!IFS for Spleewort Fern

!x' = a11*x + a12*y + b1

!y' = a21*x + a22*y + b2

!

!Four affine transformations:

!1) Scaling of 0.16 in y direction only (stem)

!2) Scaling of 0.85 in x and y + Translation of 1.6 in y ! + Rotation of -2.5 deg.

!3) Scaling of .3 in x & and .34 in y + Translation of 1.6 in y ! + Rotation of 49 deg.

!4) Scaling of .3 in x & and .37 in y + Translation of .44 in y ! + Rotation of -50 deg. + Reflexion about y.

!This affine map must be a CONTRACTION so the following conditions must verify:

!a11^2 + a21^2 >1

!a21^2 + a22^2 >1

!a11^2 + a21^2 + a21^2 + a22^2 > (a11*a22-a21*a12)^2

!

SET MODE "graphics"

SET WINDOW -6*1.33,6*1.33,-1,11

OPTION NOLET

LET x = 1

LET y = 1

FOR i = 1 to 10000

LET q = 100*rnd

IF q<3 THEN

LET x=0

LET y=0.16*y

PLOT x,y

ELSE IF q<76 and q>=3 THEN

LET x = .85*x + .04*y

LET y = -.04*x + .85*y + 1.6

PLOT x,y

ELSE IF q>76 and q<89 THEN

LET x = .2*x - .26*y

LET y = -.23*x +.22*y + 1.6

PLOT x,y

ELSE IF q>=89 THEN

LET x = -.15*x + .28*y

LET y = .26*x +.24*y + .44

PLOT x,y

END IF

NEXT i

END