Turtles and Fractals

Ron GoldmanDepartment of Computer ScienceRice University

Fractals

Sierpinski Triangle Fractal Swiss Flag

Koch Snowflake C-Curve

Basic Turtle Programs

POLY (Length, Angle)

Repeat Forever

FORWARD (Length)

TURN (Angle)

SPIRAL (Length, Angle, Scalefactor)

Repeat Forever

FORWARD (Length)

TURN (Angle)

RESIZE (Scalefactor)

Propositions

POLY Proposition

The vertices generated by the POLY program lie on a circle.

SPIRAL Proposition

The vertices generated by the SPIRAL program lie on a spiral.

Examples of POLY Procedures

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Angle = π / 5

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Angle = 3π / 7

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Angle = π 2 / 2

Proof of POLY Proposition

•R

RR

L

LL

x = R

α

αα

α βA

A

C

D

E

F

•

•

••

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G

ΔCDE ≅ ΔCEF (SSS)A + β +α = A +α +α ⇒ β =α

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ΔCEF ≅ ΔCFG (SAS)∴ x = R

Looping Lemmas

Circle Looping Lemma

Any procedure that is a repetition of the same collection of FORWARD and TURN

commands has the structure of POLY(Length, Angle), where

Angle = Total Turning within the Loop

Length = Total Distance traveled within the Loop

That is, the two programs have the same boundedness, closing, and symmetry.

In particular, if Angle ≠

€

2πk for some integer k, then all the vertices generated

by the same FORWARD command inside the loop lie on a common circle.

Looping Lemmas (continued)

Spiral Looping Lemma

Any procedure that is a repetition of the same collection of FORWARD, TURN,

and RESIZE commands has the structure of SPIRAL(Length, Angle, Scalefactor),

where

Angle = Total Turning within the Loop

Length = Total Distance traveled within the Loop

Scalefactor = Total Scaling within the Loop

That is, the two programs have the same boundedness and symmetry.

In particular, if Angle ≠

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2πk for some integer k, and Scalefactor ≠

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±1, then all

the vertices generated by the same FORWARD command inside the loop lie on a

common spiral.

Looping Program and Polygon Procedure

LoopingPr ogram

•

PolygonProcedure •

• LoopingPr ogram

Sample Looping Program

WALK

Repeat Forever

FORWARD 1

TURN

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π / 3

FORWARD

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1/ 4

TURN

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π / 4

FORWARD

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1/ 3

TURN

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−π / 2

FORWARD

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3/5

TURN

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π /6

Looping Curve

More Sample Looping Program

PolySpiral

Repeat 20

Spin[

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2π /20

Scale[1.01

Poly[5,

€

2π /5 ]

]]]

StarSpiral

Repeat 20

Spin[

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2π /20

Scale[1.1

Poly[5,

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4π /5 ]

]]]

Looping Spirals

PolySpiral StarSpiral

Recursion

Problem: How can we generate more interesting shapes from turtle programs?

Solution: Recursion

Fractals: Recursion made visible.

Motivation for Fractals

Representing Natural Objects

• Snowflakes, Mountains, Trees K

Drawing Interesting Pictures

Exploring Novel Mathematics

Performing Data Compression

Fractal Curves and Turtle Programs

Fractal Curves• Fractal curves usually cannot be represented by simple analytic expressions.• Fractal curves are generally defined by

-- Recursive Procedures-- Iterated Transformations-- Infinite Series

Turtle Programs• Recursion is used to express self-similarity.• Coordinates are irrelevant to shape.• Turtle programs are easily translated into fractal programs.

Examples• Fractal Gaskets• Bump Fractals• Other Self-Similar Fractals

Sierpinski Gasket

Sierpinski Gasket

Main Idea• Make a smaller Sierpinski gasket at one of the corners. • Then move to another corner and make another small Sierpinski gasket. • Then move to another corner and make another small Sierpinski gasket. • Then return the turtle to its initial position and heading.

Recursive Turtle ProgramSierpinski(Level)

If Level = 0, Draw a TriangleElse

Repeat 3 TimesRESIZE 1/2Sierpinski(Level–1)RESIZE 2MOVE 1TURN

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2π /3

More Fractal Gaskets

Pentagonal Gasket Fractal Swiss Flag Rectangular Gasket

Koch Curve

Koch Bump

Koch Curve

Koch Curve

Basic Idea

• Make a smaller Koch curve along the first edge of the bump.

• Then move to the next edge and make another small Koch curve.

• Then move to the next edge and make another small Koch curve.

• Then move to the next edge and make another small Koch curve.

Simpler Approach

• Replace each straight line on the bump by a Koch curve.

• Replace each FORWARD command by a recursive call.

Koch Curve

Koch Bump Koch(Level)

If Level = 0, FORWARD 1

Else

RESIZE 1/3 → RESIZE 1/3FORWARD 1 → Koch(Level-1)TURN 60 → TURN 60FORWARD 1 → Koch(Level-1)TURN −120 → TURN −120FORWARD 1 → Koch(Level-1)TURN 60 → TURN 60FORWARD 1 → Koch(Level-1)RESIZE 3 → RESIZE 3

Universal Bump–Fractal Program

Bump Bump_Fractal(Level)

If Level = 0, FORWARD 1

Else

TURN A1 TURN A1RESIZE

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S1 → RESIZE

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S1FORWARD 1 → Bump_Fractal

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(Level −1)TURN

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A2 → TURN

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A2RESIZE

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S2 → RESIZE

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S2FORWARD 1 → Bump_Fractal

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(Level −1)

M MFORWARD 1 → Bump_Fractal

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(Level −1)RESIZE

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Sn+1 → RESIZE

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Sn+1TURN

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An+1 → TURN

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An+1

Typical Bump Assumptions

1. Turtle Final Orientation = Turtle Initial Orientation• Total Turning = 0

•

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Akk=1

n+1∑ = 0

2. Turtle Returns to Initial Line• Total Y–Distance = 0

•

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SkSin Ajj=1

k∑

k=1

n∑ = 0

3. Turtle Travels Distance=Size Along Initial Line• Total X–Distance = Size

•

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SkCos Ajj=1

k∑

k=1

n∑ =1

C–Curve

C-Bump

C-Curve

C–Curve

Bump C(Level)

If Level = 0, FORWARD 1

Else

TURN 45 → TURN 45

RESIZE

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22

→ RESIZE

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22

FORWARD 1 →

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C Level −1( )TURN −90 → TURN −90FORWARD 1 →

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C Level −1( )TURN 45 → TURN 45RESIZE

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2 → RESIZE

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2

More Bumps on Bumps

Additional Examples

• Rectangular Bumps

• Pentagonal Bumps

• Space Filling Curves

Additional Extensions

• Scales and Angles can be selected randomly.

• Each FORWARD command can be replaced by a different FRACTAL program.

• Fractal programs can be connected with TURN and FORWARD transitions.

More Self-Similar Fractals

Fractal Hangman Fractal Bush Fractal Tree

More Self-Similar Fractals

Fractal Star Fractal Flower Fractal Leaf