The beauty of fractals Gwyneth Stallard The Open University LMS Summer School Leeds July 2019 Teaching Research Why fractals? Teaching Why fractals? Research Why fractals? Teaching Research The von Koch curve The von Koch curve The von Koch curve The von Koch curve The von Koch curve The von Koch curve The von Koch curve The von Koch curve The von Koch curve The von Koch curve The von Koch curve Zooming in Many natural objects are fractals A fractal is a set which has an intricate structure has a simple definition exhibits self-similarity (looks the same when you zoom in) whose geometry cannot easily be described in classical terms What is a fractal? Many natural objects are fractals has a simple definition exhibits self-similarity (looks the same when you zoom in) whose geometry cannot easily be described in classical terms What is a fractal? A fractal is a set which has an intricate structure Many natural objects are fractals exhibits self-similarity (looks the same when you zoom in) whose geometry cannot easily be described in classical terms What is a fractal? A fractal is a set which has an intricate structure has a simple definition Many natural objects are fractals whose geometry cannot easily be described in classical terms What is a fractal? A fractal is a set which has an intricate structure has a simple definition exhibits self-similarity (looks the same when you zoom in) Many natural objects are fractals What is a fractal? A fractal is a set which has an intricate structure has a simple definition exhibits self-similarity (looks the same when you zoom in) whose geometry cannot easily be described in classical terms What is a fractal? Many natural objects are fractals A fractal is a set which has an intricate structure has a simple definition exhibits self-similarity (looks the same when you zoom in) whose geometry cannot easily be described in classical terms fractal cauliflower The middle third Cantor set The middle third Cantor set The middle third Cantor set The middle third Cantor set The middle third Cantor set The middle third Cantor set A fractal is a set which has an intricate structure has a simple definition exhibits self-similarity whose geometry cannot easily be described in classical terms Length = 1 4=3 (4=3)2 (4=3)3 : : At k-th step: (4=3)k ! 1 How long is the von Koch curve? 4=3 (4=3)2 (4=3)3 : : At k-th step: (4=3)k ! 1 How long is the von Koch curve? Length = 1 (4=3)2 (4=3)3 : : At k-th step: (4=3)k ! 1 How long is the von Koch curve? Length = 1 4=3 (4=3)3 : : At k-th step: (4=3)k ! 1 How long is the von Koch curve? Length = 1 4=3 (4=3)2 : : At k-th step: (4=3)k ! 1 How long is the von Koch curve? Length = 1 4=3 (4=3)2 (4=3)3 ! 1 How long is the von Koch curve? Length = 1 4=3 (4=3)2 (4=3)3 : : At k-th step: (4=3)k How long is the von Koch curve? Length = 1 4=3 (4=3)2 (4=3)3 : : At k-th step: (4=3)k ! 1 dimension = a, 1 < a < 2 dimension = 1 dimension = b, 0 < b < 1 dimension = 0 Dimensions of fractals dimension = 2 dimension = a, 1 < a < 2 dimension = b, 0 < b < 1 dimension = 0 Dimensions of fractals dimension = 2 dimension = 1 dimension = a, 1 < a < 2 dimension = b, 0 < b < 1 Dimensions of fractals dimension = 2 dimension = 1 dimension = 0 dimension = b, 0 < b < 1 Dimensions of fractals dimension = 2 dimension = a, 1 < a < 2 dimension = 1 dimension = 0 Dimensions of fractals dimension = 2 dimension = a, 1 < a < 2 dimension = 1 dimension = b, 0 < b < 1 dimension = 0 How many boxes of side d do we need to cover a set? Box dimension Box dimension How many boxes of side d do we need to cover a set? N = 3 = 31 N = 9 = 32 N = 2 = 3log 2= log 3 = 1 = 1 If d 32 9 then N = 9 = (32)1 N = 92 = (32)2 N = 22 = (32)log 2= log 3 dimension = 1 dimension = 2 dimension = log 2/log 3 Box dimension 1 If d = 3 then N = 9 = 32 N = 2 = 3log 2= log 3 = 1 = 1 If d 32 9 then N = 9 = (32)1 N = 92 = (32)2 N = 22 = (32)log 2= log 3 dimension = 1 dimension = 2 dimension = log 2/log 3 Box dimension 1 If d = 3 then N = 3 = 31 N = 2 = 3log 2= log 3 = 1 = 1 If d 32 9 then N = 9 = (32)1 N = 92 = (32)2 N = 22 = (32)log 2= log 3 dimension = 1 dimension = 2 dimension = log 2/log 3 Box dimension 1 If d = 3 then N = 3 = 31 N = 9 = 32 = 1 = 1 If d 32 9 then N = 9 = (32)1 N = 92 = (32)2 N = 22 = (32)log 2= log 3 dimension = 1 dimension = 2 dimension = log 2/log 3 Box dimension 1 If d = 3 then N = 3 = 31 N = 9 = 32 N = 2 = 3log 2= log 3 N = 9 = (32)1 N = 92 = (32)2 N = 22 = (32)log 2= log 3 dimension = 1 dimension = 2 dimension = log 2/log 3 Box dimension 1 If d = 3 then N = 3 = 31 N = 9 = 32 N = 2 = 3log 2= log 3 = 1 = 1 If d 32 9 then N = 92 = (32)2 N = 22 = (32)log 2= log 3 dimension = 1 dimension = 2 dimension = log 2/log 3 Box dimension 1 If d = 3 then N = 3 = 31 N = 9 = 32 N = 2 = 3log 2= log 3 = 1 = 1 If d 32 9 then N = 9 = (32)1 N = 22 = (32)log 2= log 3 dimension = 1 dimension = 2 dimension = log 2/log 3 Box dimension 1 If d = 3 then N = 3 = 31 N = 9 = 32 N = 2 = 3log 2= log 3 = 1 = 1 If d 32 9 then N = 9 = (32)1 N = 92 = (32)2 dimension = 1 dimension = 2 dimension = log 2/log 3 Box dimension 1 If d = 3 then N = 3 = 31 N = 9 = 32 N = 2 = 3log 2= log 3 = 1 = 1 If d 32 9 then N = 9 = (32)1 N = 92 = (32)2 N = 22 = (32)log 2= log 3 dimension = 2 dimension = log 2/log 3 Box dimension 1 If d = 3 then N = 3 = 31 N = 9 = 32 N = 2 = 3log 2= log 3 = 1 = 1 If d 32 9 then N = 9 = (32)1 N = 92 = (32)2 N = 22 = (32)log 2= log 3 dimension = 1 dimension = log 2/log 3 Box dimension 1 If d = 3 then N = 3 = 31 N = 9 = 32 N = 2 = 3log 2= log 3 = 1 = 1 If d 32 9 then N = 9 = (32)1 N = 92 = (32)2 N = 22 = (32)log 2= log 3 dimension = 1 dimension = 2 Box dimension 1 If d = 3 then N = 3 = 31 N = 9 = 32 N = 2 = 3log 2= log 3 = 1 = 1 If d 32 9 then N = 9 = (32)1 N = 92 = (32)2 N = 22 = (32)log 2= log 3 dimension = 1 dimension = 2 dimension = log 2/log 3 Definition The escaping set is I(f ) = z : f n(z) as n : f ! 1 ! 1g Definition The Julia set J(f ) is the boundary of the escaping set. The iterates of f behave chaotically on the Julia set and most are fractals. Complex dynamics f : C C is analytic ! f n is the nth iterate of f Definition The Julia set J(f ) is the boundary of the escaping set. The iterates of f behave chaotically on the Julia set and most are fractals. Complex dynamics f : C C is analytic ! f n is the nth iterate of f Definition The escaping set is I(f ) = z : f n(z) as n : f ! 1 ! 1g The iterates of f behave chaotically on the Julia set and most are fractals. Complex dynamics f : C C is analytic ! f n is the nth iterate of f Definition The escaping set is I(f ) = z : f n(z) as n : f ! 1 ! 1g Definition The Julia set J(f ) is the boundary of the escaping set. Complex dynamics f : C C is analytic ! f n is the nth iterate of f Definition The escaping set is I(f ) = z : f n(z) as n : f ! 1 ! 1g Definition The Julia set J(f ) is the boundary of the escaping set. The iterates of f behave chaotically on the Julia set and most are fractals. I(f ) = z : z > 1 f j j g J(f ) = z : z = 1 f j j g Quadratic examples f (z) = z2 J(f ) = z : z = 1 f j j g Quadratic examples f (z) = z2 I(f ) = z : z > 1 f j j g Quadratic examples f (z) = z2 I(f ) = z : z > 1 f j j g J(f ) = z : z = 1 f j j g I(f ) is the pink and purple region J(f ) is the boundary of the black region and is a curve of infinite length dim J(f ) > 1 Quadratic examples f (z) = z2 + 0:1 J(f ) is the boundary of the black region and is a curve of infinite length dim J(f ) > 1 Quadratic examples f (z) = z2 + 0:1 I(f ) is the pink and purple region dim J(f ) > 1 Quadratic examples f (z) = z2 + 0:1 I(f ) is the pink and purple region J(f ) is the boundary of the black region and is a curve of infinite length Quadratic examples f (z) = z2 + 0:1 I(f ) is the pink and purple region J(f ) is the boundary of the black region and is a curve of infinite length dim J(f ) > 1 I(f ) is the pink and purple region J(f ) is the boundary of the black region known as the cauliflower.