Fractal dimension of profiles and surfaces using fuzzy
fractal and fractional佩普学术 -回复
fractal and fractional佩普学术-回复Fractal and Fractional PEP AcademicIntroduction:Fractals and fractions are two mathematical concepts that have significant applications in various fields, including physics, computer graphics, and finance. This article aims to provide a comprehensive understanding of fractals and fractions, exploring their basic definitions, properties, and real-life applications.I. Fractals:1. Definition:Fractals are geometric shapes that exhibit self-similarity, meaning that they contain smaller copies of themselves in a never-ending pattern. They can be generated through a mathematical process called recursion. Examples of well-known fractals include the Mandelbrot set and the Sierpinski triangle.2. Properties:Fractals possess several distinctive properties, including infinite complexity, fractional dimension, and non-integer scaling. These properties contribute to their unique visual appearance and make them applicable in various fields, such as computer graphics and image compression.3. Applications:Fractals find applications in many practical areas. In computer graphics, they are used for creating realistic landscapes, textures, and natural objects. Fractal-based algorithms are also employed in image compression techniques, enabling efficient storage and transmission of digital images. Additionally, fractal analysis is utilized in medical imaging, financial forecasting, and weather prediction.II. Fractions:1. Definition:Fractions are numerical expressions representing a part or parts ofa whole. They consist of a numerator and a denominator, with the numerator representing the number of parts involved and the denominator indicating the total number of equal parts that make up the whole. For example, 3/4 represents three parts out of four equal parts.2. Properties:Fractions possess various properties, including equivalence, addition, subtraction, multiplication, and division. Equivalent fractions represent the same part-to-whole ratio, while adding, subtracting, multiplying, or dividing fractions follow specific rules and algorithms.3. Applications:Fractions have numerous real-life applications. In cooking and baking, fractions are used to determine ingredient quantities accurately. In finances, fractions are utilized to calculate interest rates, percentages, and financial ratios. Moreover, fractions play a significant role in measurements, allowing precise representations of lengths, weights, and volumes.III. Fractals and Fractions:1. Fractional Crystals:Fractional crystals are a special type of fractal pattern that combines the concepts of fractals and fractions. They are formed by repeatedly replacing parts of a shape with smaller copies. Each iteration involves dividing the shape into fractions of the original size and replacing them with smaller-scale copies.2. Applications:Fractional crystals offer an effective way to represent complex structures with fractional dimensions. They find applications in physics, chemistry, and materials science. For instance, they are used to model the behavior of polymers, the structure of porous materials, and the properties of amorphous solids.Conclusion:Fractals and fractions are fundamental mathematical concepts withsignificant practical applications. Fractals exhibit self-similarity and possess unique properties, making them useful in computer graphics, image compression, and numerous scientific fields. Fractions, on the other hand, represent parts of a whole and find applications in cooking, finance, and measurements. The combination of fractals and fractions leads to the concept of fractional crystals, enabling the representation of complex structures with fractional dimensions. Understanding these concepts is essential for anyone interested in mathematics or its various applications.。
材料断口中的分形(英文)
Fractals in Material BreaksWhen materials break, they often leave behind jagged and irregular surfaces. These surfaces, known as fracture surfaces, contain a wealth of information about the behavior of the material leading up to the break. One particular aspect that researchers have been studying is the presence of fractals in these surfaces.Fractals are mathematical patterns that repeat at different scales. They are often found in nature, such as in the branching of trees or the coastline of a beach. However, they can also be observed in man-made materials, especially when those materials are subjected to stress and strain.Researchers have found that the presence of fractals in material fracture surfaces can give valuable insights into the material's mechanical properties. By analyzing the fractal dimension of the surface, they can determine the level of complexity and irregularity in the material's structure.Fractals can also help predict the behavior of materials under different conditions. By studying the fractal patterns of a material before and after exposure to stress, researchers can better understand how the material responds to different forces and how it might fail under certain conditions.In addition to mechanical properties, fractals in material breaks can also be used for forensic analysis. By studying the fractal patterns left behind on a surface, investigators can determine the type and amount of force that was exerted on the material, helping to identify the cause of failure.Overall, the study of fractals in material fracture surfaces is an exciting area of research with many potential applications. By understanding the patterns and behaviors of materials on a fractal level, we can better design and utilize these materials in industry, while also gaining a deeper understanding of the world around us.。
Characterization of fractal surfaces
J.-J. Wu r Wear 239 (2000) 36–47
37
where RŽt . is the auto-correlation function ŽACF. and 1
p D. sin Ž2 y D. p
The ACF and the structure function can be obtained from the spectral density by the following equations:
Keywords: Fractal surface; Root mean square; Asperity
1. Introduction
Surface roughness plays an important role in contact, friction and wear. Therefore, characterization of roughness is necessary in these studies.
Conventionally, a rough surface is assumed to be a random process. Statistical parameters, such as the standard deviations of the surface height s , slope sm and curvature sk , are used for characterizing surface roughness. These statistical parameters are used by the famous contact model established by Greenwood and Williamson w1x and other revised models w2,3x. However, these statistical parameters depend on the resolution and the scan length of the roughness-measuring instrument. They are not properties of the surface alone w4x. In spite of that, they are still valuable. Greenwood and Williamson’s contact model, which uses instrument-dependent statistical parameters, is surprisingly good at predicting elastic contact phenomena w5x.
fractal and fractional 水平 -回复
fractal and fractional 水平-回复Fractals and Fractionals in Mathematics: A Comprehensive ExplorationFractals and fractionals are fascinating concepts within the realm of mathematics. Both terms describe mathematical objects that exhibit intricate patterns and structures. In this article, we will delve into the world of fractals and fractionals, exploring their definitions, properties, and applications.To begin, let us first understand what a fractal is. A fractal is a complex geometric shape that can be split into parts, each of which is a reduced-scale replica of the whole. This self-similarity is a fundamental characteristic of fractals. The term "fractal" was coined by mathematician Benoit Mandelbrot in 1975, although the concept itself has been around for centuries.Fractals can be generated through simple mathematical equations or algorithms. One of the most well-known fractals is the Mandelbrot set, named after its creator. It consists of a collection of points in the complex plane that do not escape to infinity after iteratively applying a specific equation. The resulting shape of theMandelbrot set is incredibly intricate, revealing intricate patterns and details when zoomed in.Fractals have numerous properties that make them intriguing to mathematicians and scientists alike. One of these properties is called infinite complexity. Fractals possess structures that are infinitely detailed, regardless of the level of magnification. This attribute is seen in the self-similarity of fractals, where each part resembles the whole. The complexity of fractals often leads to a phenomenon known as "aesthetic appeal," where the observer finds beauty and harmony in the intricate patterns.Another property of fractals is their fractional dimension. Unlike traditional geometric shapes like squares or triangles, which have integer dimensions, fractals have fractional dimensions. This means that fractal objects occupy a fraction of a dimension between whole numbers. For example, the Cantor set, a simple fractal constructed by removing a middle third of a line segment and repeating the process infinitely, has a dimension of approximately 0.631.Fractionals, on the other hand, refer to mathematical objects thatinvolve fractional numbers. In mathematics, fractions represent parts of a whole. They consist of a numerator (which signifies the number of parts) and a denominator (which signifies the total number of parts required to make a whole). Fractions are essential in everyday life and are commonly used in various fields, such as physics, finance, and engineering.The relationship between fractals and fractionals lies in their shared utilization of fractions. Fractals often exhibit fractional dimensions, as discussed earlier. This fractional dimension is calculated using methods similar to those used in determining the value of a fraction. In essence, the fractional dimension of a fractal represents the part of a whole that it occupies within a given space or set.Fractals and fractionals have a wide range of applications in various scientific and technological fields. In physics, fractal theories have been used to model complex systems, such as weather patterns or the behavior of fluids. In computer graphics, fractals are employed to generate realistic and complex natural landscapes or textures. They are also used in data compression methods, allowing for efficient storage and transmission of information.In conclusion, fractals and fractionals are fascinating mathematical concepts with far-reaching applications. Fractals exhibit infinite complexity and possess fractional dimensions, while fractionals represent parts of a whole using fractions. The interplay between these two concepts allows for the exploration and understanding of complex systems and patterns in nature and science. From the mesmerizing patterns of the Mandelbrot set to the practical applications in various fields, fractals and fractionals continue to captivate our imagination and expand our understanding of the mathematical world.。
分形学pdf
分形学
分形学（Fractal Geometry）是一门研究分形（fractal）对象的几何学。
分形是一种复杂的几何形态，它们在局部和整体上具有自相似性，通常无法用传统的欧几里得几何（Euclidean geometry）来描述。
分形学的研究对象包括自然界中的许多不规则形状，如云彩、山脉、河流、海岸线等，以及人工设计的分形图案。
分形学的核心概念是自相似性（self-similarity）和分数维（fractional dimension）。
自相似性意味着分形对象在不同尺度上呈现出相似的形态，而分数维则是用来描述分形对象占据空间的方式，它们通常不是整数维，如零维的点、一维的线段、二维的平面或三维的立体。
分形学的基础是分形几何学，它由法国数学家伯纳德·曼德尔布罗特（Benoît Mandelbrot）在20世纪70年代提出。
曼德尔布罗特通过研究英国海岸线的长度发现，随着测量尺度的减小，海岸线的长度会无限增长，这种现象无法用传统的几何学来解释。
他提出了分形的概念，并定义了分形的维数。
分形学在多个领域都有广泛的应用，包括物理学、化学、生物学、地理学、环境科学、计算机科学、经济学等。
在计算机图形学中，分形学用于生成复杂的自然现象和纹理。
在金融学中，分形市场理论（fractal market hypothesis）用于解释股票市场等金融现象的不规则性和复杂性。
在地质学中，分形学用于分析地貌和地质结构。
在生物学中，分形学用于研究生物体的生长和形态。
1。
fractal and fractional 水平 -回复
fractal and fractional 水平-回复问题：分数和分形的关系是什么？分数和分形是数学中的两个重要概念，虽然看似完全不同，但实际上它们之间存在着紧密的联系。
在本文中，我们将一步一步回答这个问题，探讨分数和分形的关系。
首先，让我们来了解一下分数的基本概念。
分数是指一个整数除以另一个整数所得的结果，其中被除数称为分子，除数称为分母。
例如，1/2、3/4、5/6等都是分数。
在分数中，分子表示物体的一部分，分母表示整体的总量。
分数的大小由分子和分母的比值决定，分子越大，分数就越接近于1。
接下来，我们将介绍分形的概念。
分形是一种自相似的几何形状，即整体的局部部分与整体具有相似的结构。
分形的特点是无论怎样放大或缩小，它的结构都会重复出现。
常见的分形包括科赫曲线、斯尔皮斯基三角形和曼德勃罗集合等。
在分数和分形之间，存在着一种有趣的关系，即分数可以用分形来表示。
这是因为分数是由分子和分母所组成的，而分形的自相似性使得它们可以表示不同比例的结构。
例如，在斯尔皮斯基三角形中，每一个小三角形都是整体的缩小版本，这与分数中分子和分母的关系非常相似。
因此，我们可以将分数与分形进行类比。
进一步探究分数和分形的关系，我们可以发现一些有趣的特性。
首先，分数和分形都具有无限的重复性。
无论是分数的循环小数还是分形的自相似结构，它们都可以无限延伸下去。
其次，分数和分形都能够表示复杂的数学概念。
分数可以表示无理数和实数，分形则可以用来描述自然界中的复杂现象，如云朵、山脉和树木等。
此外，分数和分形在数学应用中也有着广泛的应用。
在几何学中，我们可以利用分数来表示比例关系，并应用到图形的相似性和比例计算中。
在计算机图像处理中，分形的特性使得它可以被用于生成逼真的自然图像，如地形地貌的模拟和纹理生成等。
在金融学中，分形理论被用来研究股票市场和经济波动等复杂系统。
总结来说，分数和分形在数学中都有着重要的地位。
它们之间不仅存在着数学上的联系，而且在应用中也有广泛的应用。
Is the Fractal Model Appropriate for Terrain
Figure 1: Fractal surface without power modification.
Figure 2: Non fractal (oscillatory) surface.
Figure 3: Power-modified fractal landsape.
dependent features. The nondifferentiability of the fractal models results in terrains that in the author’s opinion appear too rough at small scales (e.g., Figure 1 and [11]). Most of the terrain figures accompanying this article were produced using two-dimensional variations of a “Markaussian” autocorrelation R(τ ) = exp(−|τ | α ) with 1 < α < 2, which produces noises falling between the extremes of the Gaussian (too smooth) and Markovian (usually too rough) textures shown in Figure 6. The desirability of scale-dependent features for visual modeling is also obvious: most visual phenomena provide at least an approximate sense of scale (atmospheric phenomena are perhaps the exception again). As an illustration, one would be surprised to find a terrain such as Figure 3 at one’s feet, and the scale-dependent depressions in Figure 4 certainly do not detract from the realism of that terrain. The lack of scale-dependent features can be a practical problem in visual terrain simulation, since without such detail the observer cannot judge the distance to the “ground”. We conclude that, although a single descriptor such as the fractal dimension or spectrum exponent may be adequate for some classification purposes, the validity of the fractal model is not well established for many natural phenomena, and it is evident that the visual characteristics of many phenomena cannot be adequately differentiated using only this model. For example, it was found in [1] that an airport runway had a fractal dimension identical to that of some topographical data, the interpretation being that this was due to the attenuated low-frequency variation of the runway. Spectral modeling with Gaussian processes permits the description of a variety of phenomena, including fractal noises as a special case. Important perceptual characteristics of the noise, such as scale, period of oscillation, and directional tendencies are directly reflected in the noise autocorrelation function. Other characteristics such as dominant scales of detail and the small-scale or high-frequency behavior of the noise are easier to specify in the frequency domain, using the intuitive interpretation of the spectrum as the amount of detail at each scale. These comments originally appeared in section 6.1 of the academic paper [2].
fractal and fractional 水平 -回复
fractal and fractional 水平-回复水平是一个衡量事物高低、大小、长度等指标的概念，常常用于描述一个事物在垂直方向上的位置或高度。
水平可以是绝对的，也可以是相对的，它是一个相对于某种基准的概念。
在不同的领域，我们常常会遇到与水平相关的概念，比如数学中的分数和分形。
在数学中，分数是一种用来表示一个整体被均等地分割成若干份的表示方法。
分数通常由两个整数构成，分子和分母。
分子表示被分割物体中实际所分得的部分，而分母则表示被分割物体被分为几等份。
例如，1/2表示一个整体被分割成两等份，其中取了一份。
分数可以表示各种不同的比例和比率，如3/4表示一个整体被分割成四等份，其中取了三份。
分数在数学中扮演着重要的角色，它们既可以表示有理数，也可以表示实数。
不仅在算术运算中，分数还广泛应用于几何、代数和统计等领域。
例如，几何中的比例和相似性就与分数有密切的联系，而代数中的多项式分式则是由分子和分母分别为多项式的分数表示。
同时，分数还有很多实际应用，比如在日常生活中，我们经常用分数来表示时间、速度、概率等。
与分数不同，分形是一种描述自相似性的几何结构。
分形对象具有特殊的性质，无论在任何尺度下观察，其局部形态都与整体形态相似。
分形被广泛应用于自然科学、社会科学和艺术领域中。
例如，在自然科学中，分形模型被用来模拟树的生长、天气系统、地理地貌等现象。
而在社会科学中，分形理论被应用于城市规划、金融市场等领域的研究。
分形的研究始于上世纪60年代，由波兰数学家Benoit Mandelbrot提出。
他通过对一系列自然和人造对象的观察，提出了分形几何的基本概念和原理。
分形几何在数学的发展中具有重要的地位，对于解释那些传统几何无法涵盖的问题提供了新的视角。
分形的研究使我们能够更好地理解自然界的复杂性，并为许多实际问题的解决提供了有效的方法和工具。
分形与分数虽然在表面上看起来截然不同，但它们有着一些共同之处。
首先，它们都是数学中的概念，用来描述和理解我们周围的世界。
fractal and fractional的issn -回复
fractal and fractional的issn -回复【fractal and fractional的ISSN编号】是一个非常具体和专业的主题，它涉及到数学和科学领域中的Fractal（分形）和Fractional（分数）概念。
在这篇文章中，我将逐步解释Fractal和Fractional，并讨论它们的ISSN 编号的意义和应用。
首先，让我们来了解Fractal和Fractional的概念。
Fractal是一种具有自相似性的几何图形或数学对象。
它是由简单的重复过程生成的，其中每一次重复都会产生一个新的更小的自身。
换句话说，无论我们放大还是缩小这个图形，它都具有类似的形状和结构。
Fractal在自然界和人类创造的艺术品中都有广泛的应用，例如云朵的形状、树叶的结构、音乐的节奏等都可以用Fractal模型来描述。
而Fractional是指与整数不同的数值，介于整数之间的数。
它可以是一个分数，也可以是一个小数。
Fractional的概念在数学中具有广泛的应用，特别是在分数、比率和概率等领域。
许多现实世界中的问题和现象都可以用Fractional来进行建模和解释，例如物理学中的测量误差、经济学中的比率和利率等。
接下来，我们来谈谈ISSN编号。
ISSN (International Standard Serial Number)是一种用于标识连续出版物的国际标准编号。
它由8位数字组成，用于唯一地识别每个独立的期刊、报纸、杂志和其他持续出版物。
ISSN编号对于学术和研究机构非常重要，因为它可以确保各种期刊和研究文献的唯一性和可追溯性。
对于Fractal和Fractional这样的学术概念，往往会有专门的期刊和学术出版物来进行研究和讨论。
这些期刊通常都有自己的ISSN编号，以确保它们的学术产出被正确地标识和引用。
通过ISSN编号，我们可以追溯和查找特定的期刊，从而获得相关研究领域的最新进展和成果。
在数学和科学领域，有一些与Fractal和Fractional相关的期刊。
Can you hear the fractal dimension of a drum
1 CAN YOU HEAR THE FRACTAL DIMENSION OF A DRUM?* WALTER ARRIGHETTI † Electronic Engineering Department, Università degli Studi di Roma “La Sapienza”, via Eudossiana 18, Rome, 00184, Italy, www.die.uniroma1.it/strutture/labcem/. GIORGIO GEROSA Electronic Engineering Department, Università degli Studi di Roma “La Sapienza”, via Eudossiana 18, Rome, 00184, I TALY , www.die.uniroma1.it/strutture/labcem/. Electromagnetics and Acoustics on a bounded domain is governed by the Helmholtz’s equation; when such a domain is [pre-]fractal described by means of a ‘just-touching’ Iterated Function System (IFS ) spectral decomposition of the Helmholtz’s operator is self-similar as well. Renormalization of the Green’s function proves this feature and isolates a subclass of eigenmodes, called diaperiodic , whose waveforms and eigenvalues can be recursively computed applying the IFS to the initiator’s eigenspaces. The definition of spectral dimension is given and proven to depend on diaperiodic modes only for a wide class of IFS s. Finally, asymptotic equivalence between box-counting and spectral dimensions in the fractal limit is proven. As the “self-similar” spectrum of the fractal is enough to compute box-counting dimension, positive answer is given to title question. 1. Introduction 1.1. Between spectral and fractal geometry Marc Kac wondered in 1966 (in a famous paper entitled ‘Can you hear the shape of a drum?’, [1]) whether the shape of a plane domain could be inferred from the sole spectrum of its Laplace’s operator with Neumann/Dirichlet’s boundary conditions, i.e. whether the shape of a membrane can be inferred byjust “hearing” all its vibrating modes. This conjecture was confuted in 1992, when two plane domains were found to have the same spectrum, but different shapes [2]. Thereinafter Euclidean domains are usually split into equivalence classes of isospectrality , whose members have the same laplacian spectrum but different shapes. Computationally speaking, Laplace’s operator on a bounded domain has a discrete spectrum (formed by at most a countable infinity of eigenvalues); the shape of a drum has, in general, a non-countable infinity of degrees of freedom.* Published in Applied and Industrial Mathematics in Italy , World Scientific, 2005. Presented at the 7th SIMAI congress (Venice, 20-24 september 2004). † Email addresses: arrighetti@die.uniroma1.it and gerosa@die.uniroma1.it .a rXiv:math.SP/53748v131Mar252 This problem can be extended of course to arbitrary, finite-dimensional domains and to self-similar/fractal sets. Title question aims to the extraction of a single number (the “fractal dimension”) from that countable spectrum, that is computing a measure of the domain’s complexity and self-symmetry.1.2. Brief review on Iterated Function SystemsLet w 1,w 2,…,w p :I R d →I R d be p contractions with contraction ratios c j ∈]0,1[, 1≤j ≤p respectively, i.e.: ()()j j j c −≤−w x w y x y , ∀x ,y ∈I R d , 1≤j ≤p .Set {w 1,w 2,…,w p } is said to be an Iterated Function System (IFS ), or Iterated Similarity System (ISS ) if all the p contractions are similarities. It can be shown (cfr. [3], [6]) that, letting ℘©(S ) be the set of all compact subsets of a metric space S , the contraction mapping w :℘©(I R d ) → ℘©(I R d ) defined as 1():()p j j C C C ==w w ∪, ∀C ∈℘©(I R d ) (1)admits one compact set F ⊂ I R d which is a fixed point F ∈℘©(I R d ) (respect to the Hutchinson’s metric); F is called the IFS ’ attractor . Thus for any compacts E 0∈℘©(I R d ) and given the sets’ sequence (E N )N ∈I N such that ∀N ∈I N 0 E N +1=w (E N )=w N (E 0), the following limit exists:00lim lim ()N N N N N N F E E E ∞==≡=w ∩. (2) A ttractor F is often self-similar (i.e. it is made up of similar copies of itself at infinitely many smaller length-scales), does not depend on initiator E 0 and has non-integer fractal dimension s (e.g. Hausdorff and box-counting, [5]). Sets of sequence (E N )N ∈I N depend on initiator E 0 and might show self-similarity up to a finite length-scale only. E N is thus called N th prefractal of the IFS (of E 0 initiator).An IFS is said to be disconnected if the p copies are disjoint, or ‘just-touching’ if they are not overlapping (but their boundaries may partially coincide), i.e. w i (E ) ∩w j (E ) =∅, 1≤i <j ≤p .31.3. Box-counting dimension As previously said the main indicator of “fractality” is the concept of fractal dimensions, which several definitions exist for and whose values often coincide for IFS s’ attractors. Just one will be considered then and will be used in §3.2.Let (S ,d) be a metric structure, δ>0 and {U i }i ∈I be a δ-covering of S (i.e. sup diam i I i U δ∈= and ∪i ∈I U i =S ); let N δ(S ) be the minimal numer of sets δ-covering S , no matter what kind of sets they are. The box-counting dimension of S , whenever it exists (i.e. whether the following limit exists), is defined as: B 0log ()dim :lim log S S δδδ→=−N . (3) Box-counting dimension does not depend from specific δ-covering types (cfr. [5]), so its sets can belong to any subclasses, e.g. either overlapping balls and polytopes, pavements of (i.e. ‘just-touching’) polytopes, etc. A useful method to approximate it is the coarse graining , a hierarchy refinement of such coverings (usually packings of hypercubes) with a decreasing sequence of discrete diameters (δn )n ∈I N , stopped whenever desired approximation is reached.In the case of a ‘just-touching’ ISS , all the fractal dimensions coincide, depend on just the contraction ratios (cfr. §1.2) and solve the equation: B dim11p F j j c ==∑. (4)If the similarities all have the contraction ratio c , then B dim log c S p =−.2. Self-similar spectral decomposition of the Green’s function2.1. Spectral decomposition for ‘just-touching’ prefractalsLet Ε0∈℘©(I R d ), let E N be its N th prefractal under the IFS of §1.2 and G N (x ,x ′;λ)be the 20()N H E Green’s function for the Helmholtz’s operator ∇2−λid on E N with zero boundary condition on ∂E N i.e., weakly (∇2 operates on x ∈E N ), ()22()id ()N N H E G f f λ′∇−=x , 20()N f H E ∀∈. (5) Being E N compact ∀N ∈I N 0, Laplace’s operator ∇2 is self-adjoint, non-positive definite, and compact; its spectrum spec ∇2={λN ,n }n ∈I N is countable, real, non-positive and lim N λN ,n = −∞, ∀n ∈I N . Its eigenspaces are finite-dimensional and provide a complete basis for the Sobolev-Hilbert space 20()N H E of L 2 functions, zero on ∂E N , with weak II -order derivatives:4(){}220,,I N I N()Ker id span N N n N n n n H E λϕ∈∈=∇−=⊕. (6) Eigenfunction associated to eigenvalue λN ,n (counting multiplicities) is ϕN ,n .As the IFS is ‘just-touching’ and domain E N =w (E N −1) is the union of w 1(E N −1), w 2(E N −1), …, w p (E N −1), thanks to operator’s linearity, all the eigenfunctions ϕN −1,n associated to (N −1)th prefractal, once “contracted and copied” over the p copies via w j contractions, 1≤j ≤p , still are the N th prefractal’s eigenfunctions, despite not forming a complete basis . These eigenfunctions are self-similar, i.e. they are similar copies of initiatior set E 0’s eigenfunctions, rescaled down to the N th -order prefractal. They can be directly computed from eigenfunctions of the initiator and “periodically” copied on every rescaled copies of it “towards” the prefractal’s N th order. That is why they are called diaperiodic mode s, [7]. Their zero-boundary condition is copied throughout the p copies so diaperiodic eigenfunctions are zero on ∂w 1(E N −1), ∂w 2(E N −1), …, ∂w p (E N −1), i.e. on the intersections of the previous prefractal’s copies too. Unless the IFS is disconnected, additional modes will be present, nonzero on copies’ intersections and referred to as interconnective mode s.Scalar eigenfunctions associated to 0 eigenvalue are finite , different from N to N and usually depend on domain’s cohomology i.e., roughly, on the number of its “holes”. 0∈spec ∇2 whenever domains E N are simply connected. As domain’s topology will not further investigated here [6], their contribute to the Green’s function (just indicated as γN /λ, with a polar singularity in λ=0, cfr. §2.2), will not be treated.A 3-plet of indices (N ,j ,n ) is used for both eigenfunctions and (nonzero) eigenvalues: N is the prefractal order, n is the usual index counting independent eigenfunctions (or eigenvalues, although they can be equal due to degeneracy), while 0≤j ≤p separates the eigenfunctions such that j =0 refers to interconnective modes and 1≤j ≤p refers to diaperiodic modes obtained rescaling the whole set of nonzero eigenmodes of the (N −1)th prefractal, copied to the j th copy w j (E N −1)⊂E N only and letting it to be zero everywhere else on E N . Thus generic eigenfunction ϕN ,j ,n , 1≤j ≤p , is:()1,,1,()():()()j N N j n N n j E ϕϕχ−−=w x w x x (7) (characteristic functions χA ’s will be further suppressed for simplicity’s sake).Eigenfunctions’ families for cohomologic and interconnective modes should be computed for each N , and might show self-similarity features too.52.2. Renormalization of the Green’s function From the considerations of §2.1, the Green’s function G N can be decomposed as:,01(,)(,;)(,;)(,) (,;)(,;) N N N N N N G g g g γλλλγλλ′′′=+=′′′=++x x x x x x x x x x x x (8) where:,,,,,,,,,I N I N 0()()()()(,;)pN n N n N j n N j n N N n N j n n n j g ϕϕϕϕλλλλλ∈∈=′′′=≡−−∑∑∑x x x x x x . Function g N is the Green’s function expunged from cohomologic modes (not present at all whenever E N is simply linearly connected), which is furtherdecomposed in a “diaperiodic part” 0N g and an “interconnective part” 1N g :,,,,0,,I N 1()()(,;)pN j n N j n N N j n n j g ϕϕλλλ∈=′′=−∑∑x x x x ; ,0,,0,1,0,I N()()(,;)N n N n N N n n g ϕϕλλλ∈′′=−∑x x x x . Whenever an ISS is considered, eigenvalues get rescaled, as N increases, as 1,,1,N j n j N n c λλ−−≥ and this becomes an equality whenever w 1, w 2, …, w p have homogeneous ratios c j ’s (i.e. they are combinations of homotheties, reflections,translations and rotations only). In the latter case 0N g is recursivelyrenormalized [4] from the Green’s function initiator 000g g ≡ and using (7):()()()()().1,1,011,111,1,1,1111()()(,;)()() (),(); p N n j N n j N j N n j n p N n j N n j j j N n j n p j N j j j j g c c c g c ϕϕλλλϕϕλλλ∞−−−−==∞−−−==−=′′=−′=−′=∑∑∑∑∑w x w x x x w x w x w x w x (9) Figure 1 shows a sample of eigenmodes for a Šerpinskij carpet ’s 1st -step prefractal. Left to right: combination of a first ‘vertical’ diaperiodic mode and a second ‘horizontal’ interconnective mode; first square diaperiodic mode; second6square diaperiodic mode. Rightmost image shows the first square diaperiodic mode for the 2nd-step prefractal. Graphs over each image represent the wavelengths’ distribution along a vertical section of the prefractals.Figure 1. Samples of diaperiodic modes (and their symbolic wavelengths)on a square Šerpinskij carpet’s 1st- and 2nd-step prefractal.2.3.Self-similar eigenvalues’ scalingRenormalization (9) also shows the scaling of eigenvalues’ distribution as the IFS ordering increases: at N th iterate the (N−1)th-step’s whole spectrum is rescaled p times by coefficient c j; then N th-step interconnective eigenvalues are added. Figure 2 shows an example of a Šerpinskij gasket prefractal’s self-similar spectrum.Figure 2. Laplacian eigenvalues’ distribution (together with theirmultiplicities) of a Šerpinskij gasket’s 7th-step prefractal [6].7Eigenvalues’ count is reported in abscissa, versus eigenvalues; each “plateau ” in the graph stands for a multiple eigenvalue (the boarder the plateau, the higher the eigenvalue’s multiplicity). Boarder plateaux are associated to bigger triangular holes in the gasket’s prefractal, so the self-similar geometry of the set is reflected in the self-similar distribution of its eigenvalues.3. One can hear the fractal dimension of a drum3.1. Spectral dimension and its asymptoticsLet E ⊂ I R d be a compact d -dimensional domain and ∇2 be its (scalar) laplacian operator with zero boundary conditions. The spectral dimension of set E is defined as: 22spec \{0}spec \{0}log mul dim :log E λλλλ∈∇∆∈∇=∑∑. (10)It exists for every ‘just-touching’ ISS s’ prefractal and their attractor F with dim ∆F computed as the limit of its prefractals’ spectral dimension as N →∞:dim lim dim N NF E ∆∆=. Let 1<q ≤p be the number of the ISS ’ similarities with different contraction ratios, c 1, c 2, …, c q (case q =1 is ‘easier’, slightly different and will be treated later); so there are p 1 similarities with contraction ratio c 1, p 2 similarities with contraction ratio c 2, …, p q similarities with contraction ratio c q ; trivially p 1+p 2+…+p q =p . In this case mul λN ,j ,n = p j mul λN −1,n , so the numerator of(10) can be rescaled as:8 (),,,0,0,1,1,0,1,,01,0,1,0,log mul log mul log mul log mul log mul log mul log log mul log mul log mul q N n N j nj q N n j N n j q qN n N j n jj j N n N n j p q p q q p λλλλλλλλλ=−=−==−===+=++=+++∑∑∑∑()2,11,0,1,0,22,1log log mul log mul log mul (1)log q qN n j j j N n N n qN n j j p q q q p λλλ−==−−=+=+++++=∑∑∑…(11);10,,0,01 log mul log mul log N qN k n N k n j k j q q p λλ−−==⎛⎞⎟⎜⎟=++⎜⎟⎜⎟⎜⎝⎠∑∑… denominator of (10) is rescaled as: 1,,,,0,01,0,1,,012,0,1,0,2,0,1log log log log log log log log log (1)log log p qN nN j n N n jj j q qN n N j n j j j N n N n N n qj j N c qc q q q c q λλλλλλλλ−==−==−−==+=+−=+++−+==∑∑∑∑∑…….10,,(1)11log log Nq N k k n k p p n jk j q q c λλ−−++==⎛⎞⎟⎜⎟+−⋅⎜⎟⎜⎟⎜⎝⎠∑∑ (12)Putting together (11) and (12), one has:,1,1log mul dim log N nn N N n n E λλ∞=∆∞===∑∑ (13)9110,,0,0110110,,(1)0111log 1log mul log mul 1log 1log log N q N k N k j n N k n k j n k N q N k N N kn k p p n j k j n k q p q q q q q λλλλ−∞∞−−====−∞∞−−++====⎛⎞⎟⎜⎟⋅⋅++⎜⎟⎜⎟⎜⎝⎠=⎛⎞⎟⎜⎟⋅⋅++⎜⎟⎜⎟⎜⎝⎠∑∑∑∑∑∑∑∑∑∑Rightmost sums at numerator and denominator, diverging as Ο(q N −1), can be neglected respect to the other sums, increasing as Ο(q N ), thus proving that:dim lim dim N NF E ∆∆=∼ 0,110,111log 1log mul 1lim 11log 1log q N N j n j n q N N N n j j n q p q q q q λλ∞∞==∞∞==−⋅+−<+∞−⋅+−∑∑∑∑∑∑∼.This result also proves that the spectral dimension of a [pre -]fractal only depends on [diaperiodic eigenvalues and ] the ISS ’ contraction ratios .Case q =1 (p similarities with one ratio c ∈]0,1[) is not only proven to bewell-posed, but it also easily equals spectral dimension with box-counting dimension (cfr. §1.3). Estimating asymptotics as (13) leads to:(1)log 12dim limN N N p F ∞∆+⋅∑∼0,1log mul N n n q λ∞=+∑(1)1log 12N N c ∞+⋅∑0,1log N nn q λ∞=+∑.B log log lim dim log N p p F c =≡−≡ (14)Case q >1 is left with expression (13), which has no general closed-form sums, so it is better to check whether it equals some other known quantities.3.2. Spectral and box-counting dimensionsDue to the contractive nature of an ISS , an initiator B 0 ⊂ I R d is chosen such that its prefractals’ sequence (B N )N ∈I N → F can be used as a coarse graining for computing dim B F . B 0 can be chosen as a suitable hypercube [packing of hypercubes] of [maximal] side δ0; prefractals B N ’s, via IFS hypercubes of maximal side δN (of decreasing diameters 0N δ→), yet ‘just-touching’ with each other.10 The laplacian spectrum of such hypercubes is well known to be the set:{} I N d N πδ−∀∈m m ; mul (!)N d π⎛⎞⎟⎜−Ο⎟⎟⎜⎝⎠m ∼. As dim ∆Fdoes not asymptotically dependent on interconnecrtive eigenvalues (cfr. §3.1), just the algebraic multiplicities count for ()N N B δN . Considering the coarse graining properties (cfr. §1.3 and [5]), one gets:00I N \{}I N \{}log mul dim lim dim lim log d d N N N N N N F E πδπδ∈∆∆∈==∑∑m 0m 0m m ∼0 δ→=−So the spectral dimension and the box-counting dimension of such an ISS ’ attractor coincide and this gives the «yes, one can » answer to title question.3.3. ExamplesSimple examples of spectral dimension’s computation, for both standard sets and ISS ’ attractors are given.[0,1] ⊂ I R is considered first. The unit interval is the trivial attractor to a ‘just-touching’ ISS made up of 112()w x x = and 122()(1)w x x =+. Associatedprefractals all trivially equal [0,1] itself and just the N th -step nonzero diaperiodiceigenvalues {−2N n π}n ∈I N count, with mul(2N n π)=2N :()111log 2dim [0,1]lim log(2)log 21 lim log 2log 1log N n NN n N n n N N nππ∞=∆∞=∞∞∞==⋅=++∑∑∑∑∑ ,log 2 lim 1N N =∼ (16)11 where the logarithmic series ∑n log n can be neglected (although it converges, in the zeta-regularization sense, to ½log2π) as its partial sums increase as Ο(log n ), i.e. slower than any linear series, such as ∑ 1.The ISS of Cantor set ⊂ [0,1] (cfr. [3], [5]) is made up of 113()w x x = and 123()()2w x x =+. Nonzero spectrum is {−3N n π}n ∈I N , with mul(3N n π)=2N :()111log 2dim lim log(3)log 21lim log 3log 1log N n NN n N n n N N nππ∞=∆∞=∞∞∞==⋅=++∑∑∑∑∑ .B log 2log 2 lim dim N N == ∼ (17) Disconnected ISS s were chosen instead of ‘just-touching’ ones because interconnective eigenvalues were proven not to influence the spectral dimension and such examples provide an easy, closed-form computability as well. References1. M. Kac, “Can you hear the shape of a drum?”, Am. Math. Month., vol. 73,pp.1–23, 1966.2. C. Gordon, D. Webb, S. Wolpert, “You cannot hear the shape of a drum”,B. Am. Math. Soc., vol. 26, pp. 134–138, 1992, arXiv:math.DG/9207215.3. M. F. Barnsley, Fractals Everywhere (2nd ed.), Academic Press, 1993.4. M. Giona, “Analytic expression for the structure factor and for the moment-generating function of fractal sets and multifractal measures”, J. Phys. A: Math. Gen., vol. 30, no. 12, 4293–4312, 1997.5. K. Falconer, Techniques in Fractal Geometry , Wiley (1997).6. W. Arrighetti, Analisi di Strutture Elettromagnetiche Frattali , dissertationfor the Laurea degree, “La Sapienza” University of Rome (2002).7. W. Arrighetti, G. Gerosa, “Spectral analysis of Šerpinskij carpet-likeprefractal waveguides and resonators”, I EEE Microw. Wirel. Co. Lett., vol. 15, no. 1, pp. 30–32, 2005.。
分形城市与城市规划
33【摘要】分形城市是自组织城市中非常重要的内容之一，也是与城市规划关系最为密切的自组织城市研究领域。
本文首先阐述分形城市的基本概念及其测度方法，然后论证分形思想在城市规划中的应用思路和发展前景。
分形是大自然的优化结构，分形体能够最有效地占据空间。
借助分形思想规划城市和城市体系，将能使我们更为有效地利用地理空间和环境，美化人类的家园。
【关键词】分形城市；自组织城市；空间优化；城市规划FRACTAL CITIES AND CITY PLANNING CHEN Y anguangABSTRACT: The power-law associated with fractals is often regarded as a signature of feasible optimality thus yielding further support to the suggestion that optimality of the system as a whole explains the dy-namic origin of fractal forms in nature. Many empiri-cal studies have shown that cities have fractal structures, and fractal structure is the optimized struc-ture of nature. In fact, a fractal body can occupy the space in the most efficient pattern. In this sense,planned with the idea of fractals, a city will be able to make the best of the limited geographical space. This paper explains what is fractal cities, how to under-stand and measure fractal cities, and how to plan the future cities using the idea of fractals. Fractal cities is one of the significant fields of self-organized cities,the development of the principle, theory, and method of fractal city planning relies heavily on the advance of the theory of self-organized cities as a whole.KEYWORDS: fractal city; self-organized city; spa-tial optimization; city planning自组织城市（ｓｅｌｆ－ｏｒｇａｎｉｚｅｄ ｃｉｔｙ）有七大研究领域［１，２］，分形城市为其中最为重要也是与城市规划关系最为密切的研究分支。
有关分形的英文好书
有关分形的英文好书Fractals are a fascinating and complex topic that have captured the imagination of mathematicians, scientists, and artists alike. These intricate geometric patterns, characterized by their self-similar structure across different scales, have found applications in a wide range of fields, from computer graphics and image processing to biology and finance. For those interested in exploring the wonders of fractals, there are many excellent books available that delve into the subject in depth. One such book that stands out as a comprehensive and engaging introduction to the world of fractals is "Fractals Everywhere" by Michael Barnsley.Published in 1988, "Fractals Everywhere" is considered a seminal work in the field of fractal geometry. Barnsley, a renowned mathematician and pioneer in the study of fractals, presents a comprehensive and accessible exploration of the topic, making it an ideal choice for both students and general readers with an interest in the subject. The book begins by providing a solid foundation in the underlying mathematical concepts that underpin fractals, including the idea of self-similarity and the role of iteration in generating theseintricate patterns.One of the strengths of "Fractals Everywhere" is Barnsley's ability to translate complex mathematical ideas into language that is easy to understand, even for those without a strong background in mathematics. He skillfully weaves together theoretical discussions with practical examples and applications, allowing readers to grasp the significance and relevance of fractals in the real world. The book covers a wide range of topics, from the historical development of fractal geometry to the various techniques used to generate and analyze these structures.A particularly compelling aspect of "Fractals Everywhere" is the way Barnsley seamlessly integrates the visual and the analytical. The book is generously illustrated with stunning full-color images of fractals, showcasing their intricate beauty and the underlying mathematical principles that give rise to these patterns. Readers are not only treated to a comprehensive understanding of the theory but also a visually captivating exploration of the aesthetic qualities of fractals.Another notable feature of "Fractals Everywhere" is its interdisciplinary approach. Barnsley recognizes that fractals have far-reaching implications and applications, and he devotes significant attention to exploring the connections between fractals and various scientific and artistic disciplines. From the use of fractals in computergraphics and image compression to their relevance in the study of natural phenomena, the book provides a rich and diverse exploration of the topic.One of the most engaging aspects of "Fractals Everywhere" is Barnsley's ability to convey the sense of wonder and discovery that often accompanies the study of fractals. He writes with a contagious enthusiasm, inviting readers to join him on a journey of exploration and to appreciate the beauty and complexity of these mathematical structures. The book is peppered with anecdotes, historical insights, and personal reflections that add depth and personality to the presentation of the material.In addition to its comprehensive coverage of the theoretical foundations of fractals, "Fractals Everywhere" also includes a wealth of practical information and resources for those interested in further exploring the topic. The book provides detailed instructions and algorithms for generating various types of fractals, as well as suggestions for software and tools that can be used to create and manipulate these intricate patterns.Overall, "Fractals Everywhere" by Michael Barnsley stands out as an exceptional resource for anyone interested in the captivating world of fractals. Its clear and engaging writing style, combined with its comprehensive coverage of the subject and its stunning visualelements, make it an essential read for both students and general readers alike. Whether you are a mathematician, a scientist, an artist, or simply someone curious about the beauty and complexity of the natural world, this book is sure to provide a fascinating and enlightening exploration of the wonders of fractal geometry.。
还有其他蔬菜吗英语作文
还有其他蔬菜吗英语作文Title: Exploring Beyond the Usual: A Journey into the World of Vegetables。
In the realm of vegetables, there exists a vast array of options beyond the familiar ones we encounter in our daily lives. While staples like carrots, broccoli, and tomatoes often take center stage on our plates, there is a whole universe of lesser-known vegetables waiting to be discovered. Let us embark on a journey to explore some of these lesser-known yet equally delightful options.One such vegetable is kohlrabi, a member of the cabbage family with a unique appearance resembling a turnip growing above the ground. Its bulbous stem is surrounded by fan-like leaves, offering a crunchy texture and a mildly sweet flavor. Kohlrabi can be enjoyed raw in salads, pickled for added tanginess, or cooked in soups and stir-fries, adding a refreshing twist to familiar dishes.Moving on, have you ever heard of Romanesco broccoli? This mesmerizing vegetable captivates with its intricate fractal pattern, reminiscent of a natural work of art. Its taste is similar to that of traditional broccoli but with a slightly nuttier flavor. Romanesco broccoli not only adds visual appeal to any dish but also packs a nutritional punch, rich in vitamins and antioxidants.For those seeking a taste of the exotic, jicama offers a delightful surprise. Originating from Mexico, this root vegetable boasts a crisp texture and a subtly sweet taste, akin to a cross between an apple and a potato. Jicama is commonly enjoyed raw, sliced into sticks or cubes and served with a sprinkle of lime juice and chili powder for a refreshing and zesty snack.Delving deeper into the realm of leafy greens, we encounter mizuna, a Japanese mustard green with slender stems and feathery leaves. Mizuna offers a peppery flavor profile, adding a pleasant kick to salads, sandwiches, and stir-fries. Its versatility and nutritional benefits make it a valuable addition to any culinary repertoire.Venturing into the realm of roots, we discover sunchokes, also known as Jerusalem artichokes. Despitetheir name, sunchokes bear no relation to artichokes butare instead the tuberous roots of a species of sunflower. These knobby tubers boast a subtly sweet, nutty flavor anda crisp texture, making them ideal for roasting, sautéing, or pureeing into soups.As we traverse through this cornucopia of vegetables,it becomes evident that the world of culinary delights extends far beyond the confines of our familiar favorites. Each vegetable, with its unique flavors, textures, and nutritional profiles, invites us to expand our culinary horizons and embark on a journey of gastronomic exploration.In conclusion, while carrots, broccoli, and tomatoes may hold a special place in our hearts, let us not overlook the myriad of other vegetables waiting to be discovered and savored. Whether it's the crunchy sweetness of kohlrabi,the mesmerizing beauty of Romanesco broccoli, or the exotic allure of jicama, there is always something new andexciting to tantalize our taste buds. So, let us embrace the adventure and venture forth into the world of vegetables, where the possibilities are as endless as the bounty of nature itself.。
fractal and fractional 水平 -回复
fractal and fractional 水平-回复问题，并提供相关的解释和例子。
[fractal and fractional 水平]是什么意思？这两个概念之间有什么联系和区别？在数学中，"fractal"（分形）是指一类具有自相似性的几何图形，而"fractional"（分数）则是指数的一种表示形式，用于表示一个数量的部分或比例。
尽管这两个术语听起来相似，但它们描述的是不同的概念。
本文将一步一步解答这些问题。
首先，我们来探讨一下"fractal"（分形）的概念。
分形是一类几何图形，它们在不同的尺度上具有相似性。
也就是说，当我们对这些图形进行放大或缩小时，总是可以发现自相似的结构。
分形图形通常都非常复杂且具有模式重复的特点。
一个著名的分形是Mandelbrot集合，它是一个由复数构成的集合。
Mandelbrot集合的特点是，当我们对其中的每个点进行迭代计算，并根据计算结果确定该点的颜色时，会产生丰富且复杂的图案。
不管我们选择放大哪个部分，我们总是可以看到类似的图案出现。
另一个著名的分形是科赫曲线（Koch curve），它是一个由连续线段组成的图形。
科赫曲线的生成过程非常简单：我们从一个等边三角形开始，然后将每条边分成三等份，并在中间一段上加上一个等边三角形。
这样的过程可以一直进行下去，生成越来越复杂的图案。
与分形相关的一个重要概念是分形维度（fractal dimension）。
分形维度是一个描述分形图形复杂程度的指标。
与传统的欧几里得维度（integer dimension）不同，分形维度可以是一个非整数，甚至是一个分数。
这是因为分形具有自相似性，可以在多个尺度上进行测量。
接下来，我们来讨论一下"fractional"（分数）的概念。
分数是用来表示部分或比例的数学概念。
它是将一个量分成若干等分的表示方法。
分数由两个整数构成，分子（numerator）和分母（denominator），用斜杠（/）来表示。
分型面设计 英语
分型面设计英语Fractal Surface Design。
Fractal surface design is a fascinating concept that combines mathematics, art, and technology. It involves creating intricate patterns and textures that exhibit self-similarity at various scales. This article explores the principles and applications of fractal surface design, highlighting its significance in different fields.The concept of fractals was introduced by the mathematician Benoit Mandelbrot in the 1970s. Fractals are complex geometric shapes that exhibit self-similarity, meaning they have the same pattern regardless of the scale at which they are observed. This property makes fractals perfect for creating visually appealing and intricate designs.Fractal surface design finds applications in various fields, including computer graphics, architecture, textiles, and even finance. In computer graphics, fractal algorithms are used to generate realistic landscapes, clouds, and other natural phenomena. Architects incorporate fractal patterns in building facades and interior designs to create visually stunning spaces. Textile designers use fractal patterns to create unique and intricate fabrics. In finance, fractal analysis is used to model and predict complex market behavior.The key to creating successful fractal surface designs lies in understanding the underlying mathematical principles. Fractals are generated using recursive algorithms, where a simple geometric shape is repeatedly transformed and scaled down. Each iteration adds detail to the design, resulting in intricate patterns that exhibit self-similarity. The level of detail and complexity can be adjusted by varying the parameters of the algorithm.One popular fractal algorithm is the Mandelbrot set. It generates a complex and infinitely detailed fractal pattern by iterating a simple equation. The resulting image is a mesmerizing display of intricate shapes and colors. Another well-known fractal algorithm is the Julia set, which produces a different type of fractal pattern with its own unique characteristics.Fractal surface design also involves the use of color and texture to enhance the visual appeal of the patterns. Colors can be assigned based on mathematical properties of the fractal, creating a harmonious and visually pleasing composition. Textures can be added to simulate different materials or to create a tactile experience.In addition to their aesthetic value, fractal surface designs have practical applications. For example, in computer graphics, fractal algorithms are used to generate realistic terrain for video games and simulations. In architecture, fractal patterns can be used to optimize the acoustic properties of a space or to create energy-efficient designs. Textile designers can use fractal patterns to create fabrics with unique visual and tactile qualities.The future of fractal surface design holds exciting possibilities. With advancements in technology, designers can now create and manipulate fractal patterns in real-time, allowing for interactive and dynamic designs. Virtual reality and augmented reality technologies also open up new avenues for experiencing fractal surface designs in immersive environments.In conclusion, fractal surface design is a captivating field that combines mathematics, art, and technology. Its self-similar patterns and intricate details make it visually appealing and versatile. From computer graphics to architecture and textiles, fractal surface design finds applications in various domains. By understanding the underlying mathematical principles and leveraging technology, designers can create stunning and innovative fractal surface designs that push the boundaries of creativity.。
Unit47-Fractals（单元47-分形）
fractals are not so much a rigorous set of models as a
set of concepts these concepts express ideas which have been around in
new detail will be apparent at the larger scale "the closer you look, the more you see" is true of
almost all geographical data in effect the line will behave as if it had the
a fractal dimension of 2 many natural cartographic lines have fractal
dimensions between 1.15 and 1.30 a surface can have a fractal dimension anywhere from
NCGIA Core Curriculum in GIS - 1990
Page 2
Unit 47 - Fractals
Euclidean
dimension of two to define a point on a surface requires three
dimensions, therefore a surface has a Euclidean
cartography for a long time they provide a framework for understanding the way
分形维数_Fractaldimension_及其测量方法
把全体分成 1/ 2 的相似形组成 。2 、4 、8 数字还可以写成 21 、
22 、23 ,显然这里的指数与其图形的经验维数相一致 。推而广
之 ,若某图形是由把全体缩小成 1/ a 的 b 个相似形所组成 ,由
于 b = aD ,则有
D = log b/ log a 。
(1)
此 D 便是几何图形的维数 , 由于它是通过相似变换得来的 ,
1983 年 ,P. Grassberger 和 J . Procassia 给出了关联维数
的定义[1 ] :
D2
= lim ε→0
ln C (ε) lnε
。
(9)
式中
C (ε)
=
1 N2
i
,
N
∑
j=1
H
(ε-
0 xi
-
xj 0) 。
(10)
1. 6 广义维数
H. G. E. Hentschel 等提出了广义维数的概念 ,其定义
可证 Dq| q = 0 = D0 ;当 q = 1 时 , 利用罗必塔法取极限可得 D1
=
lim
ε→0
Dq
;
且
Dq| q = 2 =
D2 ,即
q = 2 时 ,广义维数
Dq 就是关键
维数 D2 。
2 分形维数的基本测量方法
2. 1 改变粗视化程度求维数的方法 这是基于盒子维数和信息维数的定义设计的一种测量分
是
Dq =
-
lim
ε→0
S q (ε) lnε
。
(11)
式中
S q (ε)
=
1
1 -
N (ε)
ln[ ∑ Piq ] q i=1
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* Corresponding author.
0013-7952/97/$17.00 0 1997 Elsevier PII SOOl3-7952(97)00039-2 Science
fractal dimension concept has been applied only to rock profiles. This study extends the fractal dimension approach to surfaces which is required for an accurate description of roughness. In this paper, we propose a new approach, based on a fuzzy morphological representation (Sinha and Dougherty, 1992; Huang and Chaparro, 1995), that applies equally well to profiles or surfaces, and is geometrically intuitive and easily implementable. We will call this method the fuzzy morphological coverings (FMC ). Traditional methods to estimate the fractal dimension (Barnsley, 1988; Falconer, 1990; Stoyan and Stoyan, 1994) have been extended. For instance, in (Maragos and Sun, 1993 ) morphological covers are used to estimate fractal dimension of profiles. Using mathematical morphological dilation and erosion operators, with preselected structuring functions at different scales, covers of
Engineering Geology 48 ( 1997) 245-253
Fractal dimension of profiles and surfaces using fuzzy morphological coverings
Chin-Pan Huang a, Luis F. Chaparro a,*, Luis E. Vallejo b
Abstract
In this paper, we propose a new method, based on fuzzy morphology coverings, to estimate the fractal dimension of profiles and surfaces. This method is geometrically intuitive and simple to implement. Algorithmically, the method fits a covering to the frames or blocks of the profile or surface using fuzzy morphology. Varying the dimension of the frame or block, estimates of the length or area covered are then used to find the fractal dimension. Validation of the proposed method is performed by comparing its results with known fractal dimensions of mathematical profiles. The method is used to obtain the fractal dimension of rock profiles and surfaces. 0 1997 Elsevier Science B.V.
Kqwords:
Fractal
dimension;
Fuzzy morphology;
Rock joints;
Rock surfaces
1. Introduction
The accurate description of rock surface roughness is a critically important topic to geotechnical engineers because roughness affects the frictional resistance of these surfaces (Barton, 1973). Recently, the roughness of surfaces has been determined using Mandelbrot’s fractal dimension concept (Mandelbrot, 1982). The importance of using fractals as a way to describe irregular or rough objects has been highlighted in articles by researchers working in fields such as powder mechanics ( Kaye, 1978), rock mechanics (Carr and Warriner, 1989; Ghosh and Daemen, 1993), soil mechanics ( Vallejo, 1995, 1996) and geo-enviromental engineering (Vallejo and Zhou, 1995). To date, the
a Department of Electrical Engineering, University of Pittsburgh, Pittsburgh, PA 1.5261, USA b Department of Civil Engineering, University of Pittsburgh, Pittsburgh, PA 15261. USA
dimension are obtained by calculating the lengths resulting from wath different yardsticks or by covering the coastline with circles or boxes (Mandelbrot, 1982) of different sizes. Thus, letting Q(r)=L(r)/r be the number of steps taken (or boxes included) in calculating L(r) we obtain from Eq. (1): log(Q(r)) = log(F) + D log( l/u). (2)
B.V. All rights reserved
246
C.-P.
Huung
et al. : Engineering
Geology
4X ( 1997)
245-253
the profiles are generated and applied to estimate their fractal dimensions. Sarkar and Chaudhuri ( 1994) and Jin and Ong ( 1995) propose improvements on the traditional box-counting method of estimating the fractal dimension of surfaces. In the FMC proposed method. a discretized profile or surface is divided into frames or blocks of varying dimension and approximated using adaptable structuring functions. This results in a polynomial-like representation in each frame or block. Using a few of the components of this representation, a tight covering for the profile or surface can be generated in each frame or block. To simplify the computations, we consider the special case of the first component of the representation from which the fractal dimension can be estimated. Our procedure is geometrically intuitive, simple to implement and applies to both profiles and surfaces. The rest of the paper is organized as follows. In Section 2, we briefly review fractal dimension and discuss recent methods proposed for its estimation. Section 3 introduces fuzzy morphology and its applications to represent profiles or surfaces. In Section 4, we discuss, in detail, the FMC method first for profiles and then for surfaces. In Section 5, we consider the validity of the FMC method and its application to rock profiles and surfaces.