프랙탈

수학노트
Pythagoras0 (토론 | 기여)님의 2020년 12월 28일 (월) 05:58 판 (→‎메타데이터: 새 문단)
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이 항목의 스프링노트 원문주소



개요

  • 다음 성질들을 가지는 도형 또는 형상
    • 소수차원
    • 부분이 전체를 닮는 자기 유사성(self-similarity)


예



생성방법

  • iterative function system
  • escape time 프랙탈



예 : 줄리아 집합

  • 복소수 \(c\in\mathbb{C}\)에 대하여 다음과 같은 점화식(iteration)을 정의하자. \[z_0=z\]\[z_{n+1} = z_n^2 + c\]
  • 이 점화식에 의한 의한 궤도가 유계가 되는 복소수 \(z\in\mathbb{C}\) 들이 이루는 집합의 경계를 복소수 \(c\in\mathbb{C}\)에 대한 줄리아 집합(Julia set)이라 한다



만델브로트 집합

  • 복소수 \(c\in\mathbb{C}\)에 대하여 줄리아 집합에서와 같은 점화식을 정의\[z_{n+1} = z_n^2 + c\]
  • 이 점화식에 의한 \(z_0=0\)의 궤도가 유계가 되는 복소수 \(c\in\mathbb{C}\)의 집합을 만델브로 집합이라 한다
  • 줄리아 집합이 연결집합이 되도록 하는 복소수 \(c\in\mathbb{C}\)



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말뭉치

  1. Fractal, in mathematics, any of a class of complex geometric shapes that commonly have “fractional dimension,” a concept first introduced by the mathematician Felix Hausdorff in 1918.[1]
  2. The term fractal, derived from the Latin word fractus (“fragmented,” or “broken”), was coined by the Polish-born mathematician Benoit B. Mandelbrot.[1]
  3. Fractals A common first step in analyzing a dynamical system is to determine which initial states exhibit similar behaviour.[1]
  4. Many fractals possess the property of self-similarity, at least approximately, if not exactly.[1]
  5. In the late 1970s and early 1980s Benoit Mandelbrot, the inventor of fractal geometry, and several others were using simple iterative equations to explore the behavior of numbers on the complex plane.[2]
  6. The points (pixels) representing the fastest-expanding numbers might be colored red, slightly slower ones magenta, very slow ones blue—whatever color scheme the fractal explorer decides.[2]
  7. The boundary area of the set is infinitely complex, therefore fractal, because it is possible to bring out finer and finer detail.[2]
  8. Homer Smith, cofounder of an independent research group based at Cornell University, produces fractal images with the aim of attracting young children to mathematics.[2]
  9. Once the basic concept of a Fractal is understood, it is shocking to see how many unique types of Fractals exist in nature.[3]
  10. Fractal Trees: Fractals are seen in the branches of trees from the way a tree grows limbs.[3]
  11. Fractals in Animal Bodies Another incredible place where Fractals are seen is in the circulatory and respiratory system of animals.[3]
  12. In the case of ice crystal formations, the starting point of the Fractal is in the center and the shape expands outward in all directions.[3]
  13. Perhaps the most famous fractal today is the Mandelbröt set (as shown below), named after its discoverer.[4]
  14. These computer programs allow you to spot a new kind of symmetry associated with fractals.[4]
  15. The infinite intricacy of fractals permits them a completely new type of symmetry that isn’t found in ordinary shapes.[4]
  16. Incredibly, zooming in on a small region of a fractal leaves you looking at the same shape you started with.[4]
  17. In mathematics, we call this property self-similarity, and shapes that have it are called fractals .[5]
  18. To create our own fractals, we have to start with a simple pattern and then repeat it over and over again, at smaller scales.[5]
  19. The plants at the beginning of this chapter look just like fractals, but it is clearly impossible to create true fractals in real-life.[5]
  20. First, let’s think about the dimension of fractals.[5]
  21. In mathematics, a fractal is a self-similar subset of Euclidean space whose fractal dimension strictly exceeds its topological dimension.[6]
  22. One way that fractals are different from finite geometric figures is the way in which they scale.[6]
  23. The term "fractal" was first used by mathematician Benoit Mandelbrot in 1975.[6]
  24. There is some disagreement among mathematicians about how the concept of a fractal should be formally defined.[6]
  25. A fractal is a never-ending pattern.[7]
  26. Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos.[7]
  27. A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales.[8]
  28. A plot of the quantity on a log-log graph versus scale then gives a straight line, whose slope is said to be the fractal dimension.[8]
  29. The prototypical example for a fractal is the length of a coastline measured with different length rulers.[8]
  30. Illustrated above are the fractals known as the Gosper island, Koch snowflake, box fractal, Sierpiński sieve, Barnsley's fern, and Mandelbrot set.[8]
  31. First, reducing the description and analysis of complex phenomena into a single fractal dimension value will always run the danger of oversimplification and overgeneralization.[9]
  32. On the other hand, one could argue that fractal dimension as a descriptor is no different from any other single statistical descriptors such as mean and standard deviation.[9]
  33. The same is true for fractals where the fractal dimension reflects the complexity of forms and patterns only.[9]
  34. The second problem, which is more specific to fractals, refers to the fundamental concept of self-similarity.[9]
  35. Due to them appearing similar at all levels of magnification, fractals are often considered to be 'infinitely complex'.[10]
  36. Objects that are now described as fractals were discovered and described centuries ago.[10]
  37. Ethnomathematics like Ron Eglash's African Fractals ( ISBN 0-8135-2613-2) describes pervasive fractal geometry in indigenous African craft work.[10]
  38. Actually, these fractals were described as curves, which is hard to realize with the well-known modern constructions.[10]
  39. The term fractal was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured.[11]
  40. Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex.[11]
  41. Images of fractals can be created using fractal-generating software.[11]
  42. In this work we present three new models of the fractal-fractional Ebola virus.[12]
  43. The first parameteris considered as the fractal dimension and the second parameteris the fractional order.[12]
  44. We evaluate the numerical solutions of the fractal-fractional Ebola virus for these operators with the theory of fractional calculus and the help of the Lagrange polynomial functions.[12]
  45. For the effect of fractal-fractional on the behavior, we study the numerical solutions for different values ofandAll calculations in this work are accomplished by using the Mathematica package.[12]
  46. Fractal provides a presentation and transformation layer for complex data output, the like found in RESTful APIs, and works really well with JSON.[13]
  47. Fractal was created by Phil Sturgeon.[13]
  48. Fractal is maintained by Graham Daniels and Jason Lewis.[13]
  49. One of the most basic repeating patterns is a fractal.[14]
  50. When people hear the word "fractal," they often think about complex mathematics.[14]
  51. Fractals are composed of five or more bars.[14]
  52. The fractals shown below are two examples of perfect patterns.[14]
  53. You may begin by entering the portal within the Mistlock to enter a fractal.[15]
  54. You will hear Dessa's voice while stabilizing the fractal.[15]
  55. The Consortium invites you to visit our asura gate near Fort Marriner to experience the spectacular scenery and awe-inspiring adventure of its newest attraction: Fractals of the Mists![15]
  56. (WARNING: some fractals may cause injury, psychic trauma, blindness, or death.[15]
  57. Fractals often start with a simple geometrical object and a rule for changing the object that leads to objects that are so complex that their dimension is not an integer.[16]
  58. According to Michael Frame, Benoit Mandelbrot (who first coined the word "fractal" and was the founding editor of this journal) considered himself above all a storyteller.[16]
  59. Fractals remind us that science has a narrative component that we too often ignore.[16]
  60. The applications of fractals range from economics to geography to medical imaging to art.[16]
  61. It could be a fraction, as it is in fractal geometry.[17]
  62. Mandelbrot began his treatise on fractal geometry by considering the question: "How long is the coast of Britain?[17]
  63. These images show diffusion limited aggregation , which is a type of fractal growth that can be analyzed with FracLac.[18]
  64. The field was developed to describe computer-generated fractals such as the diffusion limited aggregates shown on this page, but fractals are not necessarily computer-generated images.[18]
  65. Fractals are not necessarily physical forms - they can be spatial or temporal patterns, as well.[18]
  66. In general, fractals can be any type of infinitely scaled and repeated pattern.[18]
  67. Many randomly rough surfaces are assumed to belong to the random objects that exhibit the self-affine properties and they are treated self-affine statistical fractals.[19]
  68. The slope of a plot of log( N ( l )) versus log(1/ l ) gives the fractal dimension D f directly.[19]
  69. The axes in Fractal Dimension graphs always show already logarithmed quantities, therefore the linear dependencies mentioned above correspond to straight lines there.[19]
  70. Moreover, the results of the fractal analysis can be influenced strongly by the tip convolution.[19]
  71. An extremely disordered morphology, such as surface roughness and porous media having the self‐similarity property, is scrutinized by fractal geometry.[20]
  72. If microstructure formation is preferentially caused by a phenomena taking place outside of thermodynamic equilibrium, they are also characterized by fractal property.[20]
  73. Thus, a well‐known grain boundary, being the most significant element of the microstructure, is curvilinear, and this form can be described by the fractal dimension (D) correlating to 1 ≤ D ≤ 2.[20]
  74. Fractal theory thus provides a new and effective method for characterizing complex structure of the engineering materials.[20]
  75. Fractals are complex because they possess structural similarity across scales.[21]
  76. Exact fractals are built by precisely repeating a pattern at different magnifications.[21]
  77. 1C, D) fractals were used.[21]
  78. Exact midpoint displacement fractals (Fig. 1C) were generated according to an algorithm described by Fournier (Fournier et al., 1982; Bies et al., 2016a, b).[21]
  79. No wonder why so many non-mathematicians “feel” that fractals make them dwell on the meaning of life.[22]
  80. The fractals are widely attributed to mathematician Benoît Mandelbrot (1924-2010).[22]
  81. Mandelbrot called these sets fractals.[22]
  82. forme, hasard, dimension), soon updated and followed by another work in English (The Fractal Geometry of Nature, 1982).[22]
  83. The scaling for both jets and wakes extends over the entire range available; the average fractal dimension is 2.35k0.04 for both flows.[23]
  84. The notion of the nature of the stochastic layer corresponding to percolation (fractal) streamline is the foundation of percolation models.[23]
  85. In a nutshell, fractals provide a metaphor to show global 0 local links.[23]
  86. For cases where fractional kinetics can be applied, there is a fractal support in the phase-space-time description of the system.[23]
  87. For example, a fractal set called a Cantor dust can be constructed beginning with a line segment by removing its middle third and repeating the process on the remaining line segments.[24]
  88. Fractals are geometric forms that possess structure on all scales of magnification.[25]
  89. “The more I looked at fractal patterns, the more I was reminded of Pollock’s poured paintings,” he recounted in an essay.[26]
  90. Using instruments designed to measure electrical currents, Taylor examined a series of Pollocks from the 1950s and found that the paintings were indeed fractal.[26]
  91. Benoit Mandelbrot first coined the term ‘fractal’ in 1975, discovering that simple mathematic rules apply to a vast array of things that looked visually complex or chaotic.[26]
  92. As he proved, fractal patterns were often found in nature’s roughness—in clouds, coastlines, plant leaves, ocean waves, the rise and fall of the Nile River, and in the clustering of galaxies.[26]
  93. FRACTAL is a 4 year project coordinated by the Climate Systems Analysis Group at the University of Cape Town.[27]
  94. In it, her team explored how individual differences in processing styles may account for trends in fractal fluency.[28]
  95. Exact fractals are highly ordered such that the same basic pattern repeats exactly at every scale and may possess spatial symmetry such as that seen in snowflakes.[28]
  96. Statistical fractals, in contrast, repeat in a similar but not exact fashion across scale and do not possess spatial symmetry, as seen in coastlines, clouds, mountains, rivers and trees.[28]
  97. When looking at exact fractal patterns, selections involved different pairs of snowflake-like or tree-branch-like images.[28]
  98. In the first chapter, we introduce fractals and multifractals from physics and math viewpoints.[29]
  99. In what follows, in chapter 2, fragmentation process is modeled using fractals.[29]
  100. In chapter 3, the advantages and disadvantages of two- and three-phase fractal models are discussed in detail.[29]
  101. In chapter 4, two- and three-phase fractal techniques are used to develop capillary pressure curve models, which characterize pore-size distribution of porous media.[29]
  102. FRACTAL aims to understand the decision context and the climate information required to contribute to climate resilient development in nine southern African cities.[30]
  103. Decision makers and other people working in FRACTAL cities have integrated this knowledge into their resource management decisions and urban development planning.[30]

소스

  1. ↑ 1.0 1.1 1.2 1.3 Fractal | mathematics
  2. ↑ 2.0 2.1 2.2 2.3 Hunting the Hidden Dimension
  3. ↑ 3.0 3.1 3.2 3.3 What is a Fractal?
  4. ↑ 4.0 4.1 4.2 4.3 Explainer: what are fractals?
  5. ↑ 5.0 5.1 5.2 5.3 Fractals – Mathigon
  6. ↑ 6.0 6.1 6.2 6.3 Wikipedia
  7. ↑ 7.0 7.1 What are Fractals? – Fractal Foundation
  8. ↑ 8.0 8.1 8.2 8.3 Fractal -- from Wolfram MathWorld
  9. ↑ 9.0 9.1 9.2 9.3 Fractal - an overview
  10. ↑ 10.0 10.1 10.2 10.3 Fractal
  11. ↑ 11.0 11.1 11.2 Mathematics: About Fractals
  12. ↑ 12.0 12.1 12.2 12.3 Fractal and Fractional
  13. ↑ 13.0 13.1 13.2 Output complex, flexible, AJAX/RESTful data structures
  14. ↑ 14.0 14.1 14.2 14.3 A Trader's Guide to Using Fractals
  15. ↑ 15.0 15.1 15.2 15.3 Fractals of the Mists
  16. ↑ 16.0 16.1 16.2 16.3 Fractals
  17. ↑ 17.0 17.1 Fractals & the Fractal Dimension
  18. ↑ 18.0 18.1 18.2 18.3 Fractals and Complexity
  19. ↑ 19.0 19.1 19.2 19.3 Fractal Analysis
  20. ↑ 20.0 20.1 20.2 20.3 Fractal Geometry and Porosity
  21. ↑ 21.0 21.1 21.2 21.3 A shared fractal aesthetic across development
  22. ↑ 22.0 22.1 22.2 22.3 Fractals for Dummies
  23. ↑ 23.0 23.1 23.2 23.3 meaning in the Cambridge English Dictionary
  24. ↑ Definition of Fractal at Dictionary.com
  25. ↑ MA3D4 Fractal Geometry
  26. ↑ 26.0 26.1 26.2 26.3 Why Fractals Are So Soothing
  27. ↑ future resilience for african cites and lands
  28. ↑ 28.0 28.1 28.2 28.3 Study finds that by age 3 kids prefer nature's fractal patterns
  29. ↑ 29.0 29.1 29.2 29.3 Fractals Concepts and Applications in Geosciences
  30. ↑ 30.0 30.1 FRACTAL: Future Resilience for African Cities and Lands – Future Climate For Africa

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