fractals - hausdorff dimension, the koch curve, and visibility · 2019. 5. 20. · fractals...

FractalsHausdorff Dimension, the Koch Curve,and Visibility

Heidi Lei

MIT PRIMES-USAMentor: Tanya Khovanova

May 18, 2019

Koch Curve

Iteration 0:

Iteration 1:

Iteration 2:

Koch Curve

Iteration 3:

Iteration 4:

Koch Curve

Length of the Koch Curve

Iteration Length0 1

1 43

2 169

......

n(43

)n

Length of the Koch Curve

n(43

)nlimn→∞

length(K) = limn→∞

(4

3

)n

= ∞

The Scaling Property

The Scaling Property

The Scaling Property

2×4× 8×

The Scaling Property

set X scaling factor c # of piecesm dimension dimline 1/2 2 1

square 1/2 4 2cube 1/2 8 3

dimHX = logc−1 m

Dimension of the Koch Curve

13

dimHK = logc−1 m = log3 4

1 < dimHK = 1.262 · · · < 2

Dimension of the Koch Curve

13

dimHK = logc−1 m = log3 4

1 < dimHK = 1.262 · · · < 2

Dimension of the Koch Curve

13

c = 1/3m = 4

dimHK = logc−1 m = log3 4

1 < dimHK = 1.262 · · · < 2

Dimension of the Koch Curve

13

c = 1/3m = 4

dimHK = logc−1 m = log3 4

1 < dimHK = 1.262 · · · < 2

Dimension of the Koch Curve

13

c = 1/3m = 4

dimHK = logc−1 m = log3 4

1 < dimHK = 1.262 · · · < 2

Dimension of the Sierpinski Gasket

12

dimHS = logc−1 m = log2 3

1 < dimHS = 1.585 · · · < 2

Dimension of the Sierpinski Gasket

12

dimHS = logc−1 m = log2 3

1 < dimHS = 1.585 · · · < 2

Dimension of the Sierpinski Gasket

12

c = 1/2m = 3

dimHS = logc−1 m = log2 3

1 < dimHS = 1.585 · · · < 2

Dimension of the Sierpinski Gasket

12

c = 1/2m = 3

dimHS = logc−1 m = log2 3

1 < dimHS = 1.585 · · · < 2

Dimension of the Sierpinski Gasket

12

c = 1/2m = 3

dimHS = logc−1 m = log2 3

1 < dimHS = 1.585 · · · < 2

VisibilityA point P in a set X is visible from a point V if there are no other points in Xon the line segment connecting P and V.

The collection of all points in X visible from V is denoted XV.

Visibility

V

V

V

V

Koch Curve Visible from (0,∞)

dimHK(0,∞) = log3 3 = 1

Koch Curve Visible from (0,∞)

dimHK(0,∞) = log3 3 = 1

Koch Curve Visible from (0,∞)

dimHK(0,∞) = log3 3 = 1

Koch Curve Visible from (0,∞)

dimHK(0,∞) = log3 3 = 1

Koch Curve Visible from (0,∞)

dimHK(0,∞) = log3 3 = 1

Project Goal

V

V

V

V

dimHKV =?

Project Goal

V

V

V

V

dimHKV =?

Project Goal

V

V

V

V

dimHKV =?

Project Goal

V

V

V

V

dimHKV =?

Hausdorff MeasureThe s-dimensional Hausdorff measure of a set F ⊂ Rn is defined to be

Hs(F) = limδ→0

Hsδ(F),

whereHsδ(F) = inf

{ ∞∑i=1

|Ui|s : {Ui} is a δ-cover of F}.

Hausdorff DimensionThe Hausdorff dimension dimHF of a set F ∈ Rn is defined to be

dimHF = inf{s ≥ 0 : Hs(F) = 0} = sup{s : Hs(F) = ∞}.

0 s

Hs(F)∞

dimH(F)

Preliminary Results

• dimH(KV∞) = 1 when V∞ is an arbitrary point at infinity.

• dimH(KV) = 1 when V lies in the shaded regions.

Preliminary Results

• dimH(KV∞) = 1 when V∞ is an arbitrary point at infinity.• dimH(KV) = 1 when V lies in the shaded regions.

Preliminary Results

• dimH(KV∞) = 1 when V∞ is an arbitrary point at infinity.• dimH(KV) = 1 when V lies in the shaded regions.

Future Research

• Calculate the Hausdorff dimension of KV for any V ∈ R2.

• Calculate the Hausdorff dimension for other fractals with visibilityconditions.

• Generalize the results: for a fractal F, when is dimH(FV) > 1?

Future Research

• Calculate the Hausdorff dimension of KV for any V ∈ R2.

• Calculate the Hausdorff dimension for other fractals with visibilityconditions.

• Generalize the results: for a fractal F, when is dimH(FV) > 1?

Future Research

• Calculate the Hausdorff dimension of KV for any V ∈ R2.

• Calculate the Hausdorff dimension for other fractals with visibilityconditions.

• Generalize the results: for a fractal F, when is dimH(FV) > 1?

Acknowledgments

• PRIMES

• Dr. Tanya Khovanova, for mentoring this project

• Prof. Larry Guth, for suggesting the problem

• Friends and family

References[1] K. Falconer, Fractal Geometry: Mathematical Foundations andApplications. John Wiley & Sons, 2004.