Fractal Examples Handout
(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−𝑠𝑠 𝑠𝑠 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 𝜑𝜑 �𝜑𝜑 𝜑𝜑 𝜑𝜑
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 𝜑𝜑
Solution of Attractor of IFS with 3 + + 𝑠𝑠 𝑠𝑠 1.6402 similarities of ratios 1/3, 1/2 1 1 and 2/3 3 = 12 � �𝑠𝑠 � �
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+𝜑𝜑
(Adapted from “List of fractals by Hausdorff dimension”, Wikipedia)
Solution of 6 + 𝑠𝑠 1 1.8687 Monkeys tree 5 �3� = 1 1 𝑠𝑠
�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 𝑠𝑠 snowflake 6 �3� = 1 1 𝑠𝑠
�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