Fractals

Students at Central Connecticut State University taking Math 409 under the direction of jonesv@ccsu.edu created fractal designs using Logo. You can sample projects by Colleen, Dana, Daphne, Donna, Elisa, Jennifer, Jessica, Keegan, or Melissa by downloading the Logo files and loading them into Terrapin Logo (after unzipping the archive). Each file will automatically start up when you load it into Logo and will draw the fractal design created by the student.

This project illustrates fractal patterns. Click to download the self-running Logo package.

fractals.lgo

TO MINKOWSKI :LENGTH :LEVEL
    IF (:LEVEL = 0) [FD :LENGTH STOP]
    MINKOWSKI :LENGTH * (1 / 4) (:LEVEL - 1) ; draws first segment
    LT 90
    MINKOWSKI :LENGTH * (1 / 4) (:LEVEL - 1) ; draws second segment
    RT 90
    MINKOWSKI :LENGTH * (1 / 4) (:LEVEL - 1) ; draws third segment
    RT 90
    MINKOWSKI :LENGTH * (1 / 4) (:LEVEL - 1) ; draws fourth segment
    MINKOWSKI :LENGTH * (1 / 4) (:LEVEL - 1) ; draws fifth segment
    LT 90
    MINKOWSKI :LENGTH * (1 / 4) (:LEVEL - 1) ; draws sixth segment
    LT 90
    MINKOWSKI :LENGTH * (1 / 4) (:LEVEL - 1) ; draws seventh segment
    RT 90
    MINKOWSKI :LENGTH * (1 / 4) (:LEVEL - 1) ; draws eighth segment
END

TO C :LENGTH :LEVEL
    ; from Advanced Logo by Michael Friendly, page 248
    IF :LEVEL = 0 [FD :LENGTH STOP]
    LT 45
    C :LENGTH * (1 / (SQRT 2)) :LEVEL - 1
    RT 90
    C :LENGTH * (1 / (SQRT 2)) :LEVEL - 1
    LT 45
END

TO TREEP :SIDE :LEVEL
    ; this creates a Pythagorean Tree with the turtle starting
    ; and ending at the lower left corner
    IF :LEVEL < 0 [STOP]
    FD :SIDE LT 45
    TREEP :SIDE * (1 / SQRT 2) (:LEVEL - 1)
    RT 135 FD :SIDE
    PU BK :SIDE LT 45 FD :SIDE * (1 / SQRT 2) PD
    TREEP :SIDE * (1 / SQRT 2) (:LEVEL - 1)
    PU RT 90 FD :SIDE * (1 / SQRT 2) RT 45 PD
    REPEAT 2 [FD :SIDE RT 90]
END

TO HAT :LENGTH :LEVEL
    IF (:LEVEL = 0) [FD :LENGTH STOP]
    HAT :LENGTH * (1 / 3) (:LEVEL - 1)
    LT 90
    HAT :LENGTH * (1 / 3) (:LEVEL - 1)
    RT 90
    HAT :LENGTH * (1 / 3) (:LEVEL - 1)
    RT 90
    HAT :LENGTH * (1 / 3) (:LEVEL - 1)
    LT 90
    HAT :LENGTH * (1 / 3) (:LEVEL - 1)
END

TO DRAGON :LENGTH :LEVEL :ANGLE
    ;this procedure creates the standard dragon if you use
    ;90 for the angle when you call it
    ;a modification of Michael Friendly's procedure from
        ;Advanced Logo, page 632
    IF :LEVEL = 0 [FD :LENGTH STOP]
    DRAGON :LENGTH * (1 / (SQRT 2)) (:LEVEL - 1) (ABS :ANGLE)
    LT :ANGLE
    DRAGON :LENGTH * (1 / (SQRT 2)) (:LEVEL - 1) (-1 * (ABS :ANGLE))
END

TO KEEGAN
    ; created by Keegan Phelps-Franco for
    ; Math 409 Summer 2003
    CS FS HT SETBG [247 113 239] SETWIDTH 8
    PU FD 100 PD HT
    REPEAT 3 [LT 135 SETPC 12 CANTOR 99 0 PU RT 45 FD 50 SETPC 11 CANTOR 99 1 PU LT 135 FD 50 SETPC 10 CANTOR 99 2 PU RT 45 FD 50 SETPC 9 CANTOR 99 3 PU LT 135 FD 50 SETPC 8 CANTOR 99 4 PU RT 45 FD 50 SETPC 8 CANTOR 99 4 PU LT 135 FD 50 SETPC 9 CANTOR 99 3 PU RT 45 FD 50 SETPC 10 CANTOR 99 2 PU LT 135 FD 50 SETPC 11 CANTOR 99 1 PU RT 45 FD 50 SETPC 12 CANTOR 99 0]
END

TO JESSICA
    ;created by Jessica Lessard for
    ;Math 409 Summer 2003
    CS FS SETW 3 SETPC 90 HT
    PU RT 90 BK 100 LT 90 PD
    REPEAT 18 [MINK1]
END

TO DANA
    ; created by Dana Gagne for
    ;Math 409 Summer 2003
    CS FS SETW 1 SETPC 4
    PU FD 50 RT 90 BK 150 PD
    HAT 300 2 RT 90
    REPEAT 3 [HAT 300 0 RT 90]
    PU RT 90 FD 70 LT 90 FD 200 RT 180 PD
    HAT 100 1
    PU FD 100 LT 90 FD 130 LT 90 FD 50 PD
    SETPC 12 SETW 3
    HAT 200 1 RT 180
    HAT 200 1
    PU RT 180 FD 200 * 2 / 3 LT 90 FD 200 * 1 / 3 LT 90 PD
    SETPC 14
    REPEAT 2 [HAT 200 * 1 / 6 3] LT 90
    PU FD 200 * 2 / 3 LT 90 PD
    REPEAT 2 [HAT 200 * 1 / 6 3]
    PU FD 200 * 1 / 3 + 50 LT 90 FD 335 LT 90 FD 300 RT 180 PD
    SETPC 0 SETW 5 HAT 300 1
    PU SETPOS [67 67] LT 90 BK 200 * 1 / 16 PD
    SETW 1 SETPC 1 HAT 200 * 1 / 8 3 RT 180
    HAT 200 * 1 / 8 3
    PU SETPOS [-67 67] BK 200 * 1 / 16 PD
    HAT 200 * 1 / 8 3 RT 180
    HAT 200 * 1 / 8 3 HT
END

TO ELISA
    ;created by Elisa Samolyk for
    ; Math 409 Summer 2003
    CS FS HT
    SETPC [255 0 0]
    C 50 8
    LT 90
    PU SETPOS [-91 51] PD
    LT 90
    C 50 8
    PU
    SETPOS [-292 71]
    RT 115
    PD
    SETPC [255 255 8]
    C 40 8
    LT 90
    PU
    SETPOS [-365 24]
    LT 90
    PD
    C 40 8
    PU
    SETPC [0 255 0]
    SETPOS [230 -85]
    LT 65
    PD
    C 45 8
    PU
    RT 180
    SETPOS [209 5]
    PD
    C 45 8
    PU SETPOS [-251 -64]
    SETPC [0 0 255]
    PD C 35 8
    PU SETPOS [-238 -133]
    RT 180
    PD
    C 35 8
    PU
    SETPOS [128 84]
    SETPC [255 32 247]
    PD
    C 25 8
    PU
    SETPOS [118 135]
    RT 180
    PD
    C 25 8
    PU SETPOS [-89 -93] PD
    SETPC [0 0 0]
    TT [BUTTERFLIES IN FLIGHT]
END

TO DONNA
    ;created by Donna Lybarger for
    ; Math 409 Summer 2003
    CS FS SETW 1 HT
    REPEAT 6 [SETPC RANDOM 255 SIERPINSKI 200 4 RT 60]

END

TO JEN_TREE
    SETPC [231 0 107]
    TREEP 90 2 RT 180
    TREEP 90 2 LT 90
    TREEP 90 2 RT 180
    TREEP 90 2
    SETPC [231 0 90]
    REPEAT 4 [LT 90 TREEP 45 2]
    SETPC [173 56 255]
    REPEAT 4 [LT 90 TREEP 23 2]
    SETPC [173 0 81]
    REPEAT 4 [LT 90 TREEP 15 2]
    SETPC [8 182 132]
    REPEAT 4 [LT 90 TREEP 8 2]
END

TO JENNIFER
    ;created by Jennifer Schaefer for
    ;Math 409 Summer 2003
    CS FS HT SETBG 0 SETWIDTH 3
    PU SETPOS [215 0] RT 90 PD
    JEN_TREE
    PU SETPOS [-215 0] RT 90 PD
    JEN_TREE
END

TO MELISSA
    ;created by Melissa Aston for
    ; Math 409 Summer 2003
    CS FS SETW 3 SETBG [0 0 0]
    SETPC [45 45 200] HFRACTAL 100 3
    PU RT 90 FD 128 PD
    SETPC [200 45 45] HFRACTAL 100 4
    PU RT 180 FD 128 FD 128 PD
    SETPC [45 200 45] HFRACTAL 100 4
    PU RT 180 FD 128 LT 90 FD 100 PD
    SETPC [175 150 0] HFRACTAL 100 5
    PU BK 200 PD
    SETPC [120 45 120] HFRACTAL 100 5
    PU FD 200 LT 90 FD 128 RT 45 FD 64 PD
    SETW 4
    SETPC [30 75 120] HFRACTAL 150 4
    PU BK 64 LT 45 BK 256 RT 135 FD 64 PD
    SETPC [200 45 120] HFRACTAL 150 4
    PU BK 64 RT 45 BK 128 LT 90
    BK 200 RT 90 FD 128 RT 45 FD 64 PD
    SETPC [175 200 50] HFRACTAL 150 4
    PU BK 64 LT 45 BK 256 RT 135 FD 64 PD
    SETPC [255 255 255] HFRACTAL 150 4
    HT
END

TO DRAGON1
    REPEAT 4 [DRAGON 100 2 60 RT 150]
END

TO DRAGON2
    REPEAT 4 [DRAGON 50 2 60 RT 150]
    RT 120 FD 75 LT 120
    REPEAT 4 [DRAGON 50 2 60 RT 150]
END

TO MAIN
    PR "|Try one of the following procedures:|
    PR [COLLEEN DANA DAPHNE DONNA ELISA JENNIFER JESSICA KEEGAN MELISSA]
END

TO COLLEEN
    ;created by Colleen Downey for
    ;Math 409 Project 4 Summer 2003
    CS FS SETW 1 HT SETBG [0 0 0]
    REPEAT 6 [SETPC 9 PEANO 225 3 PU BK 50 PD SETPC 14 PEANO 50 4 PU BK 250 RT 30 PD SETPC 12 PEANO 225 3 PU BK 50 PD SETPC 14 PEANO 50 4 PU BK 250 PD RT 30]
END

TO MINK1
    MINKOWSKI 150 2
    RT 90 SETPC 40
    MINKOWSKI 150 2
    RT 90 SETPC 200
    MINKOWSKI 150 2
    RT 90 SETPC 90
    MINKOWSKI 150 2
    RT 90
    PU FD 10 LT 20 PD
END

TO PEANO :LENGTH :LEVEL
    IF (:LEVEL = 0) [FD :LENGTH STOP]
    PEANO :LENGTH * (1 / 3) :LEVEL - 1
    LT 90
    PEANO :LENGTH * (1 / 3) :LEVEL - 1
    RT 90
    PEANO :LENGTH * (1 / 3) :LEVEL - 1
    RT 90
    PEANO :LENGTH * (1 / 3) :LEVEL - 1
    RT 90
    PEANO :LENGTH * (1 / 3) :LEVEL - 1
    LT 90
    PEANO :LENGTH * (1 / 3) :LEVEL - 1
    LT 90
    PEANO :LENGTH * (1 / 3) :LEVEL - 1
    LT 90
    PEANO :LENGTH * (1 / 3) :LEVEL - 1
    RT 90
    PEANO :LENGTH * (1 / 3) :LEVEL - 1
END

TO DAPHNE
    ;created by Daphne Miller for
    ; Math 409 summer 2003
    CS FS HT SETBG [245 215 203] SETPC 6 SETW 2
    PU SETPOS [-200 -75] PD
    REPEAT 8 [FD 150 RT 45]
    REPEAT 4 [FD 150 RT 150 DRAGON1 LT 105 FD 150 RT 45]
    SETPC [198 2 76]
    REPEAT 4 [FD 150 RT 45 FD 75 LT 120 DRAGON2 RT 165]
    PU SETPOS [-253 128] PD
    SETPC 6
    REPEAT 4 [RT 45 FD 150 RT 45 FD 255]
END

TO SIERPINSKI :LENGTH :LEVEL
    ; this creates the "filled" version; to display the outline, make the
    ; last two lines of  this procedure into comments - by placing
    ; a semi-colon at the beginning of the lines
    IF (:LEVEL < 0) [STOP]
    REPEAT 3 [SIERPINSKI :LENGTH * (1 / 2) (:LEVEL - 1) FD :LENGTH RT 120]
    PU RT 30 FD 3 PD FILL
    PU BK 3 LT 30 PD
END

TO CANTOR :LENGTH :LEVEL
    IF (:LEVEL = 0) [PD FD :LENGTH STOP]
    CANTOR :LENGTH * (1 / 3) (:LEVEL - 1)
    PU FD :LENGTH * (1 / 3) PD
    CANTOR :LENGTH * (1 / 3) (:LEVEL - 1)
END

TO HFRACTAL :LENGTH :LEVEL
    IF (:LEVEL < 0) [STOP]
    RT 90 FD :LENGTH * (1 / 2)
    HFRACTAL :LENGTH * (1 / SQRT 2) (:LEVEL - 1)
    PU BK :LENGTH * (1 / 2) LT 180 PD
    FD :LENGTH * (1 / 2)
    HFRACTAL :LENGTH * (1 / SQRT 2) (:LEVEL - 1)
    PU BK :LENGTH * (1 / 2) PD RT 90
END

---

Procedure	COLLEEN, DANA, DAPHNE, DONNA, ELISA, JENNIFER, JESSICA, KEEGAN, MELISSA
Description	Draws various fractal images
Level	Intermediate
Tags	Fractals, Formulae