分形几何Fractal Geometry(Ch1)

In recent years this attitude has changed. Moreover, irregular sets provide a much better representation of many natural phenomena than do the figures of classical geometry. Fractal geometry provides a general framework for the study of such irregular sets.
In all his examples, Richardson found the points were well-fit by a straight line with slopes s = -0.25 for the west coast of Britain, one of the roughest in the atlas, s = -0.15 for the land frontier of Germany, s = -0.14 for the land frontier of Portugal, s = -0.02 for the South African coast, one of the smoothest in the atlas.
Lightning
Photograph of a Romanesco broccoli, showing a naturally occuring fractal
Sea shells
• The nautilus is one of the most famous examples of a fractal in nature. The perfect pattern is called a Fibonacci spiral.
The left graph shows his data for the west coast of Britain. Richardson observed that as the scale is reduced, the length increases rapidly. This and the other graphs looked like power laws, so Richardson plotted Log(length) vs Log(scale), as shown in the right figure.
Measuring the Coast of Britain
If above picture does not convince you, click here for another NASA photograph.
y sin x
A smaller measuring scale will detect more detail of the coastline, hence give a greater length. Self-similarity of coastlines casts doubt on the hope that these measurements will converge as smaller scales are used.
Lectures in Modern Mathematics
——Fractal Geometry
College of Mathematics and Computational Science Shen Chen
Chapter 1 Introduction
In the past, mathematics has been concerned largely with sets and functions to which the methods of classical calculus can be applied. Sets or functions that are not sufficiently smooth or regular have tended to be ignored as ‘pathological’ and not worthy of study.
Here is a picture from NASA's website.
y sin x
Two polygonal approximations of the coastline
y sin x
The smaller measuring scale (on the right) picks up more detail, and gives a larger length for the coastline.
Snowflake
Clouds Look outside your window - you may see a fractal cloud at any moment.
Crystals: Both chemicallyformed crystals and ice and frost crystals are breathtaking examples of fractals in nature.
The standard approach to measuring lengths of curves is to approximate the curve by straight line segments and add up the lengths of the segments.
Problems with applying standard measurement to coastlines Even a moment's reflection reveals a problem with the standard approach: a smaller measuring scale is sensitive to more details.
Vortex street in the atmosphere
Rivers and fjords
F r o m t h e midwest of the United States to the icy fjords of Norway, fractals are often viewed by airline passengers.
For the moment, assume the number N(d) of segments of length d needed to walk across the coastline is proportional to 1/dD, for some exponent D. That is, there is a constant M for which N(d) = M/dD = M· d-D and so L(d) = N(d)· d = M· d-D· d = M· d1-D . This implies Log(L(d)) = (1-D)Log(d) + Log(M) —— the equation of a straight line with slope s = 1 - D.
They’re famously found in nature. Fractals are purely a wonder - too irregular for Euclidean geometry; iterative and recursive and seemingly infinite. They turn up in food and germs, plants and animals, mountains and water and sky. Here are some examples:
For the coast of Australia and of SBaidu Nhomakorabeauth Africa, the west coast of Britain, and the land frontiers of Germany and Portugal, Richardson measured the lengths L(d) at different scales d. For most coastlines, measured length appears to increase rapidly as the measuring scale is reduced.
For circles and other smooth curves, for small enough scale the points fall on a line of slope 0.
Consequently, Richardson showed that most coastlines differ significantly from smooth curves. But what does this slope tell us? Mandelbrot's interpretation of the slope as a dimension In 1967 Mandelbrot, B. B. ("How long is the coast of Britain? Statistical self-similarity and fractional dimension," Science 156 (1967), 636-638.) interpreted the slope in Richardson's measurements.
If the data points lie along a straight line, then the assumption N(d) = M/dD is justified and the exponent D is the fractal dimension of the coastline. Richardson's data then can be interpreted as estimating the dimension D of the coastlines: D = 1.25 for the west coast of Britain D = 1.15 for the land frontier of Germany, D = 1.14 for the land frontier of Portugal, D = 1.13 for the Australian coast, and D = 1.02 for the South African coast.
We begin by looking briefly at a number of simple examples of fractals.
What are fractals and how to measure them?
1.Fractals in Nature(Natural Fractals )
Shorelines
How Long Is a Coastline ?
Coastline are natural fractals, among the first recognized. We describe the complications that arise from trying to answer the simple question, "How long is a coastline?"
Mountain ranges
A photograph of a
mountain range
One of Musgrave's multifractal forgeries
Ken Musgrave has produced some of the most convincing fractal landscape pictures