Fractal Examples Handout

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(Adapted from “List of fractals by Hausdorff dimension”, Wikipedia) Approx. Exact dimension Name Illustration dimension 2 log 3 1.12915 Contour of the Gosper island 7 ( ) 1.2083 Fibonacci word fractal 60° 3 log φ 3+√13 log� 2 � Solution of Boundary of the tame 1.2108 2 1 = 2 twindragon 2− 2 − log 4 1.2619 Triflake 3 log 4 1.2619 Koch curve 3 Boundary of Terdragon log 4 1.2619 curve 3 log 4 1.2619 2D Cantor dust 3 (Adapted from “List of fractals by Hausdorff dimension”, Wikipedia) log 4 1.2619 2D L-system branch 3 log 5 1.4649 Vicsek fractal 3 Quadratic von Koch curve log 5 1.4649 (type 1) 3 log 1.4961 Quadric cross 10 √5 � 3 � Quadratic von Koch curve log 8 = 1.5000 3 (type 2) 4 2 log 3 1.5849 3-branches tree 2 (Adapted from “List of fractals by Hausdorff dimension”, Wikipedia) log 3 1.5849 Sierpinski triangle 2 log 3 1.5849 Sierpiński arrowhead curve 2 Boundary of the T-square log 3 1.5849 fractal 2 log = 1.61803 A golden dragon � 1 + log 2 1.6309 Pascal triangle modulo 3 3 1 + log 2 1.6309 Sierpinski Hexagon 3 3 log 1.6379 Fibonacci word fractal 1+√2 Solution of Attractor of IFS with 3 + + 1.6402 similarities of ratios 1/3, 1/2 1 1 and 2/3 3 = 12 � � � � 2 �3� (Adapted from “List of fractals by Hausdorff dimension”, Wikipedia) 32-segment quadric fractal log 32 = 1.6667 5 (1/8 scaling rule) 8 3 1 + log 3 1.6826 Pascal triangle modulo 5 5 50 segment quadric fractal 1 + log 5 1.6990 (1/10 scaling rule) 10 4 log 2 1.7227 Pinwheel fractal 5 log 7 1.7712 Hexaflake 3 ( ) 1.7848 Von Koch curve 85° log 4 log 2+cos 85° log 6 1.8617 Pentaflake 1+ (Adapted from “List of fractals by Hausdorff dimension”, Wikipedia) Solution of 6 + 1 1.8687 Monkeys tree 5 �3� = 1 1 �3√3� log 8 1.8928 Sierpinski carpet 3 log 8 1.8928 3D Cantor dust 3 Cartesian product of the von log 4 + log 2 1.8928 Koch curve and the Cantor = log 8 3 3 set 3 2 2 Sierpiński curve 2 2 Hilbert curve 2 2 Peano curve 2 2 Moore curve (Adapted from “List of fractals by Hausdorff dimension”, Wikipedia) Lebesgue curve or z-order 2 2 curve log 2 = 2 2 Dragon curve √2 2 2 Terdragon curve log 4 = 2 2 Gosper curve 2 Solution of Curve filling the Koch 7 + 2 1 snowflake 6 �3� = 1 1 �3√3� log 4 = 2 2 H-fractal 2 log 2 = 2 2 Pythagoras tree (fractal) √2 log 4 = 2 2 2D Greek cross fractal 2 .

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