A dynamic Complex Transformation generating FRACTALS 北京景山学校 纪光老师 April Fractals & Complex Numbers

Generation of Julia’s “rabbit” 北京景山学校 纪光老师 April 2010 Fractals & Complex Numbers2

Generation of the set of Mendelbrot 北京景山学校 纪光老师 April 2010 Fractals & Complex Numbers3

Review 1 : Complex Numbers set 北京景山学校 纪光老师 April Fractals & Complex Numbers The complex number z = a + i b is represented in the coordinates plane by a point M(a,b) or vector (a,b) In polar coordinates z = r (cos j + i sin j) or r. e i j r is the module of z : r = |z| = j is the argument : arg(z) = j

Review 1.a : Complex Numbers set 北京景山学校 纪光老师 April Fractals & Complex Numbers The omplex number z = a + i b is represented in the coordinates plane by the point M(a,b) where a and b are eal numbers and i an imaginary square root of (-1)

Review 1.b : Complex Numbers set 北京景山学校 纪光老师 April Fractals & Complex Numbers In polar coordinates z = r (cos j + i sin j) or z = r. e i j r is the module of z : r = |z| = j is the argument : arg(z) = j

Review 2 Operations in 北京景山学校 纪光老师 April Fractals & Complex Numbers (1) Addition : if z = a + i b and z’ = a’ + i b’ then z + z’ = (a + a’) + i ( b + b’ ) (2) Multiplication : if z = r. e i j and z’ = r’. e i j ’ then z.z’ = r.r’.e i (j+j ’ )

Review 2.a Operations in 北京景山学校 纪光老师 April Fractals & Complex Numbers Construction of the Sum z = a + i b z’ = a’ + i b’ ================= z + z’ = (a + a’) + i ( b + b’) The image of the sum is the sum of the vectors associated with the vectors representing z and z’

Review 2.b Operations in 北京景山学校 纪光老师 April Fractals & Complex Numbers Construction of the product z = r. e i j z’ = r’. e i j ’ ================= z.z’ = r. r’. e i (j + j’) The module of the product is the product of the modules The argument of the product is the Sum of the arguments

Transformation in 北京景山学校 纪光老师 April Fractals & Complex Numbers Construction of the square z = r. e i j z 2 = r 2. e i 2 j The module of the square is the square of the module. The argument of the square is the double of the argument.

Transformation (1.1) in 北京景山学校 纪光老师 April Fractals & Complex Numbers Construction of z 2 z = r. e i j z 2 = r 2. e i 2 j 1 st method : 1. Square the module OM in OM 1 2. Rotate the point M 1 in M’

Transformation (1.2) in 北京景山学校 纪光老师 April Fractals & Complex Numbers Construction of z 2 z = r. e i j z 2 = r 2. e i 2 j 2 nd method : 1. Rotate the point M in M 2 2. Square the module of OM 2 in OM’

Transformation (1.3) in 北京景山学校 纪光老师 April Fractals & Complex Numbers (Demo / Cabri / Fig.2)

Transformation (2.1) in 北京景山学校 纪光老师 April Fractals & Complex Numbers Construction of z 2 + c z = r. e i j z 2 + c = r 2. e i 2 j + c c is a complex constant represented by the point C 1 st Method : 1. Square the module of OM in OM 1 2. Rotate the point M 1 (z 1 ) in M’ 3. Add the vector

Transformation (2.2) in 北京景山学校 纪光老师 April Fractals & Complex Numbers Construction of z 2 + c z = r. e i j z 2 + c = r 2. e i 2 j + c c is a complex constant represented by the point C 2 nd Method : 1. Rotate the point M(z) in M 1 2. Square the module of OM 1 in OM’ 3. Add the vector

Transformation (2.3) in 北京景山学校 纪光老师 April Fractals & Complex Numbers (Demo / Cabri / Fig.3)

Construction of “Julia’s rabbit” in by iterating the transformation 北京景山学校 纪光老师 April Fractals & Complex Numbers 1.Choose a point C of affix c in the Complex plane. 2.Choose a point M 0 (z 0 ) in the Complex plane. 3.Build the image M 1 (z 1 ) of M 0 (z 0 ) by the above transformation in the coordinates plane. 4.Build the image M 2 (z 2 ) of M 1 (z 1 ) by the above transformation in the coordinates plane.

Construction of “Julia’s rabbit” in by iterating the transformation 北京景山学校 纪光老师 April Fractals & Complex Numbers 5.Continue to apply the transformation to each new point and mark them in the plane, until you get a sequence of 10 points or more … 6.If the points get off the screen, we mark them in blue. This set of points is called the orbit ( 轨道 ) of M 0 (z 0 ) 6.if they stay inside the Unit circle we mark them in red M 0 (z 0 ), M 1 (z 1 ), M 2 (z 2 ), M 3 (z 3 ),…, M 10 (z 10 ),…,…

北京景山学校 纪光老师 April Fractals & Complex Numbers

Construction of Mendelbrot in by iterating the transformation 北京景山学校 纪光老师 April Fractals & Complex Numbers 1.Choose a point C of affix c in the Complex plane. 2.Start from M 0 (z 0 ) = O in the Complex plane. 3.Build the image M 1 (z 1 = c) of M 0 (z 0 ) by the above transformation in the coordinates plane. 4.Build the image M 2 (z 2 = c 2 + c) of M 1 (z 1 = c) by the transformation in the coordinates plane.

Construction of Mendelbrot in by iterating the transformation 北京景山学校 纪光老师 April Fractals & Complex Numbers 5.Continue to apply the transformation to each new point and mark them in the plane, until you get a sequence of 10 points or more … 6.If the points get off the screen, we mark C in red. This set of points is called the orbit ( 轨道 ) of C 6.if they stay inside the Unit circle we mark C in black. O, M 1 (z 1 = c), M 2 (z 2 = c 2 + c), M 3 (z 3 ),…, M 10 (z 10 ),…,…

北京景山学校 纪光老师 April Fractals & Complex Numbers