"프랙탈"의 두 판 사이의 차이
둘러보기로 가기
검색하러 가기
Pythagoras0 (토론 | 기여) |
|||
(사용자 2명의 중간 판 19개는 보이지 않습니다) | |||
1번째 줄: | 1번째 줄: | ||
− | + | ==이 항목의 스프링노트 원문주소== | |
− | * [[프랙탈]] | + | * [[프랙탈]] |
− | + | ||
− | + | ||
− | + | ==개요== | |
− | * 다음 성질들을 | + | * 다음 성질들을 가지는 도형 또는 형상 |
− | ** 부분이 전체를 닮는 자기 유사성(self-similarity) | + | ** 소수차원 |
+ | ** 부분이 전체를 닮는 자기 유사성(self-similarity) | ||
− | + | ||
− | + | ==예== | |
− | + | * 칸토르 집합 | |
+ | * [[코흐의 눈송이 곡선]] | ||
+ | * 시에르핀스키 삼각형(개스키 | ||
+ | * 시에르핀스키 카펫 | ||
+ | * [[서로 접하는 네 원에 대한 데카르트의 정리와 아폴로니우스 개스킷|아폴로니우스 개스킷]] | ||
+ | * 페아노 곡선 | ||
+ | * 멩거 스폰지 | ||
− | + | ||
− | + | ||
− | |||
− | |||
− | |||
− | |||
− | + | ==생성방법== | |
− | + | * iterative function system | |
+ | * escape time 프랙탈 | ||
− | + | ||
− | + | ||
− | + | ==예 : 줄리아 집합== | |
− | * | + | * 복소수 <math>c\in\mathbb{C}</math>에 대하여 다음과 같은 점화식(iteration)을 정의하자. :<math>z_0=z</math>:<math>z_{n+1} = z_n^2 + c</math> |
− | + | * 이 점화식에 의한 의한 궤도가 유계가 되는 복소수 <math>z\in\mathbb{C}</math> 들이 이루는 집합의 경계를 복소수 <math>c\in\mathbb{C}</math>에 대한 줄리아 집합(Julia set)이라 한다 | |
− | + | ||
− | + | ||
− | + | ==만델브로트 집합== | |
− | |||
− | * | + | * 복소수 <math>c\in\mathbb{C}</math>에 대하여 줄리아 집합에서와 같은 점화식을 정의:<math>z_{n+1} = z_n^2 + c</math> |
+ | * 이 점화식에 의한 <math>z_0=0</math>의 궤도가 유계가 되는 복소수 <math>c\in\mathbb{C}</math>의 집합을 만델브로 집합이라 한다 | ||
− | + | * 줄리아 집합이 연결집합이 되도록 하는 복소수 <math>c\in\mathbb{C}</math> | |
− | + | ||
− | + | ||
+ | |||
− | + | ==관련된 항목들== | |
− | * | + | * [[서로 접하는 네 원에 대한 데카르트의 정리와 아폴로니우스 개스킷]] |
− | * | + | * [[클라인군(Kleinian groups)]] |
− | + | ||
− | |||
− | + | ==사전 형태의 자료== | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
* [http://ko.wikipedia.org/wiki/%ED%94%84%EB%9E%99%ED%83%88 http://ko.wikipedia.org/wiki/프랙탈] | * [http://ko.wikipedia.org/wiki/%ED%94%84%EB%9E%99%ED%83%88 http://ko.wikipedia.org/wiki/프랙탈] | ||
* http://en.wikipedia.org/wiki/Fractal | * http://en.wikipedia.org/wiki/Fractal | ||
* http://en.wikipedia.org/wiki/Iterated_function_system | * http://en.wikipedia.org/wiki/Iterated_function_system | ||
* http://en.wikipedia.org/wiki/Mandelbrot_set | * http://en.wikipedia.org/wiki/Mandelbrot_set | ||
+ | * http://www.wolframalpha.com/input/?i=julia+set | ||
* http://www.wolframalpha.com/input/?i=mandelbrot+set | * http://www.wolframalpha.com/input/?i=mandelbrot+set | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | + | ||
− | + | ==관련도서== | |
− | + | * [http://www.amazon.com/Getting-Acquainted-Fractals-Gilbert-Helmberg/dp/3110190923 Getting Acquainted with Fractals] | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | * [http://www.amazon.com/Getting-Acquainted-Fractals-Gilbert-Helmberg/dp/3110190923 Getting Acquainted with Fractals] | ||
** Gilbert Helmberg, 2007 | ** Gilbert Helmberg, 2007 | ||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | + | == 노트 == | |
+ | ===말뭉치=== | ||
+ | # 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.<ref name="ref_ab40f7c5">[https://www.britannica.com/science/fractal Fractal | mathematics]</ref> | ||
+ | # The term fractal, derived from the Latin word fractus (“fragmented,” or “broken”), was coined by the Polish-born mathematician Benoit B. Mandelbrot.<ref name="ref_ab40f7c5" /> | ||
+ | # Fractals A common first step in analyzing a dynamical system is to determine which initial states exhibit similar behaviour.<ref name="ref_ab40f7c5" /> | ||
+ | # Many fractals possess the property of self-similarity, at least approximately, if not exactly.<ref name="ref_ab40f7c5" /> | ||
+ | # 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.<ref name="ref_1b51edd9">[https://www.pbs.org/wgbh/nova/fractals/set.html Hunting the Hidden Dimension]</ref> | ||
+ | # 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.<ref name="ref_1b51edd9" /> | ||
+ | # The boundary area of the set is infinitely complex, therefore fractal, because it is possible to bring out finer and finer detail.<ref name="ref_1b51edd9" /> | ||
+ | # Homer Smith, cofounder of an independent research group based at Cornell University, produces fractal images with the aim of attracting young children to mathematics.<ref name="ref_1b51edd9" /> | ||
+ | # Once the basic concept of a Fractal is understood, it is shocking to see how many unique types of Fractals exist in nature.<ref name="ref_c167374d">[https://iternal.us/what-is-a-fractal/ What is a Fractal?]</ref> | ||
+ | # Fractal Trees: Fractals are seen in the branches of trees from the way a tree grows limbs.<ref name="ref_c167374d" /> | ||
+ | # Fractals in Animal Bodies Another incredible place where Fractals are seen is in the circulatory and respiratory system of animals.<ref name="ref_c167374d" /> | ||
+ | # 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.<ref name="ref_c167374d" /> | ||
+ | # Perhaps the most famous fractal today is the Mandelbröt set (as shown below), named after its discoverer.<ref name="ref_54dc9198">[https://theconversation.com/explainer-what-are-fractals-10865 Explainer: what are fractals?]</ref> | ||
+ | # These computer programs allow you to spot a new kind of symmetry associated with fractals.<ref name="ref_54dc9198" /> | ||
+ | # The infinite intricacy of fractals permits them a completely new type of symmetry that isn’t found in ordinary shapes.<ref name="ref_54dc9198" /> | ||
+ | # Incredibly, zooming in on a small region of a fractal leaves you looking at the same shape you started with.<ref name="ref_54dc9198" /> | ||
+ | # In mathematics, we call this property self-similarity, and shapes that have it are called fractals .<ref name="ref_48f1de6d">[https://mathigon.org/course/fractals/introduction Fractals – Mathigon]</ref> | ||
+ | # To create our own fractals, we have to start with a simple pattern and then repeat it over and over again, at smaller scales.<ref name="ref_48f1de6d" /> | ||
+ | # The plants at the beginning of this chapter look just like fractals, but it is clearly impossible to create true fractals in real-life.<ref name="ref_48f1de6d" /> | ||
+ | # First, let’s think about the dimension of fractals.<ref name="ref_48f1de6d" /> | ||
+ | # In mathematics, a fractal is a self-similar subset of Euclidean space whose fractal dimension strictly exceeds its topological dimension.<ref name="ref_62698cf3">[https://en.wikipedia.org/wiki/Fractal Wikipedia]</ref> | ||
+ | # One way that fractals are different from finite geometric figures is the way in which they scale.<ref name="ref_62698cf3" /> | ||
+ | # The term "fractal" was first used by mathematician Benoit Mandelbrot in 1975.<ref name="ref_62698cf3" /> | ||
+ | # There is some disagreement among mathematicians about how the concept of a fractal should be formally defined.<ref name="ref_62698cf3" /> | ||
+ | # A fractal is a never-ending pattern.<ref name="ref_6ac27f32">[https://fractalfoundation.org/resources/what-are-fractals/ What are Fractals? – Fractal Foundation]</ref> | ||
+ | # Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos.<ref name="ref_6ac27f32" /> | ||
+ | # A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales.<ref name="ref_5070ce7c">[https://mathworld.wolfram.com/Fractal.html Fractal -- from Wolfram MathWorld]</ref> | ||
+ | # 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.<ref name="ref_5070ce7c" /> | ||
+ | # The prototypical example for a fractal is the length of a coastline measured with different length rulers.<ref name="ref_5070ce7c" /> | ||
+ | # Illustrated above are the fractals known as the Gosper island, Koch snowflake, box fractal, Sierpiński sieve, Barnsley's fern, and Mandelbrot set.<ref name="ref_5070ce7c" /> | ||
+ | # First, reducing the description and analysis of complex phenomena into a single fractal dimension value will always run the danger of oversimplification and overgeneralization.<ref name="ref_25c0e6a1">[https://www.sciencedirect.com/topics/earth-and-planetary-sciences/fractal Fractal - an overview]</ref> | ||
+ | # 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.<ref name="ref_25c0e6a1" /> | ||
+ | # The same is true for fractals where the fractal dimension reflects the complexity of forms and patterns only.<ref name="ref_25c0e6a1" /> | ||
+ | # The second problem, which is more specific to fractals, refers to the fundamental concept of self-similarity.<ref name="ref_25c0e6a1" /> | ||
+ | # Due to them appearing similar at all levels of magnification, fractals are often considered to be 'infinitely complex'.<ref name="ref_75db3e6d">[https://www.cs.mcgill.ca/~rwest/wikispeedia/wpcd/wp/f/Fractal.htm Fractal]</ref> | ||
+ | # Objects that are now described as fractals were discovered and described centuries ago.<ref name="ref_75db3e6d" /> | ||
+ | # Ethnomathematics like Ron Eglash's African Fractals ( ISBN 0-8135-2613-2) describes pervasive fractal geometry in indigenous African craft work.<ref name="ref_75db3e6d" /> | ||
+ | # Actually, these fractals were described as curves, which is hard to realize with the well-known modern constructions.<ref name="ref_75db3e6d" /> | ||
+ | # The term fractal was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured.<ref name="ref_9ada516f">[http://www.marywood.edu/math/fractals.html Mathematics: About Fractals]</ref> | ||
+ | # Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex.<ref name="ref_9ada516f" /> | ||
+ | # Images of fractals can be created using fractal-generating software.<ref name="ref_9ada516f" /> | ||
+ | # In this work we present three new models of the fractal-fractional Ebola virus.<ref name="ref_f99d6e13">[https://www.mdpi.com/journal/fractalfract Fractal and Fractional]</ref> | ||
+ | # The first parameteris considered as the fractal dimension and the second parameteris the fractional order.<ref name="ref_f99d6e13" /> | ||
+ | # 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.<ref name="ref_f99d6e13" /> | ||
+ | # 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.<ref name="ref_f99d6e13" /> | ||
+ | # Fractal provides a presentation and transformation layer for complex data output, the like found in RESTful APIs, and works really well with JSON.<ref name="ref_cb66a59d">[https://fractal.thephpleague.com/ Output complex, flexible, AJAX/RESTful data structures]</ref> | ||
+ | # Fractal was created by Phil Sturgeon.<ref name="ref_cb66a59d" /> | ||
+ | # Fractal is maintained by Graham Daniels and Jason Lewis.<ref name="ref_cb66a59d" /> | ||
+ | # One of the most basic repeating patterns is a fractal.<ref name="ref_552200db">[https://www.investopedia.com/articles/trading/06/fractals.asp A Trader's Guide to Using Fractals]</ref> | ||
+ | # When people hear the word "fractal," they often think about complex mathematics.<ref name="ref_552200db" /> | ||
+ | # Fractals are composed of five or more bars.<ref name="ref_552200db" /> | ||
+ | # The fractals shown below are two examples of perfect patterns.<ref name="ref_552200db" /> | ||
+ | # You may begin by entering the portal within the Mistlock to enter a fractal.<ref name="ref_b3ce51ec">[https://wiki.guildwars2.com/wiki/Fractals_of_the_Mists Fractals of the Mists]</ref> | ||
+ | # You will hear Dessa's voice while stabilizing the fractal.<ref name="ref_b3ce51ec" /> | ||
+ | # 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!<ref name="ref_b3ce51ec" /> | ||
+ | # (WARNING: some fractals may cause injury, psychic trauma, blindness, or death.<ref name="ref_b3ce51ec" /> | ||
+ | # 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.<ref name="ref_7e166227">[https://www.worldscientific.com/worldscinet/fractals Fractals]</ref> | ||
+ | # 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.<ref name="ref_7e166227" /> | ||
+ | # Fractals remind us that science has a narrative component that we too often ignore.<ref name="ref_7e166227" /> | ||
+ | # The applications of fractals range from economics to geography to medical imaging to art.<ref name="ref_7e166227" /> | ||
+ | # It could be a fraction, as it is in fractal geometry.<ref name="ref_f042d87f">[https://www.vanderbilt.edu/AnS/psychology/cogsci/chaos/workshop/Fractals.html Fractals & the Fractal Dimension]</ref> | ||
+ | # Mandelbrot began his treatise on fractal geometry by considering the question: "How long is the coast of Britain?<ref name="ref_f042d87f" /> | ||
+ | # These images show diffusion limited aggregation , which is a type of fractal growth that can be analyzed with FracLac.<ref name="ref_6e5d2b1d">[https://imagej.nih.gov/ij/plugins/fraclac/FLHelp/Fractals.htm Fractals and Complexity]</ref> | ||
+ | # 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.<ref name="ref_6e5d2b1d" /> | ||
+ | # Fractals are not necessarily physical forms - they can be spatial or temporal patterns, as well.<ref name="ref_6e5d2b1d" /> | ||
+ | # In general, fractals can be any type of infinitely scaled and repeated pattern.<ref name="ref_6e5d2b1d" /> | ||
+ | # 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.<ref name="ref_1f74beb9">[http://gwyddion.net/documentation/user-guide-en/fractal-analysis.html Fractal Analysis]</ref> | ||
+ | # The slope of a plot of log( N ( l )) versus log(1/ l ) gives the fractal dimension D f directly.<ref name="ref_1f74beb9" /> | ||
+ | # The axes in Fractal Dimension graphs always show already logarithmed quantities, therefore the linear dependencies mentioned above correspond to straight lines there.<ref name="ref_1f74beb9" /> | ||
+ | # Moreover, the results of the fractal analysis can be influenced strongly by the tip convolution.<ref name="ref_1f74beb9" /> | ||
+ | # An extremely disordered morphology, such as surface roughness and porous media having the self‐similarity property, is scrutinized by fractal geometry.<ref name="ref_6e377b07">[https://www.intechopen.com/books/fractal-analysis-applications-in-physics-engineering-and-technology/fractal-geometry-and-porosity Fractal Geometry and Porosity]</ref> | ||
+ | # If microstructure formation is preferentially caused by a phenomena taking place outside of thermodynamic equilibrium, they are also characterized by fractal property.<ref name="ref_6e377b07" /> | ||
+ | # 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.<ref name="ref_6e377b07" /> | ||
+ | # Fractal theory thus provides a new and effective method for characterizing complex structure of the engineering materials.<ref name="ref_6e377b07" /> | ||
+ | # Fractals are complex because they possess structural similarity across scales.<ref name="ref_4e0b3d1f">[https://www.nature.com/articles/s41599-020-00648-y A shared fractal aesthetic across development]</ref> | ||
+ | # Exact fractals are built by precisely repeating a pattern at different magnifications.<ref name="ref_4e0b3d1f" /> | ||
+ | # 1C, D) fractals were used.<ref name="ref_4e0b3d1f" /> | ||
+ | # Exact midpoint displacement fractals (Fig. 1C) were generated according to an algorithm described by Fournier (Fournier et al., 1982; Bies et al., 2016a, b).<ref name="ref_4e0b3d1f" /> | ||
+ | # No wonder why so many non-mathematicians “feel” that fractals make them dwell on the meaning of life.<ref name="ref_56556b6a">[https://brunomarion.com/fractals-for-dummies/ Fractals for Dummies]</ref> | ||
+ | # The fractals are widely attributed to mathematician Benoît Mandelbrot (1924-2010).<ref name="ref_56556b6a" /> | ||
+ | # Mandelbrot called these sets fractals.<ref name="ref_56556b6a" /> | ||
+ | # forme, hasard, dimension), soon updated and followed by another work in English (The Fractal Geometry of Nature, 1982).<ref name="ref_56556b6a" /> | ||
+ | # The scaling for both jets and wakes extends over the entire range available; the average fractal dimension is 2.35k0.04 for both flows.<ref name="ref_37308c38">[https://dictionary.cambridge.org/dictionary/english/fractal meaning in the Cambridge English Dictionary]</ref> | ||
+ | # The notion of the nature of the stochastic layer corresponding to percolation (fractal) streamline is the foundation of percolation models.<ref name="ref_37308c38" /> | ||
+ | # In a nutshell, fractals provide a metaphor to show global 0 local links.<ref name="ref_37308c38" /> | ||
+ | # For cases where fractional kinetics can be applied, there is a fractal support in the phase-space-time description of the system.<ref name="ref_37308c38" /> | ||
+ | # 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.<ref name="ref_d56a7fa4">[https://www.dictionary.com/browse/fractal Definition of Fractal at Dictionary.com]</ref> | ||
+ | # Fractals are geometric forms that possess structure on all scales of magnification.<ref name="ref_a4d477a7">[https://warwick.ac.uk/fac/sci/maths/undergrad/ughandbook/year3/ma3d4/ MA3D4 Fractal Geometry]</ref> | ||
+ | # “The more I looked at fractal patterns, the more I was reminded of Pollock’s poured paintings,” he recounted in an essay.<ref name="ref_1899df4f">[https://www.theatlantic.com/science/archive/2017/01/why-fractals-are-so-soothing/514520/ Why Fractals Are So Soothing]</ref> | ||
+ | # Using instruments designed to measure electrical currents, Taylor examined a series of Pollocks from the 1950s and found that the paintings were indeed fractal.<ref name="ref_1899df4f" /> | ||
+ | # 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.<ref name="ref_1899df4f" /> | ||
+ | # 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.<ref name="ref_1899df4f" /> | ||
+ | # FRACTAL is a 4 year project coordinated by the Climate Systems Analysis Group at the University of Cape Town.<ref name="ref_85a5d8bf">[https://www.sei.org/projects-and-tools/projects/future-resilience-for-african-cities-and-lands-fractal/ future resilience for african cites and lands]</ref> | ||
+ | # In it, her team explored how individual differences in processing styles may account for trends in fractal fluency.<ref name="ref_f0f87405">[https://around.uoregon.edu/content/study-finds-age-3-kids-prefer-natures-fractal-patterns Study finds that by age 3 kids prefer nature's fractal patterns]</ref> | ||
+ | # 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.<ref name="ref_f0f87405" /> | ||
+ | # 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.<ref name="ref_f0f87405" /> | ||
+ | # When looking at exact fractal patterns, selections involved different pairs of snowflake-like or tree-branch-like images.<ref name="ref_f0f87405" /> | ||
+ | # In the first chapter, we introduce fractals and multifractals from physics and math viewpoints.<ref name="ref_7ba49b7b">[https://www.routledge.com/Fractals-Concepts-and-Applications-in-Geosciences/Ghanbarian-Hunt/p/book/9781498748711 Fractals Concepts and Applications in Geosciences]</ref> | ||
+ | # In what follows, in chapter 2, fragmentation process is modeled using fractals.<ref name="ref_7ba49b7b" /> | ||
+ | # In chapter 3, the advantages and disadvantages of two- and three-phase fractal models are discussed in detail.<ref name="ref_7ba49b7b" /> | ||
+ | # 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.<ref name="ref_7ba49b7b" /> | ||
+ | # FRACTAL aims to understand the decision context and the climate information required to contribute to climate resilient development in nine southern African cities.<ref name="ref_fcd5b4f5">[https://futureclimateafrica.org/project/fractal/ FRACTAL: Future Resilience for African Cities and Lands – Future Climate For Africa]</ref> | ||
+ | # Decision makers and other people working in FRACTAL cities have integrated this knowledge into their resource management decisions and urban development planning.<ref name="ref_fcd5b4f5" /> | ||
+ | ===소스=== | ||
+ | <references /> | ||
− | + | ==메타데이터== | |
− | + | ===위키데이터=== | |
− | * [ | + | * ID : [https://www.wikidata.org/wiki/Q81392 Q81392] |
− | * | + | ===Spacy 패턴 목록=== |
− | * [ | + | * [{'LEMMA': 'fractal'}] |
− | * [ | + | * [{'LEMMA': 'fractal'}] |
− | * [ | + | * [{'LOWER': 'fractal'}, {'LEMMA': 'set'}] |
+ | * [{'LOWER': 'fractal'}, {'LEMMA': 'set'}] |
2021년 2월 17일 (수) 03:08 기준 최신판
이 항목의 스프링노트 원문주소
개요
- 다음 성질들을 가지는 도형 또는 형상
- 소수차원
- 부분이 전체를 닮는 자기 유사성(self-similarity)
예
- 칸토르 집합
- 코흐의 눈송이 곡선
- 시에르핀스키 삼각형(개스키
- 시에르핀스키 카펫
- 아폴로니우스 개스킷
- 페아노 곡선
- 멩거 스폰지
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 \(c\in\mathbb{C}\)
관련된 항목들
사전 형태의 자료
- http://ko.wikipedia.org/wiki/프랙탈
- http://en.wikipedia.org/wiki/Fractal
- http://en.wikipedia.org/wiki/Iterated_function_system
- http://en.wikipedia.org/wiki/Mandelbrot_set
- http://www.wolframalpha.com/input/?i=julia+set
- http://www.wolframalpha.com/input/?i=mandelbrot+set
관련도서
- Getting Acquainted with Fractals
- Gilbert Helmberg, 2007
노트
말뭉치
- 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]
- The term fractal, derived from the Latin word fractus (“fragmented,” or “broken”), was coined by the Polish-born mathematician Benoit B. Mandelbrot.[1]
- Fractals A common first step in analyzing a dynamical system is to determine which initial states exhibit similar behaviour.[1]
- Many fractals possess the property of self-similarity, at least approximately, if not exactly.[1]
- 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]
- 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]
- The boundary area of the set is infinitely complex, therefore fractal, because it is possible to bring out finer and finer detail.[2]
- 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]
- Once the basic concept of a Fractal is understood, it is shocking to see how many unique types of Fractals exist in nature.[3]
- Fractal Trees: Fractals are seen in the branches of trees from the way a tree grows limbs.[3]
- Fractals in Animal Bodies Another incredible place where Fractals are seen is in the circulatory and respiratory system of animals.[3]
- 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]
- Perhaps the most famous fractal today is the Mandelbröt set (as shown below), named after its discoverer.[4]
- These computer programs allow you to spot a new kind of symmetry associated with fractals.[4]
- The infinite intricacy of fractals permits them a completely new type of symmetry that isn’t found in ordinary shapes.[4]
- Incredibly, zooming in on a small region of a fractal leaves you looking at the same shape you started with.[4]
- In mathematics, we call this property self-similarity, and shapes that have it are called fractals .[5]
- 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]
- 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]
- First, let’s think about the dimension of fractals.[5]
- In mathematics, a fractal is a self-similar subset of Euclidean space whose fractal dimension strictly exceeds its topological dimension.[6]
- One way that fractals are different from finite geometric figures is the way in which they scale.[6]
- The term "fractal" was first used by mathematician Benoit Mandelbrot in 1975.[6]
- There is some disagreement among mathematicians about how the concept of a fractal should be formally defined.[6]
- A fractal is a never-ending pattern.[7]
- Driven by recursion, fractals are images of dynamic systems – the pictures of Chaos.[7]
- A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales.[8]
- 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]
- The prototypical example for a fractal is the length of a coastline measured with different length rulers.[8]
- Illustrated above are the fractals known as the Gosper island, Koch snowflake, box fractal, Sierpiński sieve, Barnsley's fern, and Mandelbrot set.[8]
- 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]
- 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]
- The same is true for fractals where the fractal dimension reflects the complexity of forms and patterns only.[9]
- The second problem, which is more specific to fractals, refers to the fundamental concept of self-similarity.[9]
- Due to them appearing similar at all levels of magnification, fractals are often considered to be 'infinitely complex'.[10]
- Objects that are now described as fractals were discovered and described centuries ago.[10]
- Ethnomathematics like Ron Eglash's African Fractals ( ISBN 0-8135-2613-2) describes pervasive fractal geometry in indigenous African craft work.[10]
- Actually, these fractals were described as curves, which is hard to realize with the well-known modern constructions.[10]
- The term fractal was coined by Benoît Mandelbrot in 1975 and was derived from the Latin fractus meaning "broken" or "fractured.[11]
- Because they appear similar at all levels of magnification, fractals are often considered to be infinitely complex.[11]
- Images of fractals can be created using fractal-generating software.[11]
- In this work we present three new models of the fractal-fractional Ebola virus.[12]
- The first parameteris considered as the fractal dimension and the second parameteris the fractional order.[12]
- 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]
- 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]
- Fractal provides a presentation and transformation layer for complex data output, the like found in RESTful APIs, and works really well with JSON.[13]
- Fractal was created by Phil Sturgeon.[13]
- Fractal is maintained by Graham Daniels and Jason Lewis.[13]
- One of the most basic repeating patterns is a fractal.[14]
- When people hear the word "fractal," they often think about complex mathematics.[14]
- Fractals are composed of five or more bars.[14]
- The fractals shown below are two examples of perfect patterns.[14]
- You may begin by entering the portal within the Mistlock to enter a fractal.[15]
- You will hear Dessa's voice while stabilizing the fractal.[15]
- 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]
- (WARNING: some fractals may cause injury, psychic trauma, blindness, or death.[15]
- 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]
- 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]
- Fractals remind us that science has a narrative component that we too often ignore.[16]
- The applications of fractals range from economics to geography to medical imaging to art.[16]
- It could be a fraction, as it is in fractal geometry.[17]
- Mandelbrot began his treatise on fractal geometry by considering the question: "How long is the coast of Britain?[17]
- These images show diffusion limited aggregation , which is a type of fractal growth that can be analyzed with FracLac.[18]
- 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]
- Fractals are not necessarily physical forms - they can be spatial or temporal patterns, as well.[18]
- In general, fractals can be any type of infinitely scaled and repeated pattern.[18]
- 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]
- The slope of a plot of log( N ( l )) versus log(1/ l ) gives the fractal dimension D f directly.[19]
- The axes in Fractal Dimension graphs always show already logarithmed quantities, therefore the linear dependencies mentioned above correspond to straight lines there.[19]
- Moreover, the results of the fractal analysis can be influenced strongly by the tip convolution.[19]
- An extremely disordered morphology, such as surface roughness and porous media having the self‐similarity property, is scrutinized by fractal geometry.[20]
- If microstructure formation is preferentially caused by a phenomena taking place outside of thermodynamic equilibrium, they are also characterized by fractal property.[20]
- 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]
- Fractal theory thus provides a new and effective method for characterizing complex structure of the engineering materials.[20]
- Fractals are complex because they possess structural similarity across scales.[21]
- Exact fractals are built by precisely repeating a pattern at different magnifications.[21]
- 1C, D) fractals were used.[21]
- 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]
- No wonder why so many non-mathematicians “feel” that fractals make them dwell on the meaning of life.[22]
- The fractals are widely attributed to mathematician Benoît Mandelbrot (1924-2010).[22]
- Mandelbrot called these sets fractals.[22]
- forme, hasard, dimension), soon updated and followed by another work in English (The Fractal Geometry of Nature, 1982).[22]
- 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]
- The notion of the nature of the stochastic layer corresponding to percolation (fractal) streamline is the foundation of percolation models.[23]
- In a nutshell, fractals provide a metaphor to show global 0 local links.[23]
- For cases where fractional kinetics can be applied, there is a fractal support in the phase-space-time description of the system.[23]
- 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]
- Fractals are geometric forms that possess structure on all scales of magnification.[25]
- “The more I looked at fractal patterns, the more I was reminded of Pollock’s poured paintings,” he recounted in an essay.[26]
- 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]
- 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]
- 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]
- FRACTAL is a 4 year project coordinated by the Climate Systems Analysis Group at the University of Cape Town.[27]
- In it, her team explored how individual differences in processing styles may account for trends in fractal fluency.[28]
- 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]
- 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]
- When looking at exact fractal patterns, selections involved different pairs of snowflake-like or tree-branch-like images.[28]
- In the first chapter, we introduce fractals and multifractals from physics and math viewpoints.[29]
- In what follows, in chapter 2, fragmentation process is modeled using fractals.[29]
- In chapter 3, the advantages and disadvantages of two- and three-phase fractal models are discussed in detail.[29]
- 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]
- FRACTAL aims to understand the decision context and the climate information required to contribute to climate resilient development in nine southern African cities.[30]
- Decision makers and other people working in FRACTAL cities have integrated this knowledge into their resource management decisions and urban development planning.[30]
소스
- ↑ 1.0 1.1 1.2 1.3 Fractal | mathematics
- ↑ 2.0 2.1 2.2 2.3 Hunting the Hidden Dimension
- ↑ 3.0 3.1 3.2 3.3 What is a Fractal?
- ↑ 4.0 4.1 4.2 4.3 Explainer: what are fractals?
- ↑ 5.0 5.1 5.2 5.3 Fractals – Mathigon
- ↑ 6.0 6.1 6.2 6.3 Wikipedia
- ↑ 7.0 7.1 What are Fractals? – Fractal Foundation
- ↑ 8.0 8.1 8.2 8.3 Fractal -- from Wolfram MathWorld
- ↑ 9.0 9.1 9.2 9.3 Fractal - an overview
- ↑ 10.0 10.1 10.2 10.3 Fractal
- ↑ 11.0 11.1 11.2 Mathematics: About Fractals
- ↑ 12.0 12.1 12.2 12.3 Fractal and Fractional
- ↑ 13.0 13.1 13.2 Output complex, flexible, AJAX/RESTful data structures
- ↑ 14.0 14.1 14.2 14.3 A Trader's Guide to Using Fractals
- ↑ 15.0 15.1 15.2 15.3 Fractals of the Mists
- ↑ 16.0 16.1 16.2 16.3 Fractals
- ↑ 17.0 17.1 Fractals & the Fractal Dimension
- ↑ 18.0 18.1 18.2 18.3 Fractals and Complexity
- ↑ 19.0 19.1 19.2 19.3 Fractal Analysis
- ↑ 20.0 20.1 20.2 20.3 Fractal Geometry and Porosity
- ↑ 21.0 21.1 21.2 21.3 A shared fractal aesthetic across development
- ↑ 22.0 22.1 22.2 22.3 Fractals for Dummies
- ↑ 23.0 23.1 23.2 23.3 meaning in the Cambridge English Dictionary
- ↑ Definition of Fractal at Dictionary.com
- ↑ MA3D4 Fractal Geometry
- ↑ 26.0 26.1 26.2 26.3 Why Fractals Are So Soothing
- ↑ future resilience for african cites and lands
- ↑ 28.0 28.1 28.2 28.3 Study finds that by age 3 kids prefer nature's fractal patterns
- ↑ 29.0 29.1 29.2 29.3 Fractals Concepts and Applications in Geosciences
- ↑ 30.0 30.1 FRACTAL: Future Resilience for African Cities and Lands – Future Climate For Africa
메타데이터
위키데이터
- ID : Q81392
Spacy 패턴 목록
- [{'LEMMA': 'fractal'}]
- [{'LEMMA': 'fractal'}]
- [{'LOWER': 'fractal'}, {'LEMMA': 'set'}]
- [{'LOWER': 'fractal'}, {'LEMMA': 'set'}]