코흐의 눈송이 곡선

개요

(1) 왼쪽 위의 삼각형의 둘레를 P1 그 옆의 삼각형을 P2, 왼쪽아래를 P3, .... 로 한다면

\(P_{n+1}=\frac{4}{3}(P_{n})\) 의 점화식이 성립되며, 따라서 n 을 무한대로 보내면 둘레는 무한으로 발산한다.

(2) (1)의 순서로 삼각형의 넓이를 S1, S2, ... 라 하자. 정확한 식을 위해 처음 한 변의 길이를 a 라고 하면, \(S_{1} = \frac{\sqrt3}{4}a^2\) 이다.

S2에서 원래 삼각형과 늘어난 삼각형의 길이비는 3:1 이고 넓이비는 9:1 이다. 따라서 \(S_{2} = S_{1} + \frac{3}{9}S_{1}\)

마찬가지로 \(S_{3} = S_{2} + \frac{12}{81}S_{1}\), \(S_{4} = S_{3} + \frac{48}{729}S_{1}\)

즉, 둘째 항부터 등비수열을 이루는 수열이다. 무한등비수열의 공식을 쓰면 \(\lim_{n \to \infty} S_{n}=\frac{8}{5}S_{1}\) 로 수렴한다.

이상의 프랙탈은 코흐의 눈송이 곡선으로, 이외에도 시어핀스키 프랙탈 등이 있다. 프랙탈의 시작은 해안선의 길이를 측정하면서부터라고 전해진다.

(3) 프랙탈의 차원(수학적 매개변수) 유클리드 차원과는 다르게 프랙탈 차원은 대개 정수가 아닌 분수로 표현. 프랙탈을 n개의 완전히 똑같은 부분으로 나누었을 때 전체 도형과 한 부분 사이의 닮음비가 m:1이면 프랙탈 도형의 차원 d는 다음과 같이 정의한다.

\(d = \frac{\log n}{\log m}\)

코흐의 눈송이 곡선을 예로 들어 차원을 계산하면, 4개의 똑같은 부분의 닮음비가 3:1 이므로 차원 d 는 \(\frac{log4}{log3}\)

노트

말뭉치

1. The Koch snowflake is noteworthy in that it is continuous but nowhere differentiable; that is, at no point on the curve does there exist a tangent line.
2. The Koch snowflake can be built up iteratively, in a sequence of stages.
3. The Koch snowflake is the limit approached as the above steps are followed indefinitely.
4. The Koch curve originally described by Helge von Koch is constructed using only one of the three sides of the original triangle.
5. In other words, three Koch curves make a Koch snowflake.
6. The Koch snowflake is a very well-known shape among mathematicians!
7. The 2D Koch snowflake is super cool … are you curious to see what it looks like in 3D?
8. The Koch snowflake can be simply encoded as a Lindenmayer system with initial string "F--F--F" , string rewriting rule "F" -> "F+F--F+F" , and angle .
9. Each fractalized side of the triangle is sometimes known as a Koch curve.
10. Some beautiful tilings, a few examples of which are illustrated above, can be made with iterations toward Koch snowflakes.
11. In addition, two sizes of Koch snowflakes in area ratio 1:3 tile the plane, as shown above.
12. d must satisfy As shown above, the Koch snowflake is self-similar with six pieces scaled by 1/3 and one piece scaled by \(1/\sqrt{3}\).
13. Notice, however, that the boundary of the Koch snowflake consists of three copies of the Koch curve, which has a fractal dimension of 1.26186.
14. Therefore the Koch snowflake has a perimeter of infinite length.
15. This variation on the Koch snowflake was created by William Gosper.
16. The Koch curve is named after the Swedish mathematician Niels Fabian Helge von Koch (25 January 1870 – 11 March 1924).
17. Here is an animation showing the effect of zooming in to a Koch curve.
18. To generate a Koch curve we start off with a line of unit length.
19. If you look closely at the formulae you will see that the limit area of a Koch snowflake is exactly 8/5 of the area of the initial triangle.
20. The Koch snowflake is sometimes called the Koch star or the Koch island.2.
21. The rule for generating the Koch snowflake: start with an equilateral triangle.
22. It is a member of the fractal family.the Koch snowflake growing here.t more about fractals.
23. If you’ve doodled in math class, you might have stumbled on a Koch snowflake accidentally.
24. The Koch curve also has no tangents anywhere, but von Koch’s geometric construction makes it a lot easier to understand.
25. Like many fractals, the Koch snowflake falls through the cracks of our normal conception of dimension.
26. Serendipitously, as I was writing this post, my pal Katie Mann, a mathematician at Brown University, shared an important practical application of the Koch snowflake: the Koch pecan pie.
27. It can be shown that the Koch curve is continuous at every point, but it is not derivable at any point.
28. Therefore we can conclude that the perimeter of the Koch curve and consequently also of the flake has an infinite value, even though the two curves are contained in a bounded region.
29. At each iteration step, each segment of the Koch curve is replaced by 4 small segments, each of length equal to \(\frac{1}{3}\) of the initial one.
30. This shape that we're describing right here is called a Koch snowflake.
31. A Koch snowflake, and it was first described by this gentleman right over here, who is a Swedish mathematician, Niels Fabian Helge von Koch, who I'm sure I'm mispronouncing it.
32. So even if you do this an infinite number of times, this shape, this Koch snowflake will never have a larger area than this bounding hexagon.
33. And this Koch snowflake will always be bounded.
34. In this post, I talk about my personal obsession: the Koch snowflake tessellation.
35. It’s no secret that the Koch snowflake is my favorite fractal and one that got me into computer graphics to start with.
36. There are also a lot of articles on the Koch snowflake out there so I’m not gonna do too much into it.
37. But did you know that the Koch snowflake shape (not just the curve) can be recursed upon?
38. Repeat the above steps on each of the twelve sides of curve 2 to get curve 3, the second iteration of the Koch snowflake.
39. You should be able to see the similarity between this third iteration of the Koch curve and an actual snowflake.
40. Repeat the process ad infinitum to complete the Koch snowflake curve.
41. Our next fractal is the Koch Snowflake, based on the Koch curve, one of the first fractals ever described.
42. Though it would be difficult, the Koch curve can be drawn as a single line, without lifting your pencil, and without connecting the two ends.
43. The Koch snowflake is the limit approached as the above steps are followed over and over again.
44. The Koch curve originally described by Helge von Koch is constructed with only one of the three sides of the original triangle.
45. A three-dimensional fractal constructed from Koch curves.
46. The Koch snowflake (also known as the Koch curve, Koch star, or Koch island) is a mathematical curve and one of the earliest fractal curves to have been described.
47. It is based on the Koch curve, which appeared in a 1904 paper titled “On a continuous curve without tangents, constructible from elementary geometry” by the Swedish mathematician Helge von Koch.
48. To create the Koch snowflake, one would use F++F++F (an equilateral triangle) as the axiom.
49. Complete this recursive procedure to draw one side of the Koch snowflake.
50. The Koch snowflake can be simply encoded as a Lindenmayer System with initial string "F-F-F" , String Rewriting rule "F" -> "F+F-F+F" , and angle 60°.
51. Some beautiful Tilings, a few examples of which are illustrated above, can be made with iterations toward Koch snowflakes.
52. In addition, two sizes of Koch snowflakes in Area ratio 1:3 Tile the Plane, as shown above (Mandelbrot).
53. Basically the Koch Snowflake are just three Koch curves combined to a regular triangle.
54. Now each line segment has become a Koch curve.
55. As such, the Koch snowflake offers a pictorial glimpse into the intrinsic unity between finite and infinite realms.
56. For more details on Koch snowflakes and what I the program does, check out my original article.
57. This tool draws Koch curves that look like snowflakes and stars.
58. The Koch curve is one of the earliest known fractals.
59. Because of its shape, it's also known as Koch island.
60. Renders a simple fractal, the Koch snowflake.
61. The one shown in the picture is a Koch snowflake of the 4th iteration.
62. In this Grasshopper tutorial, we will study the Koch snowflake Fractal Pattern and how we can model it from scratch.
63. Artwork showing the first four steps (iterations) of the sequence used to generate a Koch snowflake fractal.

메타데이터

위키데이터

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Spacy 패턴 목록

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• [{'LOWER': 'koch'}, {'LEMMA': 'curve'}]
• [{'LOWER': 'koch'}, {'LEMMA': 'star'}]
• [{'LOWER': 'koch'}, {'LEMMA': 'island'}]