Fractals have been one of the tools used in Euclidean geometry to explain the abnormal shapes in nature. Fractals are able to explain the irregular shapes that are a far cry from the normal circle or square. It is an object of symmetry that uses components to create the picture of a self-similar entity. Fractals first appeared on the scene in 1918 due to the mathematician, Felix Hausdroff. A Poland mathematician by the name of Beniot B. Mandelbrot began the term fractals. Fractals originated from the Latin term fractus meaning broken or fractured. It is a series of self-similar images repeated; The Koch snowflake, the Mandelbrot set, the Julia set and the Box fractal are many examples. The idea of a fractal is a pattern of repetitive images of the entire picture. When magnified upon, the image continues to look the same and builds upon the whole picture. “A key characteristic of fractals is fractal dimension.” [http://www.reference.com/browse/wiki/Fractal] This is the parameter of the fractal that uses fractions or nonintergers. “The table below shows the complexity of a figure as it increases its dimension.” [http://library.thinkquest.org/26242/full/index.html.] FA finite number greater than 0

IAn infinite number
DimensionNum of PointsLengthAreaVolume
D = 0F000
0 < D < 1I000
D = 1IF00
1 < D < 2II00
D = 2IIF0
2 < D < 3III0
D = 3IIIF
[http://library.thinkquest.org/26242/full/index.html.]
Fractals are most often found in nature where the patterns aren’t exactly self-similar. These fractals are known as stochastic. This can be seen in tree bark, leaves, and snowflakes. All fractals aren’t exactly self-similar, such as the Julia set, or stochastic but are random or statistical. These fractals involve a numerical measure that is “preserved across the scale.” [http://www.reference.com/browse/wiki/Fractal] Fractals are used in an assortment of fields such as computer programming, art, nature, astronomy, molecules, and the...

...Fractals
Introduction
Fractals are geometric patterns that when repeated at increasingly smaller scales they produce irregular shapes and surfaces. All fractals have a feature of ‘self-similarity’. A set is self-similar if it can be broken into arbitrary small pieces, each of which is a small copy of the entire set, for fractals the pattern reproduced must be detailed (Nuhfer 2006). Self-similarity may be demonstrated as exact self-similarity meaning the fractal is identical at all scales a fractal that demonstrates exact self-similarity is the Koch Snowflake. Other fractals exhibit quasi self-similarity. This is when fractals approximate the same pattern at different scales, they contain small copies of the entire fractal in altered or degenerate forms, and an example of this is the Mandelbrot set (Fractal 2009). Also, fractal curves are ‘nowhere differentiable’ meaning that the gradient of the curve can never be found; because of this fractals cannot be measured in traditional ways (Turner 1998). I find it interesting to note that many phenomena in nature have fractal features including clouds, mountains, fault lines and coastlines. There are also a range of mathematical structures that are fractals including, Sierpinski triangle, Koch snowflake, Peano curve and the...

...Fractal Geometry
"Fractal Geometry is not just a chapter of mathematics, but one that helps
Everyman to see the same old world differently". - Benoit Mandelbrot
The world of mathematics usually tends to be thought of as abstract. Complex and
imaginary numbers, real numbers, logarithms, functions, some tangible and others
imperceivable. But these abstract numbers, simply symbols that conjure an image,
a quantity, in our mind, and complex equations, take on a new meaning with
fractals - a concrete one. Fractals go from being very simple equations on a
piece of paper to colorful, extraordinary images, and most of all, offer an
explanation to things. The importance of fractal geometry is that it provides an
answer, a comprehension, to nature, the world, and the universe. Fractals occur
in swirls of scum on the surface of moving water, the jagged edges of mountains,
ferns, tree trunks, and canyons. They can be used to model the growth of cities,
detail medical procedures and parts of the human body, create amazing computer
graphics, and compress digital images. Fractals are about us, and our existence,
and they are present in every mathematical law that governs the universe. Thus,
fractal geometry can be applied to a diverse palette of subjects in life, and
science - the physical, the abstract, and the natural.
We were all astounded by the sudden...

...beautiful. In fact, there are some patterns that are so perfect that they self-replicate. To be technical, some patterns are fractal in nature. Fractal or not, patterns give us something more to admire and wonder about.
Introduction
Fractals never fail to fascinate. If you aren't just gazing at their unearthly beauty, you ponder the mathematics behind them... and then you can't help but wonder how such prosaic, unsensational mathematical formulae can give rise to such intricacy. What is it that makes it possible for (to some) a short, ugly equation to generate the exuberant beauty of the Mandelbrot set? Or is it all just in the way our brains are wired?
Fractals are objects with infinite lengths that occupy finite volumes, resulting in a "fractional dimension" that is not 1-, 2-, or 3-D, but a combination of all three, depending on its spatial configuration.
The Koch snowflake is the repetitive procedure of dividing the image into three equal parts and replacing the middle piece with two similar pieces.
Hypothesis
Fractals mimic nature. (true or false)
This is the basic belief of fractals, and a common concept among those who study fractals. In nature, symmetry is often remarked upon. To mimic is to be similar in to a certain object, and in this case, of a lesser proportion. Thus, we would like to propose that fractals may mimic nature.
Definitions...

...1. Introduction
The birth of every technology is the result of the quest for automation of some form of human work. This has led to many inventions that have made life easier for us. Fractal Robot is a science that promises to revolutionize technology in a way that has never been witnessed before.
The principle behind Fractal Robots is very simple. You take some cubic bricks made of metals and plastics, motorize them, put some electronics inside them and control them with a computer and you get machines that can change shape from one object to another. Almost immediately, you can now build a home in a matter of minutes if you had enough bricks and instruct the bricks to shuffle around and make a house! It is exactly like kids playing with Lego bricks and making a toy hose or a toy bridge by snapping together Lego bricks-except now we are using computer and all the work is done under total computer control. No manual intervention is required. Fractal Robots are the hardware equivalent of computer software.
1. What are Fractals?
A fractal is anything which has a substantial measure of exact or statistical self-similarity. Wherever you look at any part of its body it will be similar to the whole object.
2. Fractal Robots
A Fractal Robot physically resembles itself according to the definition above. The robot can be animated around its joints in a...

...Fractal Geometry
How would you like to take a class called geometry of chaos? Probably doesn’t sound too thrilling. A man named Benoit Mandelbrot is responsible for creating the geometry of chaos. The geometry of chaos is considered to be the fourth-dimension. It is considered to be the world in which we live in, a world where there is constant change based on feedback, an open system where everything is related to everything else. It is now recognized as the true geometry of nature. The geometric system the can describe the simple shapes of the world (Lauwerier).
Fractal geometry is a structure that provided a new key for the study of non-linear processes (Lauwerier). Benoit Mandelbrot explained that lines have a single dimension, plane figures have two dimensions and that we live in a three dimensional spatial world (Fractals Useful Beauty). In a paper published in 1967, Mandelbrot investigated the idea of measuring the length of a coastline. Mandelbrot explained that the shape of a coastline defies conventional Euclidean geometry and that rather than having a natural number dimension, it has a “fractional dimension.” The coastline is an example of a self-similar shape, which is a shape that repeats itself over and over on different scales (Fractals).
Benoit Mandelbrot was born in Warsaw in 1924 to a Lithuanian Jewish family and grew up there until they moved to Paris in 1936 (Fractals). Benoit had...

...SEMINAR FINAL DRAFT
YEAR: 2008
TOPIC: Adaptive rethinking along with fractal architecture as one of the defining solutions for the contemporary complex urban fabric.
SUBMITTED BY: Lily Tandon
SEMESTER:VIII
CHAPTERS
1. The End Of The Modern World
2. What abstraction does
3. From the modern to the complex
4. From complexity to form generation
5. Form generation and fractals
6. Fractals
7. Conclusions and findings
8. Methodology
TOPIC: Adaptive rethinking along with fractal architecture as one of the defining solutions for the contemporary complex urban fabric.
Why do we need to do this research?
New technologies (here complexity sciences) attract architectural response.
The need is to look one step ahead of modernism & the need is to produce architecture of our time. Hence, the complex urban fabric shall attempt to be more than what Modernism proposes a step ahead of what people are sticking to right now –as- plain abstraction of forms; that architecture is something more than a play of forms, should be evident from the experiences of our daily life, where architecture participates in most activities.
The intention of this paper can...

...Fractal Antenna Engineering: The Theory and
Design of Fractal Antenna Arrays
Douglas H. Werner', Randy L. Haup?,
and Pingjuan L. WerneJ
'Communications and Space Sciences Laboratory
The Pennsylvania State University
Department of Electrical Engineering
21 1A Electrical Engineering East
University Park, PA 16802
E-mail: dhw@psu.edu
2Department of Electrical Engineering
Utah State University
Logan, UT 84322-4 120
Tel: (435) 797-2840
Fax: (435) 797-3054
E-mail: randy.haupt@ece.usu.edu or haupt@ieee.org
3The Pennsylvania State University
College of Engineering
DuBois, PA 15801
E-mail: plw7@psu.edu
Keywords: Fractals; antenna arrays; antenna theory; antenna
radiation patterns; frequency-independent antennas; log-periodic
antennas; low-sidelobe antennas; array thinning; array signal
processing
1. Abstract
A fractal is a recursively generated object having a fractional
dimension. Many objects, including antennas, can be designed
using the recursive nature of a fractal. In this article, we provide a
comprehensive overview of recent developments in the field of
fractal antenna engineering, with particular emphasis placed on the
theory and design of fractal arrays. We introduce some important
properties of fractal arrays, including the frequency-independent
multi-band characteristics, schemes for realizing low-sidelobe
designs,...

...The Application of Fractal Geometry to Ecology
Principles of Ecology 310L
Victoria Levin
7 December 1995
Abstract
New insights into the natural world are just a few of the results from the use
of fractal geometry. Examples from population and landscape ecology are used to
illustrate the usefulness of fractal geometry to the field of ecology. The
advent of the computer age played an important role in the development and
acceptance offractal geometry as a valid new discipline. New insights gained
from the application of fractal geometry to ecology include: understanding the
importance of spatial and temporal scales; the relationship between landscape
structure and movement pathways; an increased understanding of landscape
structures; and the ability to more accurately model landscapes and ecosystems.
Using fractal dimensions allows ecologists to map animal pathways without
creating an unmanageable deluge of information. Computer simulations of
landscapes provide useful models for gaining new insights into the coexistence
of species. Although many ecologists have found fractal geometry to be an
extremely useful tool, not all concur. With all the new insights gained through
the appropriate application of fractal geometry to natural sciences, it is clear
that fractal geometry a useful and valid tool.
New insight into the natural world is just one...