 ,     ( x , y ) n + 1 = ( 0 . 17 x n − 0 . 215 y n + 0 . 408  0 . 222 x n + 0 . 176 y n + 0 . 0893 ) , 0 . 17 < r ≤ 0 . 3 ,      ( 0 . 781 x n + 0 . 034 y n + 0 . 1075 ,      − 0 . 032 x n + 0 . 739 y n + 0 . 27 ) , 0 . 3 < r < 1 .  8 / 1

Self-Affine Trees (Tree.py)  ( 0 . 05 x n , 0 . 6 y n ) , 10% probability ,     ( 0 . 05 x n , − 0 . 5 y n + 1 . 0 ) , 10% probability ,       ( 0 . 46 x n − 0 . 15 y n , 0 . 39 x n + 0 . 38 y n + 0 . 6 ) , 20% probability ,  ( x n + 1 , y n + 1 ) = ( 0 . 47 x n − 0 . 15 y n , 0 . 17 x n + 0 . 42 y n + 1 . 1 ) , 20% probability ,      ( 0 . 43 x n + 0 . 28 y n , − 0 . 25 x n + 0 . 45 y n + 1 . 0 ) , 20% probability ,      ( 0 . 42 x n + 0 . 26 y n , − 0 . 35 x n + 0 . 31 y n + 0 . 7 ) , 20% probability .  9 / 1

Good Time for a Break 10 / 1