This page is a resource for the mathematical concept of
fractals, or fractal geometry. So much information is available
on the topic that rather than attempt to explain the concept
here, I aim to provide links to various starting points for those
who want to know more about fractals. The applications of
fractal concepts in art and music are given their own pages.
Fractal Geometry (The Story of Benoit B. Mandelbrot and the
Geometry of Chaos)
"The story of Chaos begins in number, specifically in the
mathematics and geometry of the fourth dimension."
The basic fractal property of self-similarity at varying scales of
magnitude, the concepts of iteration and recursiveness, the
contributions of the mathematicians Lorenz and Keyserling, the
science and philosophy of chaos, its extension into New Age
spiritual concepts, and more..
"Wherever we look in nature we find fractals with self similarity
over scales. It is in every snow flake, every bolt of lightening,
every tree, every branch; it is even in our very blood with its
veins, and in our Galaxies with their clusters."
(Fractal Geometry links continue below...)
Featured piece: Sufi (304C)
Fractal Geometry (Crystalinks)
Fractal geometry discussed as part of "Sacred Geometry":
"Sacred geometry is geometry that is sacred to the observer or discoverer. This meaning is sometimes
described as being the language of the God of the religion of the people who discovered or used it. Sacred
geometry can be described as attributing a religious or cultural value to the graphical representation of the
mathematical relationships and the design of the man-made objects that symbolize or represent these
Fractal Geometry (Yale University course)
Benoit Mandelbrot himself is one of the co-authors of this class.
This is one of the best online resources about fractals, and is "meant to support a first course in fractal
geometry for students without especially strong mathematical preparation." The site is incredibly deep,
consisting of over 100 sections in the course alone under 13 main topics, and with a "Panorama" section which
covers the application of fractal concepts in nearly a hundred areas from "African architecture" to "Wuorinen."
Fractal (Wikipedia, the free encyclopedia)
The wonder that is Wikipedia is at its best here:
CONTENTS: 1 History; 2 Examples; 2.1 The fractional dimension of the boundary of the Koch snowflake;
2.2 Generating fractals; 2.3 Classification of fractals; 3 Fractals in nature; 4 Applications; 5 See also;
6 References; 7 External links; 7.1 Multiplatform generator programs; 7.2 Linux generator programs;
7.3 Windows generator programs; 7.4 Mac generator programs; 7.5 MorphOS generator programs;
7.6 Fractal Art Galleries
Cynthia Lanius' Lessons: A Fractals Lesson - Introduction
A Fractals Unit for Elementary and Middle School Students - That Adults are Free to Enjoy
"They're everywhere, those bright, weird, beautiful shapes called fractals. But what are they, really?
Fractals are geometric figures, just like rectangles, circles and squares, but fractals have special properties that
those figures do not have.
There's lots of information on the Web about fractals, but most of it is either just pretty pictures or very
high-level mathematics. So this fractals site is for kids, to help them understand what the weird pictures are all
about - that it's math - and that it's fun! "
Chaos Theory and Fractal Geometry
This site has been designed to offer educators a course in Chaos Theory and Fractal Geometry. The course
encompasses the following topics: The mathematics and history of Chaos Theory; Iterations and recursions;
Dynamical systems and how they relate to real world situations; Graphing non-linear equations and the creation
of strange attractors; The mathematics of fractal, including fractal generators for students to create their own
fractals; Mandelbrot and Julia sets; Measurement and Scale
"In this document, we discuss briefly the basic ideas of fractal geometry. We begin with a brief introduction to the
ideas involved in a so-called ``fractal dimension'', and discuss methods for computing this quantity. Standard
examples are used to illustrate the ideas. Several current areas of research in the application of fractal
geometry are then discussed. Finally, we close with a discussion of what may be the most well-known examples
of fractals, namely the Julia and Mandelbrot sets."
Fractal (from Wolfram MathWorld)
Wolfram MathWorld bills itself as "the world's most extensive mathematics resource." (As of 07/20/2006, there
were 12,610 entries.) Links to over 50 related math topics, plus nearly 40 references both printed and online.
"A fractal is an object or quantity that displays self-similarity, in a somewhat technical sense, on all scales. The
object need not exhibit exactly the same structure at all scales, but the same "type" of structures must appear on
all scales. 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. The prototypical example for a fractal is the length of a coastline measured with
different length rulers. The shorter the ruler, the longer the length measured, a paradox known as the coastline
Welcome to Fractal World!
"On this web page, you will be introduced to the mysterious world of fractal geometry. You have the choice to
learn about three major topics of fractal (see chart below).By clicking on "What is a Fractal?", you will find basic
information on the formulas and origins of fractal geometry. Under "What is a Fractal? 2", I've included more
complicated information like the Mandelbrot function, and imaginary numbers. Click on "Fractal Links", and you
will find a list of other fractal related sites across the Internet. Finally, view some very appealing computer
generated fractals under "Sample Pictures". So, prepare to be shocked as you enter the great Fractal World!"
"Fractal World" is a web page created as a project for Mr. Gordon's Honors Algebra 2 class at Lock Haven (Pa)