Fractal Web Quest分形网络探索.ppt
合集下载
FRACTALES - Inicio分形开始18页PPT
Pero no del valor crítico de
crecimiento
¿Y quépodemos concluir?
• Las figuras logradas cumplieron con, al menos, algunas de las características fractales.
UNIVERSIDAD DE BUENOS AIRES Facultad de Ciencias Exactas y Naturales
Materia: Física II Cátedra: Mariano Sigman Mra. Fernanda D’Jonsiles - Jésica Daniela Fey
para el crecimiento de las plantas.
Las hojas presentan una morfología similar a la pequeña rama de la que forman parte que, a su vez, presenta una forma similar a la rama, que a su vez es similar a la forma del árbol.
Tener tu dimensión de Hausdorff mayor a tu dimensión topológica, si o si…
La estructura fractal es ideal para aprovechar al máximo el
contacto con el exterior, como mecanismo más efectivo
Fractales Fractales
Fractales
crecimiento
¿Y quépodemos concluir?
• Las figuras logradas cumplieron con, al menos, algunas de las características fractales.
UNIVERSIDAD DE BUENOS AIRES Facultad de Ciencias Exactas y Naturales
Materia: Física II Cátedra: Mariano Sigman Mra. Fernanda D’Jonsiles - Jésica Daniela Fey
para el crecimiento de las plantas.
Las hojas presentan una morfología similar a la pequeña rama de la que forman parte que, a su vez, presenta una forma similar a la rama, que a su vez es similar a la forma del árbol.
Tener tu dimensión de Hausdorff mayor a tu dimensión topológica, si o si…
La estructura fractal es ideal para aprovechar al máximo el
contacto con el exterior, como mecanismo más efectivo
Fractales Fractales
Fractales
Fractal Geometry - The Center for Mathematics and Science 分形几何的数学与科学中心 14页PPT
• Triangles • Circles • Squares • Rectangles • Trapezoids • Pentagons • Hexagons • Octagons • Cyclinders
Can nature be described in terms of Euclidean Geometry?
Try drawing nature using Euclidean
Geometry:
• A tree using cylinders??? • A mountain range using triangles and
pyramids???
• Clouds using circles??? • Leaves??? • Rocks??? • Humans and animals with rectangles and
and non-uniform shapes.
• Euclidean geometry describes ideal shapes
which rarely occur in nature.
• Why then do we even bother with
Euclidean geometry?
1. Historically…
circles???
Look outside… Do you see any shapes in Euclidean Geometry?
If so, they were more than likely man made. For example…
The point is this…
• Our world is fashioned with rough edges
Can nature be described in terms of Euclidean Geometry?
Try drawing nature using Euclidean
Geometry:
• A tree using cylinders??? • A mountain range using triangles and
pyramids???
• Clouds using circles??? • Leaves??? • Rocks??? • Humans and animals with rectangles and
and non-uniform shapes.
• Euclidean geometry describes ideal shapes
which rarely occur in nature.
• Why then do we even bother with
Euclidean geometry?
1. Historically…
circles???
Look outside… Do you see any shapes in Euclidean Geometry?
If so, they were more than likely man made. For example…
The point is this…
• Our world is fashioned with rough edges
探索分形的奥秘ppt课件
• 在数学方面，康托集是由德国数学家康托于1883年引入的（但在1875年 就由Henry John Stephen Smith发现了），它是一个取自简单直线段上的点 集，它有若干非凡而又深刻的性质。通过对它的思考，康托和其他助手 奠定了现代一般拓扑学基础。虽然康托自己用抽象的方法定义了这个集 合，但一般而言，现代最流行的构造是康托三分集，它是通过将一条线 段的中间部分去掉而获得的。康托自己只是顺便提及了三重构造，作为 无处稠密的完备集的一般例子。
此比赛是由美国IBM公司数学实验 室举办
比赛一年举办一次
每年都会吸引大量的分形爱好 者参加比赛
什么是分形（Fractal）
Fractal本意是不规则的、破碎的、分数的， 来源于拉丁文Frangere。
曼德勃罗用Fractal来形容自然界中传统欧氏 几何学所不能描述的一大类复杂无规则的 几何对象
探索分形的奥秘
Explore the Mysteries of Fractal
数
目录
• 欣赏曼德勃罗分形艺 术大赛获奖作品
• 什么是分形 • 自然界中的分形 • 分形理论的诞生
• 分形树 • Koch雪花 • Cantor三分集 • 分形几何与欧氏几何 • 生活中的分形
下面您欣赏的作品来自 曼德勃罗分形艺术大赛获奖作品
，而是在越来越细微的尺度上不断自我重 复，是一项研究不规则性的科学。
• 1986年，曼德勃罗曾给分形下过这样一个 定义：组成部分和整体以某种方式相似的 形。
• 分形是一种粗糙的或破碎的几何图形，它 的组成部分可以被无限细分，而且它的局 部的形状一般与整体相似。
自然界中的分形
蜿蜒曲折的海岸线
起伏不定的山脉
• 曼德勃罗(Benoit B. Mandelbrot)，数学 家，经济学家，分形理论的创 始人。
Fractal Web Quest分形网络探索19页PPT
• It is geared towards a fifth grade mathematics class.
• You will need internet access in order to complete this WebQuest.
Introduction
You have entered the world of strange and unusual shapes!
and for what purpose?
Go to the resources page to start searching for the answers!
Introduction
Although all of the pictures from the previous slide look different, they are all fractals, which seems like a fitting name for these images of small broken pieces all put together.
role to withstand. You will meet with your group once you have completed your role and share the
information that you found with your group. Click on the image below your individual task to
Person 2 Process
Write the answers in your note book so they can be handed in after your group presents.
• You will need internet access in order to complete this WebQuest.
Introduction
You have entered the world of strange and unusual shapes!
and for what purpose?
Go to the resources page to start searching for the answers!
Introduction
Although all of the pictures from the previous slide look different, they are all fractals, which seems like a fitting name for these images of small broken pieces all put together.
role to withstand. You will meet with your group once you have completed your role and share the
information that you found with your group. Click on the image below your individual task to
Person 2 Process
Write the answers in your note book so they can be handed in after your group presents.
分形科普教学课件(一)
分形科普教学课件(一)分形科普教学教学内容•什么是分形•分形的特点和应用•分形的种类和形式教学准备•实物展示：分形图案的打印件或示意图•教学演示软件：例如Fractal Explorer等•白板或投影仪教学目标1.理解分形的概念和特点2.掌握分形的常见种类和形式3.了解分形在科学、艺术、建筑等领域的应用设计说明本节课采用展示、讲解和操作的方式进行分形科普教学。
首先通过实物展示和讲解引入分形的概念和特点，然后通过教学演示软件展示分形的种类和形式，最后讲解分形在各个领域的应用。
教学过程1.引入（5分钟）–展示分形图案的实物或示意图，引起学生的兴趣和好奇心。
–提问：你们是否见过这样的图案？它们有什么特点？2.讲解概念（10分钟）–讲解分形的定义：图形的某个部分与整体具有相似之处，即自相似性。
–引导学生理解自相似性的概念：无论放大还是缩小，图形的细节都保持相似。
–举例说明分形的概念：如科赛雪花、谢尔宾斯基三角形等。
3.展示种类和形式（15分钟）–使用教学演示软件展示常见的分形种类和形式，如曼德布罗集合、茱利亚集合等。
–讲解每种分形的生成原理和特点，与学生进行互动讨论。
4.应用展示（10分钟）–分享分形在科学、艺术、建筑等领域的应用案例，如分形在生物学中的应用、分形艺术品欣赏等。
–引导学生思考：为什么分形在各个领域都有广泛的应用？5.总结和拓展（5分钟）–概括分形的概念、特点和应用。
–提醒学生可以自己通过软件或其他方式制作分形图案，并进行拓展和创新。
课后反思本节课通过实物展示、讲解和教学演示软件的方式，引导学生了解分形的概念、特点和应用。
学生们对分形产生了浓厚的兴趣，积极参与讨论和互动。
但在教学过程中有些学生存在一定的理解困难，下次可以适当增加示意图和例子的数量，加强概念的理解和掌握。
鼓励学生进行拓展和创新，提高课后作业的引导性，促使学生自主学习和实践。
分形科普教学（续）教学内容•什么是分形•分形的特点和应用•分形的种类和形式教学准备•实物展示：分形图案的打印件或示意图•教学演示软件：例如Fractal Explorer等•白板或投影仪教学目标1.理解分形的概念和特点2.掌握分形的常见种类和形式3.了解分形在科学、艺术、建筑等领域的应用设计说明本节课采用展示、讲解和操作的方式进行分形科普教学。
Fractals and L-Systems分形L系统
x + i*y, where i = sqrt( - 1 ) • Iterate on the function Zn = Zn-1 2 + C
• Where Z0 = 0, and Z1 = C2 + C
• What does Z do? It either
• Remains near the origin, or • Eventually goes toward infinity
• Say what?
Self-Similarity and Iteration
• Simple
• Shapes repeat themselves exactly at different scales
• Koch curves and snowflakes
• Statistical
• Statistics of a random geometry are similar at different scales
• Use mathematical iteration and simple logic to produce amazing images
• One kind, you have already seen in the Koch and Sierpinski fractals
• Another kind is more numerical
• Pick equally spaced points and draw lines between them. Then measure the lines
• Now you know the length, right?
• Well, sort of ...
• Where Z0 = 0, and Z1 = C2 + C
• What does Z do? It either
• Remains near the origin, or • Eventually goes toward infinity
• Say what?
Self-Similarity and Iteration
• Simple
• Shapes repeat themselves exactly at different scales
• Koch curves and snowflakes
• Statistical
• Statistics of a random geometry are similar at different scales
• Use mathematical iteration and simple logic to produce amazing images
• One kind, you have already seen in the Koch and Sierpinski fractals
• Another kind is more numerical
• Pick equally spaced points and draw lines between them. Then measure the lines
• Now you know the length, right?
• Well, sort of ...
高考数学选修课课件：数学史选讲 分形概述 (共55张PPT)
数学史选讲-分形概述
分形(fractal)
分形几何理论诞生于20世纪70年代中期， 创始人是美国数学家---曼德布罗特 (B.B.Mandelbrot)，他1982年出 版的 《大自然的分形几何学》 (The Fractal Geometry of Nature)是这一学科经典之作。
分形(fractal)是20多年来科学前沿领域提出的 一个非常重要的概念，
科赫曲线F的自相似维数为
ln 2 dimF ln 3
波兰著名数学家谢尔宾斯基在1915-1916 年期间构造了几个典型的例子， 这些怪物
常称作“谢氏地毯”、“谢氏三角”、“谢 氏海绵” 。如今，讲分形都要提到。它们 不但有趣，而且有助于形象地理解分形。
图3 谢尔宾斯基三角形
分形
将分形看作具有如下性质的集合: 1.F具有精细结构，即在任意小的比例尺度内包含
分形理论已经对方法论和自然观产生强烈影响，从 分形的观点看世界，我们发现，这个世界是以分形 的方式存在和演化着的世界。
分形的特性
英国数学家Falconer在《分形几何的数学基 础及应用》一书中认为：
分形的定义应该以生物学家给出“生命”定 义的类似方法给出，即不寻求分形的确切简 明的定义，而是寻求分形的特性，将分形看 作具有某些性质的集合。
分形几何的历史(续)
发展期：二十世纪八十年代至今. 1. Hutchinson, 1981, 分形与自相似. 给出了自相似集合的数学理论基础. 2. Mandelbrot, 1982, 《自然界的分形几何》. 3. Barnsley, 1988, 《Fractal everywhere》. 4. Falconer, 1990, 《分形几何——数学基础 及其应用》.
分形(fractal)
分形几何理论诞生于20世纪70年代中期， 创始人是美国数学家---曼德布罗特 (B.B.Mandelbrot)，他1982年出 版的 《大自然的分形几何学》 (The Fractal Geometry of Nature)是这一学科经典之作。
分形(fractal)是20多年来科学前沿领域提出的 一个非常重要的概念，
科赫曲线F的自相似维数为
ln 2 dimF ln 3
波兰著名数学家谢尔宾斯基在1915-1916 年期间构造了几个典型的例子， 这些怪物
常称作“谢氏地毯”、“谢氏三角”、“谢 氏海绵” 。如今，讲分形都要提到。它们 不但有趣，而且有助于形象地理解分形。
图3 谢尔宾斯基三角形
分形
将分形看作具有如下性质的集合: 1.F具有精细结构，即在任意小的比例尺度内包含
分形理论已经对方法论和自然观产生强烈影响，从 分形的观点看世界，我们发现，这个世界是以分形 的方式存在和演化着的世界。
分形的特性
英国数学家Falconer在《分形几何的数学基 础及应用》一书中认为：
分形的定义应该以生物学家给出“生命”定 义的类似方法给出，即不寻求分形的确切简 明的定义，而是寻求分形的特性，将分形看 作具有某些性质的集合。
分形几何的历史(续)
发展期：二十世纪八十年代至今. 1. Hutchinson, 1981, 分形与自相似. 给出了自相似集合的数学理论基础. 2. Mandelbrot, 1982, 《自然界的分形几何》. 3. Barnsley, 1988, 《Fractal everywhere》. 4. Falconer, 1990, 《分形几何——数学基础 及其应用》.
fractals - Joe Rainshadow分形-乔rainshadow-PPT精品文档
Star fractals
• The Star Fractal is formed by taking a 5corner star and connecting stars that are 3 times smaller to every corner • If you do the same to every one of the smaller stars and continue it infinitely, you will get a Star Fractal • You can use different shapes to make star fractals also.
Sierpinkis triangle
• it is a mathematically generated pattern that can be reproducible at any magnification or reduction. • Comparing the Sierpinski triangle or the Sierpinski carpet to equivalent repetitive tiling arrangements, it is evident that similar structures can be built into any reptile arrangements. • Also called Sierpinski gasket or the Sierpinski Sieve
Piano curves
• This is a curve that range that contains the entire 2 dimensional square unit or more represented by x dimension discovered by Giuseppe Piano that’s why its called Pianos curve • this curve can transform simple fractals into complex fractals.
分形科普教学课件
分形科普教学课件分形是一种数学形态，展现了自相似的特性。
它们可以在自然界中广泛观察到，如树木的分支结构、云朵的形状和山脉的地形等。
本次科普教学课件将详细介绍分形的概念、特点和应用，并提供相应的例子进行示范。
课件整体流程如下：第一部分：引言和概述1. 引入分形的概念和定义2. 提出分形的重要性和应用领域3. 激发学生对分形的兴趣，引入下一部分第二部分：分形的基本特征与性质1. 自相似性：解释分形的核心特性2. 尺度不变性：解释分形的尺度特性和其意义3. 分形维度：定义分形维度及其计算方法4. 分形的几种经典形状及其描述第三部分：分形的生成方法1. 德国麦德尔布洛特集（Mandelbrot Set）：使用数学公式生成著名分形图形2. 迭代函数系统（IFS）：介绍IFS的原理和应用3. 分形的递归构造方法第四部分：分形的应用领域1. 自然界中的分形：通过例子展示分形在自然界中的存在2. 艺术与设计中的分形：介绍分形在艺术、设计和建筑中的应用3. 数据压缩与编码：解释分形编码的原理和优势4. 分形的科学研究和计算机模拟：介绍分形在科学研究和计算机模拟中的应用第五部分：分形实践与探索1. 分形图形的绘制与生成：教授学生如何使用矢量绘图软件生成分形图形2. 程序编写与交互设计：指导学生使用编程语言编写分形生成程序，并实现交互性设计3. 学生展示与分享：让学生展示他们自己制作的分形图形，并分享经验和感悟第六部分：总结与展望3. 展望分形在未来的发展和应用前景每个环节中会有详细描述，包括相关公式的解释、图形的展示和实例的说明。
可以加入一些交互式环节，如让学生亲自操作生成分形图形或设计分形应用。
这样的教学课件能够帮助学生更好地理解分形的概念和应用，激发他们对数学和科学的兴趣。
Fractales Un nuevo punto de vista分形理论的新观点-42页PPT资料
A. Douady y J.H. Hubbard demostraron que el fractal de Mandelbrot Mo reune en su interior a todos los puntos C a los que corresponde un fractal de Julia conexo y que además, el mismo Mo, era conexo.
Nota: Siento haber entregado este trabajo tan tarde, pero deseo que al menos lo disfrute. Atentamente: Enrique Casielles Lapeira
A continuación vamos a observar cuatro progresiones de imágenes, las cuales son fractales de Mandelbrot y de Julia a los cuales se les ha ido haciendo variaciones en las componentes iniciales de manera que se observa esta propiedad.
En la siguiente progresión veremos una que es bastante famosa, ya que es fácil encontrarla por Internet: los fractales de Julia generados al ir por la parte negativa del eje real asociado al fractal Mo. Los valores son:
Fractals - San Jose State University分形-圣何塞州立大学
float x1, x2, w;
do { x1 = 2.0 * 0.001*(float)(rand()%1000) - 1.0; x2 = 2.0 * 0.001*(float)(rand()%1000) - 1.0; w = x1 * x1 + x2 * x2;
} while ( w >= 1.0 );
• Suppose that the two end-points are a and b. Suppose the height is in the y direction, such that the height at a is y(a), and the height at b is y(b).
• If we have the same terrain-generation algorithm for both mountains and valleys, it will result in unrealistic, alien-looking landscapes.
• Therefore, use different parameters for valleys and mountains.
• Then, the height at the mid-point will be:
ymid = (y(a)+y(b))/2 + r, where
– r is the random offset
• This is how to generate the random offset r:
r = srg|b-a|, where
glPushMatrix(); glTranslatef(0.0,h,0.0); // translate upwards by h glRotatef(30.0,1.0,0.0,0.0); // rotate about the x axis by 30 degrees FractalTree(0.8*r,0.8*h,iter-1); // draw the next iteration fractal tree glPopMatrix();
do { x1 = 2.0 * 0.001*(float)(rand()%1000) - 1.0; x2 = 2.0 * 0.001*(float)(rand()%1000) - 1.0; w = x1 * x1 + x2 * x2;
} while ( w >= 1.0 );
• Suppose that the two end-points are a and b. Suppose the height is in the y direction, such that the height at a is y(a), and the height at b is y(b).
• If we have the same terrain-generation algorithm for both mountains and valleys, it will result in unrealistic, alien-looking landscapes.
• Therefore, use different parameters for valleys and mountains.
• Then, the height at the mid-point will be:
ymid = (y(a)+y(b))/2 + r, where
– r is the random offset
• This is how to generate the random offset r:
r = srg|b-a|, where
glPushMatrix(); glTranslatef(0.0,h,0.0); // translate upwards by h glRotatef(30.0,1.0,0.0,0.0); // rotate about the x axis by 30 degrees FractalTree(0.8*r,0.8*h,iter-1); // draw the next iteration fractal tree glPopMatrix();
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• It is geared towards a fifth grade mathematics class.
• You will need internet access in order to complete this WebQuest.
Introduction
You have entered the world of strange and unusual shapes!
and for what purpose?
Go to the resources page to start searching for the answers!
role to withstand. You will meet with your group once you have completed your role and share the
information that you found with your group. Click on the image below your individual task to
Does anyone have an idea of what these images below are? Or how they were created? We are going to explore pictures like this and others to learn about what they are and what their purpose in math is.
Go to the resources page to start searching for the answers!
Click the button above to go to the resources page.
Person 3 Process
• How are fractals used in mathematics? • Where else are fractals present besides in mathematics? • In what other fields besides math are fractals being used,
• Click
at any
time to return to this
page
• Click
to go to
the next page and
to go back
Start
Overview
• This lesson is a WebQuest about fractals and what they are and how they relate to the world.
view some specific questions that you will have to answer. Write your answers on a
Person 1:
separate piece of paper that will be handed in.
Person 2:
-Your task will be to focus on fractals and what they have to do with math.
By: Leonora Spyros Click to Begin!!
Main Menu
• Introduction • Task • Process:
-Person 1 -Person 2 -Person 3 -Person 4 • Evaluation • Resources
• Teacher’s Page
-Your task will be to focus on the history behind fractals.
Person 3:
-Your task will be to focus on how fractals are used.
Person 4:
-Your task will be to focus on how fractals relate to art.
Introduction
Although all of the pictures from the previous slide look different, they are all fractals, which seems like a fitting name for these images of small broken pieces all put together.
This WebQuest is developed to get a better understanding of what fractals are and what
their purpose is in mathg to the number that you have been assigned in your group, you will all have a different
discovered fractals? Give a brief biography of him and show some examples of his work. • Who are some other mathematicians who have worked with fractals? Give some examples of their work.
Person 2 Process
Write the answers in your note book so they can be handed in after your group presents.
• How long have fractals been around? • Who is the “father of fractals” or in other words, who
• You will need internet access in order to complete this WebQuest.
Introduction
You have entered the world of strange and unusual shapes!
and for what purpose?
Go to the resources page to start searching for the answers!
role to withstand. You will meet with your group once you have completed your role and share the
information that you found with your group. Click on the image below your individual task to
Does anyone have an idea of what these images below are? Or how they were created? We are going to explore pictures like this and others to learn about what they are and what their purpose in math is.
Go to the resources page to start searching for the answers!
Click the button above to go to the resources page.
Person 3 Process
• How are fractals used in mathematics? • Where else are fractals present besides in mathematics? • In what other fields besides math are fractals being used,
• Click
at any
time to return to this
page
• Click
to go to
the next page and
to go back
Start
Overview
• This lesson is a WebQuest about fractals and what they are and how they relate to the world.
view some specific questions that you will have to answer. Write your answers on a
Person 1:
separate piece of paper that will be handed in.
Person 2:
-Your task will be to focus on fractals and what they have to do with math.
By: Leonora Spyros Click to Begin!!
Main Menu
• Introduction • Task • Process:
-Person 1 -Person 2 -Person 3 -Person 4 • Evaluation • Resources
• Teacher’s Page
-Your task will be to focus on the history behind fractals.
Person 3:
-Your task will be to focus on how fractals are used.
Person 4:
-Your task will be to focus on how fractals relate to art.
Introduction
Although all of the pictures from the previous slide look different, they are all fractals, which seems like a fitting name for these images of small broken pieces all put together.
This WebQuest is developed to get a better understanding of what fractals are and what
their purpose is in mathg to the number that you have been assigned in your group, you will all have a different
discovered fractals? Give a brief biography of him and show some examples of his work. • Who are some other mathematicians who have worked with fractals? Give some examples of their work.
Person 2 Process
Write the answers in your note book so they can be handed in after your group presents.
• How long have fractals been around? • Who is the “father of fractals” or in other words, who