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