1. A dynamic Complex Transformation generating FRACTALS 北京景山学校 纪光老师 April 2010 1Fractals & Complex Numbers

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

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

4. Review 1 : Complex Numbers set 北京景山学校 纪光老师 April 2010 4Fractals & 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

5. Review 1.a : Complex Numbers set 北京景山学校 纪光老师 April 2010 5Fractals & 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)

6. Review 1.b : Complex Numbers set 北京景山学校 纪光老师 April 2010 6Fractals & 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

7. Review 2 Operations in 北京景山学校 纪光老师 April 2010 7Fractals & 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 ’ )

8. Review 2.a Operations in 北京景山学校 纪光老师 April 2010 8Fractals & 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’

9. Review 2.b Operations in 北京景山学校 纪光老师 April 2010 9Fractals & 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

10. Transformation in 北京景山学校 纪光老师 April 2010 10Fractals & 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.

11. Transformation (1.1) in 北京景山学校 纪光老师 April 2010 11Fractals & 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’

12. Transformation (1.2) in 北京景山学校 纪光老师 April 2010 12Fractals & 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’

13. Transformation (1.3) in 北京景山学校 纪光老师 April 2010 13Fractals & Complex Numbers (Demo / Cabri / Fig.2)

14. Transformation (2.1) in 北京景山学校 纪光老师 April 2010 14Fractals & 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

15. Transformation (2.2) in 北京景山学校 纪光老师 April 2010 15Fractals & 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

16. Transformation (2.3) in 北京景山学校 纪光老师 April 2010 16Fractals & Complex Numbers (Demo / Cabri / Fig.3)

17. Construction of “Julia’s rabbit” in by iterating the transformation 北京景山学校 纪光老师 April 2010 17Fractals & 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.

18. Construction of “Julia’s rabbit” in by iterating the transformation 北京景山学校 纪光老师 April 2010 18Fractals & 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 ),…,…

19. 北京景山学校 纪光老师 April 2010 19Fractals & Complex Numbers

20. Construction of Mendelbrot in by iterating the transformation 北京景山学校 纪光老师 April 2010 20Fractals & 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.

21. Construction of Mendelbrot in by iterating the transformation 北京景山学校 纪光老师 April 2010 21Fractals & 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 ),…,…

22. 北京景山学校 纪光老师 April 2010 22Fractals & Complex Numbers