Terrapin Resources

# 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