The Koch Snowflake Curve Monica J. Agana Boise State University
Fall 2014
Monica J. Agana (Boise State University)
Fall 2014
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Motivation
To introduce the audience to the Koch snowflake construction, and discuss several of its main properties.
Monica J. Agana (Boise State University)
Fall 2014
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Main Properties
Infinite Perimeter
Monica J. Agana (Boise State University)
Fall 2014
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Main Properties
Infinite Perimeter Finite Area
Monica J. Agana (Boise State University)
Fall 2014
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Main Properties
Infinite Perimeter Finite Area Continuous Everywhere
Monica J. Agana (Boise State University)
Fall 2014
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Main Properties
Infinite Perimeter Finite Area Continuous Everywhere Nowhere Differentiable
Monica J. Agana (Boise State University)
Fall 2014
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Koch Curve
Start with a line segment of any length.
Monica J. Agana (Boise State University)
Fall 2014
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Koch Curve
Monica J. Agana (Boise State University)
Fall 2014
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Koch Snowflake
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Fall 2014
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Historical Background
(1900’s) Niels Fabian Helge von Koch, a Swedish mathematician [2].
Monica J. Agana (Boise State University)
Fall 2014
7 / 16
Historical Background
(1900’s) Niels Fabian Helge von Koch, a Swedish mathematician [2]. (1904) Koch snowflake: Sur une courbe continue sans tangente, obtenue par une construction g´eom´etrique ´el´ementaire” [3].
Monica J. Agana (Boise State University)
Fall 2014
7 / 16
Historical Background
(1900’s) Niels Fabian Helge von Koch, a Swedish mathematician [2]. (1904) Koch snowflake: Sur une courbe continue sans tangente, obtenue par une construction g´eom´etrique ´el´ementaire” [3]. (1906) Koch curve: Une m´ethode g´eom´etrique ´el´ementaire pour l’´etude de certaines questions de la th´eorie des courbes plane [2].
Monica J. Agana (Boise State University)
Fall 2014
7 / 16
Historical Background
(1900’s) Niels Fabian Helge von Koch, a Swedish mathematician [2]. (1904) Koch snowflake: Sur une courbe continue sans tangente, obtenue par une construction g´eom´etrique ´el´ementaire” [3]. (1906) Koch curve: Une m´ethode g´eom´etrique ´el´ementaire pour l’´etude de certaines questions de la th´eorie des courbes plane [2]. Continuous everywhere but nowhere differentiable functions [2].
Monica J. Agana (Boise State University)
Fall 2014
7 / 16
Infinite length
Sketch of Proof: [1] Enough to show that one side of the equilateral triangle has infinite length. At stage 0 we begin with a line segment with length, L representing one side.
Monica J. Agana (Boise State University)
Fall 2014
8 / 16
Infinite length
Sketch of Proof: [1] Enough to show that one side of the equilateral triangle has infinite length. At stage 0 we begin with a line segment with length, L representing one side. At stage 1, apply the first iteration. We should have 4 segments each of length L3 . Then the total length of the sides is 43 · L.
Monica J. Agana (Boise State University)
Fall 2014
8 / 16
Infinite length
Sketch of Proof: [1] Enough to show that one side of the equilateral triangle has infinite length. At stage 0 we begin with a line segment with length, L representing one side. At stage 1, apply the first iteration. We should have 4 segments each of length L3 . Then the total length of the sides is 43 · L. At stage 2, apply second iteration. We should have 16 segments each of length 13 · L3 = L9 . Total length is 43 · 43 = ( 43 )2 .
Monica J. Agana (Boise State University)
Fall 2014
8 / 16
Repeat process up to n stages. So the total length, Ltotal is ( 43 )n .
Monica J. Agana (Boise State University)
Fall 2014
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Repeat process up to n stages. So the total length, Ltotal is ( 43 )n . Final curve: 4 limn→∞ ( )n = ∞, 3
Monica J. Agana (Boise State University)
Fall 2014
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Finite Area Find area of equilateral triangle, and build the formula √ 2 3s Aequilateral = , 4 where s is the (finite) length of one side of this equilateral triangle.
Monica J. Agana (Boise State University)
Fall 2014
10 / 16
Finite Area Find area of equilateral triangle, and build the formula √ 2 3s Aequilateral = , 4 where s is the (finite) length of one side of this equilateral triangle. Using this formula, we build the area for the Koch snowflake as √ 2 √ √ 3s 3 s 2 3 s 2 Akoch = +3· ( ) + 12 · ( ) + ··· 4 4 3 4 9
Monica J. Agana (Boise State University)
Fall 2014
10 / 16
Finite Area Find area of equilateral triangle, and build the formula √ 2 3s Aequilateral = , 4 where s is the (finite) length of one side of this equilateral triangle. Using this formula, we build the area for the Koch snowflake as √ 2 √ √ 3s 3 s 2 3 s 2 Akoch = +3· ( ) + 12 · ( ) + ··· 4 4 3 4 9 Use algebra to modify and simplify the series:
Monica J. Agana (Boise State University)
Fall 2014
10 / 16
Finite Area Find area of equilateral triangle, and build the formula √ 2 3s Aequilateral = , 4 where s is the (finite) length of one side of this equilateral triangle. Using this formula, we build the area for the Koch snowflake as √ 2 √ √ 3s 3 s 2 3 s 2 Akoch = +3· ( ) + 12 · ( ) + ··· 4 4 3 4 9 Use algebra to modify and simplify the series:
Akoch =
Monica J. Agana (Boise State University)
8 · Aequilateral 5
Fall 2014
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Continuous everywhere but nowhere differentiable
Nowhere differentiable - no tangent line at any point.
Monica J. Agana (Boise State University)
Fall 2014
11 / 16
Continuous everywhere but nowhere differentiable
Nowhere differentiable - no tangent line at any point. The Koch Curve: A Geometric Proof by Sˆıme Ungar [4].
Monica J. Agana (Boise State University)
Fall 2014
11 / 16
Continuous everywhere but nowhere differentiable
Suffices to assume [0, 1] is the interval under consideration [under a linear transformation, a finite interval can be put into (0, 1), preserving the continuity of f .
Monica J. Agana (Boise State University)
Fall 2014
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Continuous everywhere but nowhere differentiable
Suffices to assume [0, 1] is the interval under consideration [under a linear transformation, a finite interval can be put into (0, 1), preserving the continuity of f . Define a continuous mapping f : [0, 1] → R2 by f := limn→∞ fn . and let K be the Koch curve.
Monica J. Agana (Boise State University)
Fall 2014
12 / 16
Continuous everywhere but nowhere differentiable Consider the sequence of piecewise linear functions [0, 1] → R2 .
Monica J. Agana (Boise State University)
Fall 2014
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Continuous everywhere but nowhere differentiable Consider the sequence of piecewise linear functions [0, 1] → R2 .
Monica J. Agana (Boise State University)
Fall 2014
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Continuity everywhere
To obtain K, limit of the fn ’s w.r.t. the Hausdorff metric: C([0, 1], R2 ), the set of all continuous maps from [0, 1] into the plane with, kgk := supt∈[0,1] kf (t)k
Monica J. Agana (Boise State University)
Fall 2014
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Continuity everywhere
To obtain K, limit of the fn ’s w.r.t. the Hausdorff metric: C([0, 1], R2 ), the set of all continuous maps from [0, 1] into the plane with, kgk := supt∈[0,1] kf (t)k Uses this to show K is a Jordan arc (i.e. f is an injection).
Monica J. Agana (Boise State University)
Fall 2014
14 / 16
Nowhere differentiable
Idea behind it: Iterate each side up to a point, and it becomes nondifferentiable. You can do this at all points.
Monica J. Agana (Boise State University)
Fall 2014
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Nowhere differentiable
Idea behind it: Iterate each side up to a point, and it becomes nondifferentiable. You can do this at all points. For a proof of this refer to [2] or [4].
Monica J. Agana (Boise State University)
Fall 2014
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Applications of Series. N.p., n.d. Web. 27 Nov. 2014. H. von Koch, Une m´ethode g´eom´etrique ´el´ementaire pour l’´etude de certaines questions de la th´eorie des curves plane, Acta Math. 30. (1906) 145-174. “Niels Fabian Helge Von Koch.” Koch Biography. N.p., n.d. Web. 27 Nov. 2014. Ungar, Sime. “The Koch Curve: A Geometric Proof.” The American Mathematical Monthly 114.1 (2007): 61-66. JSTOR. Web. 11 Dec. 2014.
Monica J. Agana (Boise State University)
Fall 2014
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