## Which Mathematician Was A Pioneer In The Study Of Fractals?

Benoit Mandelbrot, whose pioneering work on fractal geometry made him one of the few modern mathematicians to approach widespread fame, died October 14 at the age of 85.

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Contents

- 1 Who is the pioneer of fractals?
- 2 Who is the mathematician known for fractals?
- 3 Who was the first person who used the word fractal in 1975?
- 4 Who is the father of geometry?
- 5 What are 3 well known fractals?
- 6 What are the 4 types of fractals?
- 7 What famous paintings have fractals?
- 8 Who introduced the theory of fractals and published the fractal geometry of nature in 1982?
- 9 What is the most famous fractal?
- 10 What are 3 well known fractals?

## Who is the pioneer of fractals?

Remembering the father of fractals By Carrie Arnold, Inside Science News Service The late mathematician Benoit Mandelbrot. Benoit Mandelbrot, the mathematics professor at Yale who coined the word “fractal,” passed away on October 14 at the age of 85. His death recalls the complicated history of his life’s work – the details of which, like fractals themselves, depend on how closely one looks. Some varieties of cauliflower are fractals. Fractals are geometric shapes that can be broken down into smaller parts, each of which resembles the whole – like broccoli florets or branches on a bolt of lightning. Mandelbrot first developed the mathematics behind fractals in order to answer a simple question: how long is the coastline of Britain? He imagined measuring the coastline by laying yardsticks end-to-end around the perimeter of the island and counting how many would be needed to encircle it.

- He then imagined repeating the same process but measuring with a stick only 2 inches long.
- This second measurement would be longer because a shorter stick can measure smaller indentations in the coastline.
- Mandelbrot’s revelation, published in a 1967 paper, was this: You can’t accurately measure the coastline.

Its length depends on how closely you look. Out of this paradox, he created a new way of looking at mathematically-difficult phenomena that researchers have continued to explore and develop. Fractals repeat the same pattern over and over again on ever-smaller scales. Credit: Nevit Dilmen “You get this mysterious combination of variability and organization that is mathematically describable,” said Goldberger. Goldberger uses fractals to help define a healthy human heartbeat.

- Physicians once thought that a healthy heartbeat should be as steady as a metronome, but heart traces, or EKGs, have revealed that healthy hearts are actually much more irregular.
- Instead of a metronome, Goldberger said, they’re more like a symphony.
- Irregularity, he said, “is where physiology meets fractality.” He believes that in order to adapt to the environment, our bodies can’t be locked into one mode of functioning.

Variability in heartbeats, he said, is essential to life and repeats itself across different scales. The peaks and valleys of an EKG look the same over 10 minutes as they do over 10 milliseconds. This similarity of fractals over different timescales also exists in patterns found in landscapes and geography.

That’s why photographs of rocky terrain often contain a reference object, such as a Swiss Army knife, to provide a sense of scale. Without the knife, it would be impossible to tell the size of the rock. “That’s what ‘scale invariance’ is,” said geologist Donald Turcotte of the University of California, Davis.

“Everything looks the same,” and you can’t tell whether you are looking at a one square centimeter of rock or a one kilometer landscape, he said. Besides describing the appearance of the earth’s surface, the mathematics of fractals also help scientists to better predict the frequency of earthquakes, floods, and other natural disasters.

- Mathematical models that don’t use fractals tend to forecast far fewer major natural disasters than actually occur, said Turcotte.
- By using a fractal models, however, geologists have been able to obtain more accurate predictions of the frequency and severity of such natural disasters.
- Yahya Rahmat-Samii, an electrical engineer at the University of California, Los Angeles, uses fractals to improve cell phones’ ability to pick up signals.

Mobile phone antenna once picked up only one radio wave frequency. In order to pick up rather faint signals, the antenna itself had to be rather large. In the mid-1990s, though, engineers discovered that bending an antenna into a fractal-like shape enabled a miniature antenna to pick up an array of signals.

Each portion of the fractal could be designed to pick up a different frequency, which has allowed cell phone companies to provide Bluetooth and Wi-Fi capabilities (all of which operate at different radio frequencies) on the same phone. The next step in fractal antenna design, said Rahmat-Samii, is continued miniaturization.

“There’s a lot of room for miniaturization,” Rahmat-Samii said, “because fractals come with so many different features, and there are likely some that have not been exploited effectively.” All three of these researchers have met Benoit Mandelbrot, and all three describe the late mathematician as having had a defining influence on their life’s work.

They also acknowledge that he was a character who at times could be as complicated as his mathematics. “A lot of people didn’t like him,” Turcotte said. “He was extremely arrogant and a bit prickly. But he introduced this concept in the 1960s, and he didn’t receive recognition for another 20 years. So if you’ve been fighting a battle for 20 years, I think you have the right to resent things a bit.” Provided by Inside Science News Service Citation : Remembering the father of fractals (2010, October 25) retrieved 25 April 2023 from https://phys.org/news/2010-10-father-fractals.html This document is subject to copyright.

Apart from any fair dealing for the purpose of private study or research, no part may be reproduced without the written permission. The content is provided for information purposes only. : Remembering the father of fractals

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## Who is the mathematician known for fractals?

Mandelbrot set zoom – Watch for the repeating structures contained in the Mandelbrot set. Instantly, Mandelbrot knew he was onto something. He saw unquestionably organic structures in the details of this shape and quickly published his findings. This shape and structure, later known as the Mandelbrot set, was an extraordinarily complex and beautiful example of a “fractal” object, fractal being the name coined by Mandelbrot in 1975 to describe such repeating or self-similar mathematical patterns.

- But it wasn’t until his 1982 book, The Fractal Geometry of Nature, that Mandelbrot would receive public attention and widespread legitimacy.
- In this book, Mandelbrot highlighted the many occurrences of fractal objects in nature.
- The most basic example he gave was a tree.
- Each split in a tree—from trunk to limb to branch and so forth—was remarkably similar, he noted, yet with subtle differences that provided increasing detail, complexity and insight into the inner-workings of the tree as a whole.

True to his academic roots, Mandelbrot went beyond identifying these natural instances and presented the sound mathematical theories and principles upon which his newly coined “fractal geometry” was based. What emerged was a geometry of the cosmos—one that broke all Euclidean laws of the man-made world and deferred to the properties of the natural world.

- If one identified an essential structure in nature, Mandelbrot claimed, the concepts of fractal geometry could be applied to understand its component parts and make postulations about what it will become in the future.
- This new way of viewing our surroundings, this new perception of reality, has since led to a number of remarkable discoveries about the worlds of nature and man, and has shown that they are not as disconnected as once thought.

Take biology, for example. Fractal patterns have appeared in almost all of the physiological processes within our bodies. For ages, the human heart was believed to beat in a regular, linear fashion, but recent studies have shown that the true rhythm of a healthy heart fluctuates radically in a distinctively fractal pattern.

Blood is also distributed throughout the body in a fractal manner. Researchers in Toronto are using ultrasound imaging to identify the fractal characteristics of blood flows in both healthy and diseased kidneys. The hope is to measure the fractal dimensions of these blood flows and use mathematical models to detect cancerous cell formations sooner than ever before.

In the fractal approach, doctors won’t need sharper medical images or more powerful machines to see these miniscule pre-cancerous structures. Math, rather than microscopes, will provide the earliest detection. Biology and healthcare are only some of the latest applications of fractal geometry.

- The developments arising from the Mandelbrot set have been as diverse as the alluring shapes it generates.
- Fractal-based antennas that pick up the widest range of known frequencies are now used in many wireless devices.
- Graphic design and image editing programs use fractals to create beautifully complex landscapes and life-like special effects.

And fractal statistical analyses of forests can measure and quantify how much carbon dioxide the world can safely process. Today, we have merely scratched the surface of what fractal geometry can teach us. Weather patterns, stock market price variations and galaxy clusters have all proven to be fractal in nature, but what will we do with this insight? Where will the rabbit hole take us? The possibilities, like the Mandelbrot set, are infinite.

- Benoit Mandelbrot was an intellectual jack-of-all-trades.
- While he will always be known for his discovery of fractal geometry, Mandelbrot should also be recognized for bridging the gap between art and mathematics, and showing that these two worlds are not mutually exclusive.
- His creative approach to complex problem solving has inspired peers, colleagues and students alike, and instilled in IBM a strong belief in the power of perspective.

Decades after his discovery of the Mandelbrot set, data visualization continues to provide fresh and unexpected insights into some of the world’s most difficult problems by altering our perspective, challenging our preconceptions and revealing connections previously invisible to the eye.

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#### Who defined fractals?

The term ‘fractal’ was coined by Benoit Mandelbrot in 1975. It comes from the Latin fractus, meaning an irregular surface like that of a broken stone. Fractals are the kind of shapes we see in nature.

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## Who was the first person who used the word fractal in 1975?

The term ‘fractal’ was coined by the mathematician Benoît Mandelbrot in 1975. Mandelbrot based it on the Latin frāctus, meaning ‘broken’ or ‘fractured’, and used it to extend the concept of theoretical fractional dimensions to geometric patterns in nature.

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#### What are 2 famous artists known to use fractals?

Artists – Notable fractal artists include Desmond Paul Henry, Hamid Naderi Yeganeh, and musician Bruno Degazio, British artists include William Latham, who has used fractal geometry and other computer graphics techniques in his works. and Vienna Forrester who creates flame fractal art using data extracted from her photographs.

Greg Sams has used fractal designs in postcards, T-shirts, and textiles. American Vicky Brago-Mitchell has created fractal art which has appeared in exhibitions and on magazine covers. Scott Draves is credited with inventing flame fractals. Carlos Ginzburg has explored fractal art and developed a concept called “homo fractalus” which is based around the idea that the human is the ultimate fractal.

Merrin Parkers from New Zealand specialises in fractal art. Kerry Mitchell wrote a “Fractal Art Manifesto”, claiming that. In Italy, the artist Giorgio Orefice wrote the “Fractalism” manifesto, founding a Fractalism cultural mouvement in 1999. Fractal Art is a subclass of two-dimensional visual art, and is in many respects similar to photography—another art form that was greeted by skepticism upon its arrival.

Fractal images typically are manifested as prints, bringing fractal artists into the company of painters, photographers, and printmakers. Fractals exist natively as electronic images. This is a format that traditional visual artists are quickly embracing, bringing them into Fractal Art’s digital realm.

Generating fractals can be an artistic endeavor, a mathematical pursuit, or just a soothing diversion. However, Fractal Art is clearly distinguished from other digital activities by what it is, and by what it is not. According to Mitchell, fractal art is not computerized art, lacking in rules, unpredictable, nor something that any person with access to a computer can do well.

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### Who is considered to be the father of fractals and where was he born?

Benoit Mandelbrot, (born November 20, 1924, Warsaw, Poland—died October 14, 2010, Cambridge, Massachusetts, U.S.), Polish-born French American mathematician universally known as the father of fractals, Fractals have been employed to describe diverse behaviour in economics, finance, the stock market, astronomy, and computer science,

- Mandelbrot was educated at the École Polytechnique (1945–47) in Paris and at the California Institute of Technology (1947–49).
- He studied for a doctorate in Paris between 1949 and 1952 and then did research for a year under John von Neumann at the Institute for Advanced Study in Princeton, New Jersey,

From 1958 to 1993 he worked for IBM at its Thomas J. Watson Research Center in New York, becoming a research fellow there in 1974. From 1987 he taught at Yale University, becoming the Sterling Professor of Mathematical Sciences in 1999. Britannica Quiz Numbers and Mathematics As set out in his highly successful book The Fractal Geometry of Nature (1982) and in many articles, Mandelbrot’s work is a stimulating mixture of conjecture and observation, both into mathematical processes and their occurrence in nature and in economics.

- In 1980 he proposed that a certain set governs the behaviour of some iterative processes in mathematics that are easy to define but have remarkably subtle properties.
- He produced detailed evidence in support of precise conjectures about this set and helped to generate a substantial and continuing interest in the subject.

Many of these conjectures have since been proved by others. The set, now called the Mandelbrot set, has the characteristic properties of a fractal: it is very far from being “smooth,” and small regions in the set look like smaller-scale copies of the whole set (a property called self-similarity).

- Mandelbrot’s innovative work with computer graphics stimulated a whole new use of computers in mathematics.
- Mandelbrot won a number of awards and honorary degrees.
- He became a Fellow of the American Academy of Arts and Sciences in 1982 and of the National Academy of Sciences in 1987.
- He was awarded the Wolf Foundation Prize for Physics in 1993 for his work on fractals, and in 2003 he shared the Japan Prize of the Science and Technology Foundation of Japan for “a substantial contribution to the advance of science and technology.” Mandelbrot’s memoir, The Fractalist, was published posthumously in 2012.

This article was most recently revised and updated by John M. Cunningham,

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### Which mathematician is best known for developing what is known as fractal geometry?

While researching properties of turbulence over telephone lines at the IBM Thomas J. Watson Research Center in Yorktown Heights, NY, Benoit Mandelbrot discovered the principles that would later form the new field of fractal geometry. This discovery made it possible, for the first time in the history of mankind, to describe nature with math. These are Benoit Mandelbrot’s words.

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### Why do mathematicians study fractals?

Beautiful math of fractals (PhysOrg.com) – What do mountains, broccoli and the stock market have in common? The answer to that question may best be explained by fractals, the branch of geometry that explains irregular shapes and processes, ranging from the zigs and zags of coastline to Wall Street market risk.

Because fractal geometry is relatively new – the term was coined in 1975 by the late Benoit Mandelbrot, – it is a concept not well understood by a portion of the population. Casey Donoven, one of Montana State University’s newest recipients of the prestigious Goldwater Scholarship for excellence in science and math, uses in his research to understand variations in heartbeats.

Donoven, who hails from a family farm outside Kremlin-Gilford, first learned about fractals while a student at Havre High School. At MSU, he has been studying under math professor Lukas Geyer. What is a fractal? A fractal is a geometric pattern that repeats at every level of magnification.

Another way to explain it might be to use Mandelbrot’s own definition that “a fractal is a geometric shape that can be separated into parts, each of which is a reduced-scale version of the whole.” Think of Russian nesting dolls. Fractals are common in nature and are found nearly everywhere. An example is broccoli.

Every branch of broccoli looks just like its parent stalk. The surface of the lining of your lungs has a fractal pattern that allows for more oxygen to be absorbed. Such complex real-world processes can be expressed in equations through fractal, Even to the everyday person, fractals are generally neat to look at even if you don’t understand what a fractal is.

- But to a mathematician, it is a neat, neat subject area.
- Why are fractals important? Fractals help us study and understand important scientific concepts, such as the way bacteria grow, patterns in freezing water (snowflakes) and brain waves, for example.
- Their formulas have made possible many scientific breakthroughs.

Wireless cell phone antennas use a fractal pattern to pick up the signals better, and pick up a wider range of signals, rather than a simple antenna. Anything with a rhythm or pattern has a chance of being very fractal-like. Why do we hear so much about fractals now? Actually, some fractals were understood long before Mandelbrot coined the term.

He popularized the concept with computer graphics and pictures of fractal patterns in nature. While, the Mandelbrot set and Julia sets (two well-known fractals) were investigated in the early 20th century, they never left the mathematical/physical “ghetto” until fast computers and good computer graphics came along, which in turn led to a wave of new research and better understanding.

More information: If you would like to learn more about fractals, an easy place to begin is: Citation : Beautiful math of fractals (2011, October 13) retrieved 25 April 2023 from https://phys.org/news/2011-10-beautiful-math-fractals.html This document is subject to copyright.

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### What inspired him to discover fractals?

Fractals There is no doubt that just about everyone on this planet has seen a fractal in their lifetime, even though many do not even know it. Every natural thing on our planet can be described in mathematical terms. This is where the subject of fractals comes into play.

Fractals occur everywhere in nature and can sometimes be depicted as forming the core of our lives. Being able to comprehend the structure of a mathematical fractal opens up the ability to understand how everything in this world was formed. A fractal is defined to be a rough or fragmented shape that can be broken up into smaller parts, which can be seen as a smaller copy of the original shape.

It is almost impossible to describe the natural things in our world, such as the clouds, trees, plants and so on, as geometric images. This is why fractals are such an important part of how nature is structured. This history of fractals is quite interesting, considering that the “father” of fractals has just passed away a few weeks ago.

- Also looking at different kinds of sets that describe fractals is of great importance in the overall application of them.
- Finally, I will write about the computer programming of fractals and how to create them on your own.
- Benoit Mandelbrot is considered to be the father of fractal geometry.
- He has said that the first thing that made him start to even think about the idea of fractals was when he was trying to figure out how long the coast of Britain was.

What he discovered was that if you look at a map and keep on zooming in on it, repeated patterns will appear. The idea that he used to get the most accurate measure of the coastline of Britain was determined by what length of ruler he would use. He showed that smaller rulers are more accurate because they can fit better into the irregular patterns of the coast, rather than using one large ruler.

- He concluded that as the scale of measurement he used decreased in size, the actual length of the coastline increased,
- This shows that we can zoom into the coastline an infinite number of times, using a smaller unit of measurement and keep getting a more accurate estimate.
- Mandelbrot always said to not think about what you see, but what it took to make what you see.

“The key to fractal geometryis that if you look on the surface, you see complexity and it looks very non-mathematical”, His studies about the Britain coastline lead into one of the main ideas of fractals, known as self-similarity. “A set S is called self-similar if S can be subdivided into k congruent subsets, each of which can be magnified by a constant factor M to yield the whole set S”,

By looking at the coastline of Britain from a far distance and then zooming up extremely close, the images would look similar. Self-similarity is one huge principal idea when classifying what a fractal is. Although Mandelbrot was the one to coin the term “fractal geometry” in 1975, there were many mathematicians before his time that noticed this property of self-similarity.

A simple start to understanding the formation of fractals is to look at the Sierpinski Triangle. WacLaw Sierpinski was a polish mathematician whose most important work was in the fields of set theory, number theory, and point set topology, What Sierpinski came up with was to first look at a large equilateral triangle.

- He then started to divide that large triangle into four smaller equilateral triangles.
- This action is repeated over and over again leaving the center triangle open each time,
- Looking at this triangle and dividing it up an infinite number of times displays the idea of self- similarity, upon which fractals are based upon.

By looking at the design of the Sierpinski triangle, it can be concluded that if the number of triangles is increased, the length and area of the triangles will decrease. If we let $N_k$ denote how many triangles we have within the main triangle, $L_k$ denote the length of the sides of each triangle, and $A_k$ be the area of the triangle, we have the following equations: : \beginequation\begin N_k=3^k\nonumber\\ L_k=(1/2)^k=2^(-k)\nonumber\\ A_k=L_k^2\timesN_k=(3/4)^k\nonumber\\\ \end \endequation We have to understand that we can take one portion of this divided up triangle and it will look exactly like the whole triangle itself, and as the number of iterations tends to infinity, the area of each triangle tends to zero, however will never equal zero.

Another type of fractal image is called the Koch curve. Helge von Koch was a mathematician who studied curves without any tangents. He came up with the idea that any line segment can be described as being infinitely long. The idea of the Koch curve, or fractal, is to take one line segment with some length, $$l$$.

Then take that line segment and divide it into three different segments of equal length. By taking the middle segment and splitting it into two and configuring an equilateral triangle out of it, we expand the length of the entire line segment when added together.

For example, if we took a line that had length 1 and took out the middle third of the segment and added in an equilateral triangle, we would have four smaller line segments of length, 1/3. So the original line of length 1 has turned into length 4/3. Doing this repeatedly will keep expanding the length of that original line segment,

If we were to apply this process to a particular shape, the equilateral triangle for instance, the result would be a figure that looks like a snowflake. This Koch snowflake is shown to have infinite perimeter but finite area. In order to see this we let $$N_k$$ be the number of sides the snowflake has after completing the kth step of the process.

We start out with three sides because it is an equilateral triangle: \beginequation\begin N_0=3\nonumber\\ N_1 =4×3=12\nonumber\\ N_2 =4×12=4^2 ×3\nonumber\\N_3 = 4^3 × 3\nonumber\\ N_k = 4N_(k−1) = 4^k × 3\nonumber\\ \end \endequation For each step in the process, each line segment is divided three times, so if we nlet L_k define the length of each segment after the kth step, then we have $$ L_k=1/3^k$$.

Then, for the perimeter we have to multiply the number of sides by the length of each side so we let $$P_k$$ be the perimeter after the kth step. We have $$P_k=N_kL_k=3(4/3)^k$$ By looking at this equation we can see that as k tends to infinity, so does the perimeter.

We can then conclude that the Koch snowflake as finite area but infinite perimeter, The German mathematician, Greg Cantor, who was one of the founders of set theory, discovered the Cantor fractal. This fractal is similar to the Koch Curve. Similar to the Koch curve, we start out with one straight-line segment and divide it into three parts.

Then remove the middle third, but keep the end points. So we started out with one line and two endpoints, and turned it into two lines and four endpoints. This process is to be repeated over and over, and the outcome after many times, would be points. \begin \begin \hline Steps & Number of Line Segments& Length of Line Segments\\ \hline 1& 2^1=1 & 1/3^(1)=1/3\\ \hline 2& 2^2=4 & 1/3^(2)=1/9\\ \hline 3& 2^3=8 & 1/3^(3)=1/27\\ \hline n & 2^n& 1/3^(-n)\\ \hline \end \end As this table shows, the length of the lines tends to approach zero, as we increase the amount of steps we apply.

- Therefore this is why the Cantor Fractal is also known as the Cantor point-set.
- Another way of drawing the Cantor set to make it more clear, is to use horizontal bars rather than just straight lines.
- By repeating the steps of removing the middle third, we get a picture that looks more like a comb,
- In 1977, when Benoit Mandelbrot was exploring the concept of what a fractal was, he also discovered that fractals have dimension.

He explained that fractals have capacity and it can be defined by a formula. “The concept of fractal dimension provides a way to measure how rough fractal curves are. The more jagged and irregular a curve is, the higher ts fractal dimension, a value between one and two.

- Fractional dimension is related to self-similarity in that the easiest way to create a figure that has fractional dimension is through self-similarity”,
- Before explaining what the formula depends on, we need to be sure that when we are zoomed into a fractal image, the boundary to what we are zoomed in at must match that of the entire fractal.

After that is confirmed, the number of pieces or line segments that fit into the larger one must be defined. We will call this number n. We also need to know the scale or how many times we have magnified into the entire fractal itself, which we will call M.

- We can then define the fractal dimension to be: \beginequation\begin D= (log(n)) /logM/nonumber//\end \endequation Using this equation, we can apply it to the fractal that was just discussed, the Koch curve.
- First we can look at the dimension of the Koch curve after doing the process just one time.
- So, there is one line fragment that is divided up into four with an equilateral triangle in the middle.

Now we have four line segments and we can say $$n=4$$. Because these four pieces are 1/3 the length of the original line segment, then the scale or magnification is 3. So using the definition for fractal dimension, we have $$D=log4/log3$$, which gives us D= 1.26185 Because this number has a dimension greater than 1, and is an integer, then we can conclude that the Koch curve is indeed a fractal.

Gaston Julia was a French mathematician who was interested in looking the behavior of the orbit of a complex number when it is iterated under a function. This means that for a function, namely f, we apply it to a complex number. Whatever the result is, apply the same function to the new value. Julia repeated this action over and over again to see how the results acted.

He came up with the idea of the prisoner set and the escape set by studying how iterating certain functions will give bounded or unbounded sets. A prisoner set refers to all the complex numbers in its orbit is bounded and an escape set refers to the complex numbers that are unbounded in its orbit under a certain function.

- An example of a prisoner is the value $$z_0 = 2$$ under the function $$ f(z)=z^2-z+1.
- By iterating this function starting with $$(1+i)$$ we get: \beginequation\begin z_0=1+\imath\nonumber\\ z_1=f(1+\imath)=(1+\imath)^2-(1+\imath)+1=\imath \nonumber \\ z_2=f(\imath)=\imath^2-\imath+1=-\imath\nonumber\\ z_3=f(-\imath)=(-\imath)^2-(-\imath)+1=\imath\nonumber\\ z_4=f(\imath)=-\imath\nonumber\\ \end \endequation Because the outputs switch back and forth, we call the value $$z_0=1+\imath$$ a prisoner.

If we were to pick a value to start with and iterate it under a function and the values were to get infinitely large or small, then we would have an escapee, Understanding what prisoner and escape sets are gives a better understanding of what Julia sets are.

- The Julia set is defined to be the boundary between the prisoner set and the escape set”,
- The functions that Julia looked at were of the form $$f(z)=z^2 +c$$ where c represents a complex constant.
- For each different c that we use in this function we will get a different Julia set.
- We choose $$z_0$$ to start with whatever we choose for c.

It has been shown that if the orbit of the starting value, $$z_0$$, is ever located outside of the circle which has radius 2, then the rest of the orbit will be unbounded and therefore an escapee, After choosing a value of c that we want to use in our function we can generate the Julia set on computer.

We use whatever complex number we have chosen as the starting value of $$z_0$$. For each pixel on the computer screen, the color black is assigned if the value is bounded or a prisoner and is colored white if it is not bounded and tends to infinity, This is where all these colorful art images that we see comes from.

Many use different color schemes, instead of black and white, to get these vibrant images. Another famous set that in a way relates to the Julia set was discovered by Benoit Mandelbrot and is called the Mandelbrot set. Mandelbrot also used the same exact function that Julia used, $$f(z)=z^2 +c$$.

Mandelbrot wanted to find the values of c that made the set bounded and similarly the values of c that made the orbit unbounded when this function was iterated always starting with $$z=0$$. However, what was drastic difference between what Gaston Julia could do with his data and what Mandelbrot could do with his.

The time that Mandelbrot was studying fractals was at a much later date than when Julia was, at a time when computers were in existence. As a result, Mandelbrot could do hundreds of thousands of iterations of a function using a computer, where Julia could only compute a small number of them by hand.

- After Mandelbrot gathered his Julia sets, he plotted them on a graph, which results in the Mandelbrot set,
- Following is a set of steps, or an algorithm, for developing a Mandelbrot set by computer: \begin \item Choose a part of the complex plane and divide it up into a grid of c values.
- Item Define n to be how many points of each orbit it will take you to decide whether it is bounded or unbounded.

\item Use the function $$f(z)=z^2+c to iterate the first n points of the orbit starting with the value, 0. \item If the orbit is unbounded, then color the corresponding c value on the grid a certain color.\item If the orbit is bounded, then color the corresponding c value on the grid a different color.

\item Move on to the next c value and keep repeating until all c values are accounted for\end There are many algorithms for creating your own fractals on computer and many websites that can show the properties of self-similarity in the Mandelbrot sets, and in creating original fractals by just choosing certain complex numbers.

There are many advantages in knowing and understanding what a fractal is, no matter what subject area you are working in. Fractals have been observed in just about every living thing in nature from trees in the rainforest to our human bodies. But fractals have also helped us propel forward immensely in the field of technology and entertainment.

- Fractals, or more specifically the Koch snowflake, were used to make the antennas of our cell phones smaller while increasing the amount of frequencies they can receive,
- Fractals are also used vastly in the medical field.
- One such example is that the healthy heartbeat when recorded on paper has a fractal pattern.

This has helped doctors anticipate people who might have heart problems in the future, One other example of where fractals are used is in computer science to help increase the believability of special effects and graphics in today’s movies, One specific application of fractals in special effects in the movies involves the movie, Star Wars: Episode III.

The part of the movie is when the two heroes run onto the end of a giant mechanical platform and a huge substance of lava comes crashing down in front of them. The initial process that they used to produce this lava was to make it appear that the lava is being shot up from a jet down below the mechanical platform.

At first the graphics of the lava looked extremely unrealistic, and emerged as just a straight cylinder of lava flowing up through the air. The creators wanted this to look more realistic, so they took the idea of the fractal and applied it to this cylinder spiral shape of the lava.

- They took the original shape, shrunk it down and reapplied it.
- They repeated this over and over again to get a extremely realistic huge ball of fire and lava \cite,
- Many may not know or understand how movies that are solely computer generated suddenly started.
- The first movie to have a complete computer generated sequence was Star Trek II: The Wrath of Khan\cite,

This movie was created in 1982, less than ten years after Mandelbrot made the idea of fractals public. Fractals is what makes computer generated imagery possible, and without Mandelbrot’s discoveries, we would not have the amazing graphics in our movies and games today.

- The grand discovery that Benoit Mandelbrot had made about fractals has changed the way we make movies, video games, computer games, etc.
- Fractals has made it possible to make real life images through a computer.
- Loren Carpenter, who is now works at Pixar animation studios, is known for his computer science work taking fractals and using them to create computer generated images.

He used to work for Boeing aircraft and his job was to see how the planes they were idealizing and creating looked while in flight. He wanted to make mountains to put in the background to make it look as realistic as possible. So after reading Fractals: Form, Chance, and Dimension by Benoit Mandelbrot, he learned that every surface can be broken down into smaller simpler shapes.

He was then able to create images triangles into four smaller ones and kept on iterating until he had the jagged form of mountains, In conclusion, fractals are geometrical figures that have identical repeating patterns on a scale that reduces infinitely. At first, when looking at the colorful picture of a fractal, one might think that it is just a creative piece of artwork.

However after studying the mathematical background behind them, they have so much more depth than being just a piece of art. Benoit Mandelbrot is considered to be the father of fractal geometry and coined the term, “fractal.” He showed that every living and non-living thing in this world can be broken down into mathematical terms using fractal geometry.

His first experiment of trying to calculate the length of Britain’s coastline led to this theory. The two main principles that define a fractal are self-similarity and fractal dimension. Self-similarity simply means that a pattern looks the same no matter how much it magnified. Fractal dimension is important because it shows that fractals have capacity and that they are not just flat images.

Along with Benoit Mandelbrot, WacLaw Sierpinski,Gaston Julia, Helge von Koch Greg Cantor are all very important mathematicians in the history of fractal geometry. All of these mathematicians have their own forms or types of fractals that they have discovered.

One of the most popular is the Mandelbrot set, which describes the iteration of a function using complex numbers. Even though the subject of fractal geometry is purely math, many people have found ways to take it outside math into computer science, health science, and even fashion and the arts. Fractals are used to make our movies look more amazing than life, and to make our computer and video games feel like we are right there in them.

The discovery of fractals have allowed us to decrease the size of our cell phones every year and at the same time they have helped doctors anticipate heart problems in our bodies way before they happen. Without the discovery of fractals, our technology, entertainment, our health, etc.

Would not be where it is today. References Bourke, P. Julia Set Fractal (2D)(2001). http://local.wasp.uwa.edu.au/~pbourke/fractals/juliaset/ Hoffman,J. Benoit Mandelbrot, Novel Mathematician, Dies at 85. The New York Times (2010). http://www.nytimes.com/2010/10/17/us/17mandelbrot.html?_r=1 Jersey,B.,And Shwarz,M.Hunting the Hidden Dimension.(2008).

http://www.pbs.org/wgbh/nova/fractals/ Laubender, P. “What is a Fractal?” Fractaline (1999). http://www.peter-laubender.de/fractaline/what_is_a_fractal.htm Lauwerier, H.Fractals Endlessly Repeated Geometrical Figures.Princeton, New Jersey. pgs 13,15-16, 32-33,(2010).

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## Who is the father of geometry?

Euclid, The Father of Geometry.

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### What is that one famous fractal?

Strange attraction – But a major importance of the set may be that it has become a strange attractor for scientists, artists, and the public, though each may be drawn to it for quite different reasons. Scientists have found themselves attracted—often with childlike delight—to a new aesthetic that involves the artistic choices of color and detail they must make when exploring the set.

- Artists and the public have been attracted by the set’s haunting beauty and the idea of abstract mathematics turned into tangible pleasures.
- Homer Smith, cofounder of an independent research group based at Cornell University, produces fractal images with the aim of attracting young children to mathematics.

“We hope that fractals show up in early classrooms, to get kids interested in mathematics very early,” Smith says, “because it really opens the eyes of children who haven’t been turned off by education. We hope by the time they get up to the tenth grade, they’ll have seen these things and say, ‘There’s something here in math, science, and computers that I want to learn.'” In this detail of the Mandelbrot set, the set itself appears in black, with the fractal boundary alive with color. Because an infinite number of points exist between any two points on the number plane, the Mandelbrot set’s detail is infinite. This image is a tiny part of the previous image magnified many thousands of times over. “A flaming boundary of filigreed detail” is how Briggs aptly describes the border of the Mandelbrot set. The “self-similar” nature of fractals means that particular elements, such as the Mandelbrot set, reappear over and over again, no matter how “deep” one goes into the image through magnification. John Briggs is author of Fractals: The Patterns of Chaos (Simon & Schuster, 1992), from which this article was excerpted with kind permission of the author and publisher.

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## What are 3 well known fractals?

Generating fractals – Three common techniques for generating fractals are:

Escape-time fractals — These are defined by a recurrence relation at each point in a space (such as the complex plane). Examples of this type are the Mandelbrot set, Julia set, the Burning Ship fractal and the Lyapunov fractal. Iterated function systems — These have a fixed geometric replacement rule. Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake, Harter-Heighway dragon curve, T-Square, Menger sponge, are some examples of such fractals. Random fractals — Generated by stochastic rather than deterministic processes, for example, trajectories of the Brownian motion, Lévy flight, fractal landscapes and the Brownian tree. The latter yields so-called mass- or dendritic fractals, for example, diffusion-limited aggregation or reaction-limited aggregation clusters.

### Who is the Swedish mathematician fractals?

Niels Fabian Helge von Koch, (born January 25, 1870, Stockholm, Sweden—died March 11, 1924, Stockholm), Swedish mathematician famous for his discovery of the von Koch snowflake curve, a continuous curve important in the study of fractal geometry. Von Koch was a student of Gösta Mittag-Leffler and succeeded him as professor of mathematics at Stockholm University in 1911. Britannica Quiz Numbers and Mathematics Von Koch is remembered primarily, however, for a 1906 paper in which he gave a very attractive description of a continuous curve that never has a tangent. Continuous, “nowhere differentiable ” functions had been rigorously introduced into mathematics by the German Karl Weierstrass in the 1870s, following suggestions by the German Bernhard Riemann and, even earlier, by the Bohemian Bernhard Bolzano, whose work was not well known.

- Von Koch’s example is perhaps the simplest.
- Starting with an equilateral triangle, it replaces the middle third of each segment with an equilateral triangle having the deleted portion of the segment as its base (the base is erased).
- This replacement operation is continued indefinitely, with the result that the limiting curve is continuous but nowhere differentiable.

If the new triangles always face outward, the resulting curve will have a striking resemblance to a snowflake, and so the curve is often called von Koch’s snowflake. This article was most recently revised and updated by William L. Hosch,

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## What are the 4 types of fractals?

Fractals are exquisite structures produced by nature, hiding in plain sight all around us. They are tricky to define precisely, though most are linked by a set of four common fractal features: infinite intricacy, zoom symmetry, complexity from simplicity and fractional dimensions – all of which will be explained below.

The next fern you encounter will provide a great illustration of these features if you pause for a closer look. First, notice that the shape of the fern is intricately detailed. Remarkably, you can see that the leaves are shaped like little copies of the branches. In fact, the entire fern is mostly built up from the same basic shape repeated over and over again at ever smaller scales.

Most astonishing of all, fractal mathematics reveals that this humble fern leaf is neither a one- nor an two-dimensional shape, but hovers somewhere in-between. A fern displaying its fractal features. The same shape is repeated in the branches, the fronds and the leaves – and even the veins inside each leaf. Wikimedia Commons Exactly what shape does this fern have? The classical Euclidean geometry taught in high-school leaves us at a loss to answer this simple question. The International Space Station, an engineering wonder whose shape can be modelled by classical Euclidean geometry. Such regular shapes are extremely rare in nature. Wikimedia Commons How can we describe a fern as a precise mathematical shape? How can we build a mathematical model of this wonderful object? Enter a completely new world of beautiful shapes: a branch of mathematics known as fractal geometry.

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#### Is the Fibonacci sequence a fractal?

Just like the exhibits at San Francisco’s Exploratorium that inspired Ned Kahn’s artwork, Kahn’s own work involves numerous scientific concepts and applications. Because Kahn’s work focuses on the scientific natural processes in the world, most, if not all of his displays require a basic understanding of related scientific concepts. Ned Kahn’s “Prism Tunnel,” located in San Marino, CA http://nedkahn.com/portfolio/prism-tunnel/ Pink Floyd’s famous “Dark Side of The Moon” album cover http://www.whiz.se/backgrounds/dsotm/ Most people encounter light refraction at some point or another during their lifetime, whether in the form of Pink Floyd’s “The Dark Side of The Moon” album cover, or the rainbow that appears when light shines through a faceted piece of glass.

- Light refraction is simply the bending of light by a foreign medium, which changes the speed of the wave 6,
- The most well known example of this phenomenon is the dispersion of white light into its constituent colors.
- Ahn uses a material called a diffraction grating in his “Prism Tunnel” sculpture in order to create a fascinating, ever-changing light display.

The position and intensity of the light changes depending on the time of day during which it is observed. Many of Ned Kahn’s other sculptures incorporate suspended fog or clouds, and I was curious about the process of their creation. Ned Kahn’s “Cloud Arbor,” located in Pittsburgh, PA https://www.codaworx.com/project/cloud-arbor-pittsburgh-children-s-museum The sheer absurdity of a single cloud suspended among an artistic sculpture is fascinating, and the science behind it makes it exceptionally more interesting. One of Berndnaut Smilde’s cloud displays http://www.sfartscommission.org/gallery/2013/prints-for-sale-clouds-in-san-francisco/ To create this transient work of art, Smilde mists water into the air, and releases a burst of chemical “fog” from a smoke machine into the mist.

- Simply put, the water droplets and smoke particles stick to each other, and a cloud forms.
- While seemingly simple in process, this art form is complex, and requires a substantial amount of patience and practice.
- Like most of the other “hard-to-manage phenomena” 5 that Ned Kahn works with, the creation of artificial clouds is no exception in terms of difficulty.

However, this difficulty and value of innovation and creative experimentation are key to the development of an intellectually interesting display that engages viewers to interact and explore. To say the least, difficulty and creative experimentation have played a pretty key role in the design and construction of my own creative sculpture. Ned Kahn’s “Fire Vortex,” located in Winterthur, Switzerland http://nedkahn.com/portfolio/fire-vortex/ Because this display functions so much like a tornado, I researched the characteristics of vortex flow structure and how they compare to the properties of a fire vortex.

- I came across a peer-reviewed journal article that discussed the results of a study, which examined this very relationship.
- The study involved the usage of velocity measurements to determine that the basic flow structure is identical among fire and tornado-like vortices 7,
- Ahn’s “Fire Vortex” display is basically a tornado of fire! This insight into the physics of this particular display provoked my curiosity in the physical shapes and patterns formed by tornados and other “spiraling” natural phenomena (the formation of hurricanes, the shape of spiral galaxies, etc.).

The extreme similarity between the shape of the spirals within these various phenomena led to my research of the mathematics behind this omnipresent spiral. From this point on, I focused my following research on the mathematics of the Fibonacci sequence, its connection to fractal art and mathematics, and its ubiquitous presence in the natural world.

My research about fractals in general began with a course reading about fractal complexity, its presence in nature, and its application to the drip-painting art of American painter, Jackson Pollock 9, This article describes a study conducted by four scientists, one of whom is Dr. Richard Taylor from the University of Oregon.

The study concluded that the works of American painter, Jackson Pollock, display fractal patterns. Additionally, it explored the connection between the characteristics of fractals, and the long-term aesthetic appeal of Pollock’s splatter paintings. Dr. Richard Taylor from the University of Oregon https://around.uoregon.edu/content/uo-idea-bio-inspired-implant-wins-900000-grant American painter, Jackson Pollock, standing in front of one his drip paintings https://www.pinterest.com/farkas74/111-jackson-pollock/ Simply put, fractals are infinite patterns – mathematical phenomena that display a constantly repeating pattern on every scale. A major idea explored in this study is the concept that consistent mathematical patterns and ratios, such as fractals, exist within seemingly chaotic places. The complexity (also known as the “roughness” 10 ) of such patterns can be quantified by their fractal dimension (signified by D ). The value of D lies within the range of 1 and 2, where 1 displays no complexity, and 2 displays infinite complexity. Taylor’s research, which he explained during our class visit to University of Oregon’s Camcor Lab 12, found that Jackson Pollock’s early work (1943-1945) displayed values of around D = 1.10. Between 1945 and 1947, this value jumped to 1.7, and even later, he painted a piece with D = 1.89. Pollock immediately erased this most complex work, an re-stabilized his paintings at approximately D = 1.7. As noted in Taylor’s study, this suggests that Pollock’s years of refinement were “motivated by the desire to create fractal patterns with D approximately equal to 1.7.” 9 Further investigation using eye-tracking and EEG measurements confirmed that humans visually prefer images in the range of D = 1.3, to D = 1.5. But WHY ? Taylor’s article references another study, which suggests that this preference is a result of the complexity of our immediate environment. The fractals in nature that we most frequently experience are found in this same range of complexity! This concept is extremely fascinating, and it directly relates to the idea that was provoked in my mind by Lisa Freinkel 11 (another guest speaker): our perceptions are collectively shaped by perspective and awareness. Our visual preference for images in the range of D = 1.3 to D = 1.5 is based upon our perceptions of what we visualize. These perceptions are influenced by the mindsets we adopt when observing something (perspective), and how deeply we observe it (awareness). Taylor extensively references the revolutionary work of mathematician Benoit Mandelbrot in his explanation of fractal concepts. This sparked my curiosity about his work, and led to the discovery of a TED Talk that Mandelbrot presented early in 2010 10, In this talk, Mandelbrot emphasizes the underlying concept that fractals are able find order within extreme chaos. These fractals can be infinitely complex, yet can often be described with utmost simplicity. Using his own Mandelbrot Set to exemplify this, Mandelbrot displays an image of the Mandelbrot Set, accompanied by its extremely simple defining expression: z –> z 2 + c. The paradoxical nature of fractals is absolutely intriguing: fractals are infinitely complex, yet can be described so simply. “A fractal is a way of seeing infinity.” – Benoit Mandelbrot During this course, I chose to direct my focus towards a single type of fractal pattern: the Fibonacci sequence. While applications of this mathematical sequence can be extremely complex, the sequence itself is very simple. As defined in Designa: Technical Secrets of the Traditional Visual Arts 13, the Fibonacci sequence is a “cumulative progression where each number is the sum of the preceding two.” The sequence starts with 0 and 1, and theoretically, goes on forever. The Fibonacci Sequence: Leonardo of Pisa (“Fibonacci”) https://www.fibonicci.com/fibonacci/ Golden Fibonacci numbers in the Parthenon http://www.enarmour.com/blog/2015/8/18/fibonacci-capturing-a-mathematical-enigma-in-jewelry While its exact origin is not known, the Fibonacci sequence is believed to have been discovered by ancient Egyptians, and their Greek students.

There is architectural evidence of this in the Great Egyptian Pyramids 14 and in Greek architecture, such as the Parthenon 15, Its discovery, however, is formally attributed to Leonardo of Pisa (Leonardo Pisano Bigollo), later named “Fibonacci” for short 1, Fibonacci brought this number sequence to the world’s attention through his famous rabbit breeding problem.

This problem, published in his book called Liber Abaci (“Book of Calculation”), showed how to determine the number of rabbit breeding pairs after a given number of generations 13, Fibonacci numbers were officially discovered by Leonardo of Pisa, but have existed in the universe for as long as we know. Pigeons standing in a Fibonacci sequence https://mathblag.files.wordpress.com/2011/11/fibonacci_pigeons.jpg Often more relevant than the actual numbers in the sequence, is the phi (Φ) ratio 13, which is equal to ≈1.618. This ratio is the relationship that exists between consecutive numbers in the Fibonacci sequence – each number is approximately 1.618 times the quantity of the previous number in the sequence. The “Golden” (phi) ratio in the Fibonacci spiral http://aetherforce.com/fibonacci/ The Fibonacci Spiral, which is my key aesthetic focus of this project, is a simple logarithmic spiral based upon Fibonacci numbers, and the golden ratio, Φ. Because this spiral is logarithmic, the curve appears the same at every scale, and can thus be considered fractal.

As we have seen, numbers in the Fibonacci sequence have several ways of appearing in natural examples. The Fibonacci spiral, however, is a particularly fascinating application because of its mathematical perfection, aesthetic appeal, and biological importance. One notable example of natural Fibonacci spiraling, is the arrangement of leaves, seeds, and petals in plants.

For instance, the seeds in the center of a sunflower follow a perfect Fibonacci spiraling pattern, which allows for an efficient usage of space, and maximal seed packing. Looking at the leaf arrangement around the stem in many herbaceous plants (a study known as phyllotaxis 13 ) reveals that the leaves are arranged in an identical spiraling pattern. The Fibonacci spiral in cauliflower http://from-bedroom-to-study.blogspot.com/2012/09/the-swirly-spirals-of-natures-numbers.html The Fibonacci spiral arrangement of leaves around a plant stem http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html The Fibonacci sequence is fundamental to life on Earth, but goes much beyond our own world; the spiral that exists in the center of sunflowers, heads of cabbage, and chambered nautilus shells, is the same spiral that shapes spiral galaxies. Fibonacci spiral and the phi ratio in planet orbiting patterns https://www.pinterest.com/pin/291608144600944480/ From all of these examples, it is not difficult to see that the Fibonacci sequence is EVERYWHERE, Its omnipresence in the universe is fascinating and extremely powerful: through a simple sequence of numbers and a related ratio, we share an inherent connection with all living beings, and the entirety of the universe.

This is not only intriguing on a biological level, but also in a an intellectual and spiritual sense. We all seek answers about the universe and our place within it; the Fibonacci spiral has the power to both illustrate our relationship to others, and to give us a unique perspective about our place within the universe.

By delving into the deeply interconnected nature of the world, my intention is to illuminate the connections that humans share with their surroundings. In highlighting these connections, I hope to heighten both people’s awareness of the natural environment, and their concern about its well-being.

In our environment’s critical state, it has never been more important for people to care about their own impact on the world. Knowledge is powerful because it has the ability to elevate our own personal perceptions and environmental awareness. In this regard, my intention is to provide people with information, provoke their sense of curiosity and concern for the environment, and ultimately encourage collective motivation to take action.

One of Ned Kahn’s core intentions in creating his sculpture art, is to enliven peoples’ senses of curiosity about the world. This aspect of his artistic goal is one that we both share – provoking curiosity is not only an intellectual and artistic goal that I aspire to achieve, for it is also the first step in taking action.

Curiosity is a crucial first step in the direction of action. In addition to emphasizing my goal of provoking curiosity, Ned Kahn’s work has effectively inspired me to focus on generating environmental awareness via my creative display. Also, while I originally considered different 2-dimensional media for my artistic display, I decided to create a 3-dimensional sculpture in order to emulate Ned Kahn’s style.

I didn’t have much previous experience with 3D media, so this form of creative expression was definitely a new one!

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## What famous paintings have fractals?

Artists intuit the appeal of fractals – It’s therefore not surprising to learn that, as visual experts, artists have been embedding fractal patterns in their works through the centuries and across many cultures. Fractals can be found, for example, in Roman, Egyptian, Aztec, Incan and Mayan works.

- My favorite examples of fractal art from more recent times include da Vinci’s Turbulence (1500), Hokusai’s Great Wave (1830), M.C.
- Escher’s Circle Series (1950s) and, of course, Pollock’s poured paintings,
- Although prevalent in art, the fractal repetition of patterns represents an artistic challenge.

For instance, many people have attempted to fake Pollock’s fractals and failed. Indeed, our fractal analysis has helped identify fake Pollocks in high-profile cases. Recent studies by others show that fractal analysis can help distinguish real from fake Pollocks with a 93 percent success rate.

How artists create their fractals fuels the nature-versus-nurture debate in art: To what extent is aesthetics determined by automatic unconscious mechanisms inherent in the artist’s biology, as opposed to their intellectual and cultural concerns? In Pollock’s case, his fractal aesthetics resulted from an intriguing mixture of both.

His fractal patterns originated from his body motions (specifically an automatic process related to balance known to be fractal). But he spent 10 years consciously refining his pouring technique to increase the visual complexity of these fractal patterns. The Rorschach inkblot test relies on what you read in to the image. Hermann Rorschach

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## Who introduced the theory of fractals and published the fractal geometry of nature in 1982?

The Fractal Geometry of Nature is a 1982 book by the Franco-American mathematician Benoît Mandelbrot.

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#### Which French American mathematician showed how the mathematics of fractals could create plant growth patterns?

Patterns in nature are visible regularities of form found in the natural world, These patterns recur in different contexts and can sometimes be modelled mathematically, Natural patterns include symmetries, trees, spirals, meanders, waves, foams, tessellations, cracks and stripes.

- Early Greek philosophers studied pattern, with Plato, Pythagoras and Empedocles attempting to explain order in nature.
- The modern understanding of visible patterns developed gradually over time.
- In the 19th century, the Belgian physicist Joseph Plateau examined soap films, leading him to formulate the concept of a minimal surface,

The German biologist and artist Ernst Haeckel painted hundreds of marine organisms to emphasise their symmetry, Scottish biologist D’Arcy Thompson pioneered the study of growth patterns in both plants and animals, showing that simple equations could explain spiral growth.

In the 20th century, the British mathematician Alan Turing predicted mechanisms of morphogenesis which give rise to patterns of spots and stripes. The Hungarian biologist Aristid Lindenmayer and the French American mathematician Benoît Mandelbrot showed how the mathematics of fractals could create plant growth patterns.

Mathematics, physics and chemistry can explain patterns in nature at different levels and scales. Patterns in living things are explained by the biological processes of natural selection and sexual selection, Studies of pattern formation make use of computer models to simulate a wide range of patterns.

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#### What is the surname of the mathematician who pioneered fractal geometry and the mathematics of chaos?

Biography – Benoit Mandelbrot was largely responsible for the present interest in fractal geometry. He showed how fractals can occur in many different places in both mathematics and elsewhere in nature. Mandelbrot was born in Poland in 1924 into a family with a very academic tradition. His father, however, made his living buying and selling clothes while his mother was a doctor. As a young boy, Mandelbrot was introduced to mathematics by his two uncles. Mandelbrot’s family emigrated to France in 1936 and his uncle Szolem Mandelbrojt, who was Professor of Mathematics at the Collège de France and the successor of Hadamard in this post, took responsibility for his education. In fact the influence of Szolem Mandelbrojt was both positive and negative since he was a great admirer of Hardy and Hardy ‘s philosophy of mathematics. This brought a reaction from Mandelbrot against pure mathematics, although as Mandelbrot himself says, he now understands how Hardy ‘s deep felt pacifism made him fear that applied mathematics, in the wrong hands, might be used for evil in time of war. Mandelbrot attended the Lycée Rolin in Paris up to the start of World War II, when his family moved to Tulle in central France. This was a time of extraordinary difficulty for Mandelbrot who feared for his life on many occasions. In the effect of these years on his education was emphasised:- The war, the constant threat of poverty and the need to survive kept him away from school and college and despite what he recognises as “marvellous” secondary school teachers he was largely self taught. Mandelbrot now attributed much of his success to this unconventional education. It allowed him to think in ways that might be hard for someone who, through a conventional education, is strongly encouraged to think in standard ways. It also allowed him to develop a highly geometrical approach to mathematics, and his remarkable geometric intuition and vision began to give him unique insights into mathematical problems. After studying at Lyon, Mandelbrot entered the École Normale in Paris. It was one of the shortest lengths of time that anyone would study there, for he left after just one day. After a very successful performance in the entrance examinations of the École Polytechnique, Mandelbrot began his studies there in 1944, There he studied under the direction of Paul Lévy who was another to strongly influence Mandelbrot. After completing his studies at the École Polytechnique, Mandelbrot went to the United States where he visited the California Institute of Technology. After a Ph.D. granted by the University of Paris, he went to the Institute for Advanced Study in Princeton where he was sponsored by John von Neumann, Mandelbrot returned to France in 1955 and worked at the Centre National de la Recherche Scientifique. He married Aliette Kagan during this period back in France and Geneva, but he did not stay there too long before returning to the United States. Clark gave the reasons for his unhappiness with the style of mathematics in France at this time :- Still deeply concerned with the more exotic forms of statistical mechanics and mathematical linguistics and full of non standard creative ideas he found the huge dominance of the French foundational school of Bourbaki not to his scientific tastes and in 1958 he left for the United States permanently and began his long standing and most fruitful collaboration with IBM as an IBM Fellow at their world renowned laboratories in Yorktown Heights in New York State. IBM presented Mandelbrot with an environment which allowed him to explore a wide variety of different ideas. He has spoken of how this freedom at IBM to choose the directions that he wanted to take in his research presented him with an opportunity which no university post could have given him. After retiring from IBM, he found similar opportunities at Yale University, where he is presently Sterling Professor of Mathematical Sciences. In 1945 Mandelbrot’s uncle had introduced him to Julia ‘s important 1918 paper claiming that it was a masterpiece and a potential source of interesting problems, but Mandelbrot did not like it. Indeed he reacted rather badly against suggestions posed by his uncle since he felt that his whole attitude to mathematics was so different from that of his uncle. Instead Mandelbrot chose his own very different course which, however, brought him back to Julia ‘s paper in the 1970 s after a path through many different sciences which some characterise as highly individualistic or nomadic. In fact the decision by Mandelbrot to make contributions to many different branches of science was a very deliberate one taken at a young age. It is remarkable how he was able to fulfil this ambition with such remarkable success in so many areas. With the aid of computer graphics, Mandelbrot who then worked at IBM’s Watson Research Center, was able to show how Julia ‘s work is a source of some of the most beautiful fractals known today. To do this he had to develop not only new mathematical ideas, but also he had to develop some of the first computer programs to print graphics. The Mandelbrot set is a connected set of points in the complex plane. Pick a point z 0 z_ in the complex plane. Calculate: z 1 = z 0 2 + z 0 z_ = z_ ^ + z_ z 2 = z 1 2 + z 0 z_ = z_ ^ + z_ z 3 = z 2 2 + z 0 z_ = z_ ^ + z_,, If the sequence z 0, z 1, z 2, z 3,,, z_, z_, z_, z_,, remains within a distance of 2 of the origin forever, then the point z 0 z_ is said to be in the Mandelbrot set. If the sequence diverges from the origin, then the point is not in the set. You can see the Mandelbrot Set at The Mandelbrot set ” href=”https://localhost/././Diagrams/MandelbrotSet/” target=”_blank”>THIS LINK, His work was first put elaborated in his book Les objets fractals, forn, hasard et dimension Fractals, shape, chance and dimension “>Ⓣ (1975) and more fully in The fractal geometry of nature in 1982, On 23 June 1999 Mandelbrot received the Honorary Degree of Doctor of Science from the University of St Andrews. At the ceremony Peter Clark gave an address in which he put Mandelbrot’s achievements into perspective. We quote from that address:-, at the close of a century where the notion of human progress intellectual, political and moral is seen perhaps to be at best ambiguous and equivocal there is one area of human activity at least where the idea of, and achievement of, real progress is unambiguous and pellucidly clear. That is mathematics. In 1900 in a famous address to the International Congress of mathematicians in Paris David Hilbert listed some 25 open problems of outstanding significance. Many of those problems have been definitively solved, or shown to be insoluble, culminating as we all know most recently in the mid-nineties with the discovery of the proof of Fermat’s Last Theorem, The first of Hilbert ‘s problems concerned a thicket of issues about the nature of the continuum or the real line, a major concern of 19 th and indeed of 20 th century analysis. The problem was both one of geometry concerning the nature of the line thought of as built up of points and of arithmetic thought of as the theory of the real numbers. The integration of those two fields was one of the great achievements of Richard Dedekind and Georg Cantor, the latter of whom we were intelligent enough to honour in 1911, Now lurking about so to speak in the undergrowth of that achievement lay certain very extraordinary geometric objects indeed. To all at the time, they seemed strange, indeed rather pathological monsters. Odd indeed they were, there were curves – one dimensional lines in effect – which filled two dimensional spaces, there were curves which were well behaved, that is nice and continuous but which had no slope at any point ( not just some points, ANY points ) and they went by strange names, the Peano Space filling curve, the Sierpiński gasket, the Koch curve, the Cantor Ternary set. Despite their pathological qualities, their extraordinary complexity, especially when viewed in greater and greater detail, they were often very simple to describe in the sense that the rules which generated them were absurdly simple to state. So odd were these objects that mathematicians set about barring these monsters and they were set aside as too strange to be of interest. That is until our honorary graduand created out of them an entirely new science, the theory of fractal geometry: it was his insight and vision which saw in those objects and the many new ones he discovered, some of which now bear his name, not mathematical curiosities, but signposts to a new mathematical universe, a new geometry with as much system and generality as that of Euclid and a new physical science. As well as IBM Fellow at the Watson Research Center Mandelbrot was Professor of the Practice of Mathematics at Harvard University. He also held appointments as Professor of Engineering at Yale, of Professor of Mathematics at the École Polytechnique, of Professor of Economics at Harvard, and of Professor of Physiology at the Einstein College of Medicine. Mandelbrot’s excursions into so many different branches of science was, as we mention above, no accident but a very deliberate decision on his part. It was, however, the fact that fractals were so widely found which in many cases provided the route into other areas :- I should not, give the impression that we have here before us a mathematician alone. Let me explain why. The first of his great insights was the discovery that the extraordinarily complex almost pathological structures, which had been long ignored, exhibited certain universal characteristics requiring a new theory of dimension to treat them adequately which he had generalised from earlier work of Hausdorff and Besicovitch but the second great insight was that the fractal property so discovered, the general theory of which he had provided, was present almost universally in Nature. What he saw was that the overwhelming smoothness paradigm with which mathematical physics had attempted to describe Nature was radically flawed and incomplete. Fractals and pre-fractals once noticed were everywhere. They occur in physics in the description of the extraordinarily complex behaviour of some simple physical systems like the forced pendulum and in the hugely complex behaviour of turbulence and phase transition. They occur as the foundations of what is now known as chaotic systems. They occur in economics with the behaviour of prices and as Poincaré had suspected but never proved in the behaviour of the Bourse or our own Stock exchange in London. They occur in physiology in the growth of mammalian cells. Believe it or not, they occur in gardens. Note closely and you will see a difference between the flower heads of broccoli and cauliflower, a difference which can be exactly characterised in fractal theory. Mandelbrot has received numerous honours and prizes in recognition of his remarkable achievements. For example, in 1985 Mandelbrot was awarded the Barnard Medal for Meritorious Service to Science, The following year he received the Franklin Medal, In 1987 he was honoured with the Alexander von Humboldt Prize, receiving the Steinmetz Medal in 1988 and many more awards including the Légion d’Honneur in 1989, the Nevada Medal in 1991, the Wolf prize for physics in 1993 and the 2003 Japan Prize for Science and Technology, A full list of his prizes and honours is available ( as a download ) at THIS LINK,

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## What is the most famous fractal?

Strange attraction – But a major importance of the set may be that it has become a strange attractor for scientists, artists, and the public, though each may be drawn to it for quite different reasons. Scientists have found themselves attracted—often with childlike delight—to a new aesthetic that involves the artistic choices of color and detail they must make when exploring the set.

- Artists and the public have been attracted by the set’s haunting beauty and the idea of abstract mathematics turned into tangible pleasures.
- Homer Smith, cofounder of an independent research group based at Cornell University, produces fractal images with the aim of attracting young children to mathematics.

“We hope that fractals show up in early classrooms, to get kids interested in mathematics very early,” Smith says, “because it really opens the eyes of children who haven’t been turned off by education. We hope by the time they get up to the tenth grade, they’ll have seen these things and say, ‘There’s something here in math, science, and computers that I want to learn.'” In this detail of the Mandelbrot set, the set itself appears in black, with the fractal boundary alive with color. Because an infinite number of points exist between any two points on the number plane, the Mandelbrot set’s detail is infinite. This image is a tiny part of the previous image magnified many thousands of times over. “A flaming boundary of filigreed detail” is how Briggs aptly describes the border of the Mandelbrot set. The “self-similar” nature of fractals means that particular elements, such as the Mandelbrot set, reappear over and over again, no matter how “deep” one goes into the image through magnification. John Briggs is author of Fractals: The Patterns of Chaos (Simon & Schuster, 1992), from which this article was excerpted with kind permission of the author and publisher.

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#### What inspired him to discover fractals?

Fractals There is no doubt that just about everyone on this planet has seen a fractal in their lifetime, even though many do not even know it. Every natural thing on our planet can be described in mathematical terms. This is where the subject of fractals comes into play.

- Fractals occur everywhere in nature and can sometimes be depicted as forming the core of our lives.
- Being able to comprehend the structure of a mathematical fractal opens up the ability to understand how everything in this world was formed.
- A fractal is defined to be a rough or fragmented shape that can be broken up into smaller parts, which can be seen as a smaller copy of the original shape.

It is almost impossible to describe the natural things in our world, such as the clouds, trees, plants and so on, as geometric images. This is why fractals are such an important part of how nature is structured. This history of fractals is quite interesting, considering that the “father” of fractals has just passed away a few weeks ago.

Also looking at different kinds of sets that describe fractals is of great importance in the overall application of them. Finally, I will write about the computer programming of fractals and how to create them on your own. Benoit Mandelbrot is considered to be the father of fractal geometry. He has said that the first thing that made him start to even think about the idea of fractals was when he was trying to figure out how long the coast of Britain was.

What he discovered was that if you look at a map and keep on zooming in on it, repeated patterns will appear. The idea that he used to get the most accurate measure of the coastline of Britain was determined by what length of ruler he would use. He showed that smaller rulers are more accurate because they can fit better into the irregular patterns of the coast, rather than using one large ruler.

He concluded that as the scale of measurement he used decreased in size, the actual length of the coastline increased, This shows that we can zoom into the coastline an infinite number of times, using a smaller unit of measurement and keep getting a more accurate estimate. Mandelbrot always said to not think about what you see, but what it took to make what you see.

“The key to fractal geometryis that if you look on the surface, you see complexity and it looks very non-mathematical”, His studies about the Britain coastline lead into one of the main ideas of fractals, known as self-similarity. “A set S is called self-similar if S can be subdivided into k congruent subsets, each of which can be magnified by a constant factor M to yield the whole set S”,

- By looking at the coastline of Britain from a far distance and then zooming up extremely close, the images would look similar.
- Self-similarity is one huge principal idea when classifying what a fractal is.
- Although Mandelbrot was the one to coin the term “fractal geometry” in 1975, there were many mathematicians before his time that noticed this property of self-similarity.

A simple start to understanding the formation of fractals is to look at the Sierpinski Triangle. WacLaw Sierpinski was a polish mathematician whose most important work was in the fields of set theory, number theory, and point set topology, What Sierpinski came up with was to first look at a large equilateral triangle.

- He then started to divide that large triangle into four smaller equilateral triangles.
- This action is repeated over and over again leaving the center triangle open each time,
- Looking at this triangle and dividing it up an infinite number of times displays the idea of self- similarity, upon which fractals are based upon.

By looking at the design of the Sierpinski triangle, it can be concluded that if the number of triangles is increased, the length and area of the triangles will decrease. If we let $N_k$ denote how many triangles we have within the main triangle, $L_k$ denote the length of the sides of each triangle, and $A_k$ be the area of the triangle, we have the following equations: : \beginequation\begin N_k=3^k\nonumber\\ L_k=(1/2)^k=2^(-k)\nonumber\\ A_k=L_k^2\timesN_k=(3/4)^k\nonumber\\\ \end \endequation We have to understand that we can take one portion of this divided up triangle and it will look exactly like the whole triangle itself, and as the number of iterations tends to infinity, the area of each triangle tends to zero, however will never equal zero.

Another type of fractal image is called the Koch curve. Helge von Koch was a mathematician who studied curves without any tangents. He came up with the idea that any line segment can be described as being infinitely long. The idea of the Koch curve, or fractal, is to take one line segment with some length, $$l$$.

Then take that line segment and divide it into three different segments of equal length. By taking the middle segment and splitting it into two and configuring an equilateral triangle out of it, we expand the length of the entire line segment when added together.

- For example, if we took a line that had length 1 and took out the middle third of the segment and added in an equilateral triangle, we would have four smaller line segments of length, 1/3.
- So the original line of length 1 has turned into length 4/3.
- Doing this repeatedly will keep expanding the length of that original line segment,

If we were to apply this process to a particular shape, the equilateral triangle for instance, the result would be a figure that looks like a snowflake. This Koch snowflake is shown to have infinite perimeter but finite area. In order to see this we let $$N_k$$ be the number of sides the snowflake has after completing the kth step of the process.

We start out with three sides because it is an equilateral triangle: \beginequation\begin N_0=3\nonumber\\ N_1 =4×3=12\nonumber\\ N_2 =4×12=4^2 ×3\nonumber\\N_3 = 4^3 × 3\nonumber\\ N_k = 4N_(k−1) = 4^k × 3\nonumber\\ \end \endequation For each step in the process, each line segment is divided three times, so if we nlet L_k define the length of each segment after the kth step, then we have $$ L_k=1/3^k$$.

Then, for the perimeter we have to multiply the number of sides by the length of each side so we let $$P_k$$ be the perimeter after the kth step. We have $$P_k=N_kL_k=3(4/3)^k$$ By looking at this equation we can see that as k tends to infinity, so does the perimeter.

We can then conclude that the Koch snowflake as finite area but infinite perimeter, The German mathematician, Greg Cantor, who was one of the founders of set theory, discovered the Cantor fractal. This fractal is similar to the Koch Curve. Similar to the Koch curve, we start out with one straight-line segment and divide it into three parts.

Then remove the middle third, but keep the end points. So we started out with one line and two endpoints, and turned it into two lines and four endpoints. This process is to be repeated over and over, and the outcome after many times, would be points. \begin \begin \hline Steps & Number of Line Segments& Length of Line Segments\\ \hline 1& 2^1=1 & 1/3^(1)=1/3\\ \hline 2& 2^2=4 & 1/3^(2)=1/9\\ \hline 3& 2^3=8 & 1/3^(3)=1/27\\ \hline n & 2^n& 1/3^(-n)\\ \hline \end \end As this table shows, the length of the lines tends to approach zero, as we increase the amount of steps we apply.

- Therefore this is why the Cantor Fractal is also known as the Cantor point-set.
- Another way of drawing the Cantor set to make it more clear, is to use horizontal bars rather than just straight lines.
- By repeating the steps of removing the middle third, we get a picture that looks more like a comb,
- In 1977, when Benoit Mandelbrot was exploring the concept of what a fractal was, he also discovered that fractals have dimension.

He explained that fractals have capacity and it can be defined by a formula. “The concept of fractal dimension provides a way to measure how rough fractal curves are. The more jagged and irregular a curve is, the higher ts fractal dimension, a value between one and two.

- Fractional dimension is related to self-similarity in that the easiest way to create a figure that has fractional dimension is through self-similarity”,
- Before explaining what the formula depends on, we need to be sure that when we are zoomed into a fractal image, the boundary to what we are zoomed in at must match that of the entire fractal.

After that is confirmed, the number of pieces or line segments that fit into the larger one must be defined. We will call this number n. We also need to know the scale or how many times we have magnified into the entire fractal itself, which we will call M.

- We can then define the fractal dimension to be: \beginequation\begin D= (log(n)) /logM/nonumber//\end \endequation Using this equation, we can apply it to the fractal that was just discussed, the Koch curve.
- First we can look at the dimension of the Koch curve after doing the process just one time.
- So, there is one line fragment that is divided up into four with an equilateral triangle in the middle.

Now we have four line segments and we can say $$n=4$$. Because these four pieces are 1/3 the length of the original line segment, then the scale or magnification is 3. So using the definition for fractal dimension, we have $$D=log4/log3$$, which gives us D= 1.26185 Because this number has a dimension greater than 1, and is an integer, then we can conclude that the Koch curve is indeed a fractal.

- Gaston Julia was a French mathematician who was interested in looking the behavior of the orbit of a complex number when it is iterated under a function.
- This means that for a function, namely f, we apply it to a complex number.
- Whatever the result is, apply the same function to the new value.
- Julia repeated this action over and over again to see how the results acted.

He came up with the idea of the prisoner set and the escape set by studying how iterating certain functions will give bounded or unbounded sets. A prisoner set refers to all the complex numbers in its orbit is bounded and an escape set refers to the complex numbers that are unbounded in its orbit under a certain function.

An example of a prisoner is the value $$z_0 = 2$$ under the function $$ f(z)=z^2-z+1. By iterating this function starting with $$(1+i)$$ we get: \beginequation\begin z_0=1+\imath\nonumber\\ z_1=f(1+\imath)=(1+\imath)^2-(1+\imath)+1=\imath \nonumber \\ z_2=f(\imath)=\imath^2-\imath+1=-\imath\nonumber\\ z_3=f(-\imath)=(-\imath)^2-(-\imath)+1=\imath\nonumber\\ z_4=f(\imath)=-\imath\nonumber\\ \end \endequation Because the outputs switch back and forth, we call the value $$z_0=1+\imath$$ a prisoner.

If we were to pick a value to start with and iterate it under a function and the values were to get infinitely large or small, then we would have an escapee, Understanding what prisoner and escape sets are gives a better understanding of what Julia sets are.

The Julia set is defined to be the boundary between the prisoner set and the escape set”, The functions that Julia looked at were of the form $$f(z)=z^2 +c$$ where c represents a complex constant. For each different c that we use in this function we will get a different Julia set. We choose $$z_0$$ to start with whatever we choose for c.

It has been shown that if the orbit of the starting value, $$z_0$$, is ever located outside of the circle which has radius 2, then the rest of the orbit will be unbounded and therefore an escapee, After choosing a value of c that we want to use in our function we can generate the Julia set on computer.

- We use whatever complex number we have chosen as the starting value of $$z_0$$.
- For each pixel on the computer screen, the color black is assigned if the value is bounded or a prisoner and is colored white if it is not bounded and tends to infinity,
- This is where all these colorful art images that we see comes from.

Many use different color schemes, instead of black and white, to get these vibrant images. Another famous set that in a way relates to the Julia set was discovered by Benoit Mandelbrot and is called the Mandelbrot set. Mandelbrot also used the same exact function that Julia used, $$f(z)=z^2 +c$$.

Mandelbrot wanted to find the values of c that made the set bounded and similarly the values of c that made the orbit unbounded when this function was iterated always starting with $$z=0$$. However, what was drastic difference between what Gaston Julia could do with his data and what Mandelbrot could do with his.

The time that Mandelbrot was studying fractals was at a much later date than when Julia was, at a time when computers were in existence. As a result, Mandelbrot could do hundreds of thousands of iterations of a function using a computer, where Julia could only compute a small number of them by hand.

- After Mandelbrot gathered his Julia sets, he plotted them on a graph, which results in the Mandelbrot set,
- Following is a set of steps, or an algorithm, for developing a Mandelbrot set by computer: \begin \item Choose a part of the complex plane and divide it up into a grid of c values.
- Item Define n to be how many points of each orbit it will take you to decide whether it is bounded or unbounded.

\item Use the function $$f(z)=z^2+c to iterate the first n points of the orbit starting with the value, 0. \item If the orbit is unbounded, then color the corresponding c value on the grid a certain color.\item If the orbit is bounded, then color the corresponding c value on the grid a different color.

Item Move on to the next c value and keep repeating until all c values are accounted for\end There are many algorithms for creating your own fractals on computer and many websites that can show the properties of self-similarity in the Mandelbrot sets, and in creating original fractals by just choosing certain complex numbers.

There are many advantages in knowing and understanding what a fractal is, no matter what subject area you are working in. Fractals have been observed in just about every living thing in nature from trees in the rainforest to our human bodies. But fractals have also helped us propel forward immensely in the field of technology and entertainment.

- Fractals, or more specifically the Koch snowflake, were used to make the antennas of our cell phones smaller while increasing the amount of frequencies they can receive,
- Fractals are also used vastly in the medical field.
- One such example is that the healthy heartbeat when recorded on paper has a fractal pattern.

This has helped doctors anticipate people who might have heart problems in the future, One other example of where fractals are used is in computer science to help increase the believability of special effects and graphics in today’s movies, One specific application of fractals in special effects in the movies involves the movie, Star Wars: Episode III.

The part of the movie is when the two heroes run onto the end of a giant mechanical platform and a huge substance of lava comes crashing down in front of them. The initial process that they used to produce this lava was to make it appear that the lava is being shot up from a jet down below the mechanical platform.

At first the graphics of the lava looked extremely unrealistic, and emerged as just a straight cylinder of lava flowing up through the air. The creators wanted this to look more realistic, so they took the idea of the fractal and applied it to this cylinder spiral shape of the lava.

- They took the original shape, shrunk it down and reapplied it.
- They repeated this over and over again to get a extremely realistic huge ball of fire and lava \cite,
- Many may not know or understand how movies that are solely computer generated suddenly started.
- The first movie to have a complete computer generated sequence was Star Trek II: The Wrath of Khan\cite,

This movie was created in 1982, less than ten years after Mandelbrot made the idea of fractals public. Fractals is what makes computer generated imagery possible, and without Mandelbrot’s discoveries, we would not have the amazing graphics in our movies and games today.

- The grand discovery that Benoit Mandelbrot had made about fractals has changed the way we make movies, video games, computer games, etc.
- Fractals has made it possible to make real life images through a computer.
- Loren Carpenter, who is now works at Pixar animation studios, is known for his computer science work taking fractals and using them to create computer generated images.

He used to work for Boeing aircraft and his job was to see how the planes they were idealizing and creating looked while in flight. He wanted to make mountains to put in the background to make it look as realistic as possible. So after reading Fractals: Form, Chance, and Dimension by Benoit Mandelbrot, he learned that every surface can be broken down into smaller simpler shapes.

- He was then able to create images triangles into four smaller ones and kept on iterating until he had the jagged form of mountains,
- In conclusion, fractals are geometrical figures that have identical repeating patterns on a scale that reduces infinitely.
- At first, when looking at the colorful picture of a fractal, one might think that it is just a creative piece of artwork.

However after studying the mathematical background behind them, they have so much more depth than being just a piece of art. Benoit Mandelbrot is considered to be the father of fractal geometry and coined the term, “fractal.” He showed that every living and non-living thing in this world can be broken down into mathematical terms using fractal geometry.

- His first experiment of trying to calculate the length of Britain’s coastline led to this theory.
- The two main principles that define a fractal are self-similarity and fractal dimension.
- Self-similarity simply means that a pattern looks the same no matter how much it magnified.
- Fractal dimension is important because it shows that fractals have capacity and that they are not just flat images.

Along with Benoit Mandelbrot, WacLaw Sierpinski,Gaston Julia, Helge von Koch Greg Cantor are all very important mathematicians in the history of fractal geometry. All of these mathematicians have their own forms or types of fractals that they have discovered.

One of the most popular is the Mandelbrot set, which describes the iteration of a function using complex numbers. Even though the subject of fractal geometry is purely math, many people have found ways to take it outside math into computer science, health science, and even fashion and the arts. Fractals are used to make our movies look more amazing than life, and to make our computer and video games feel like we are right there in them.

The discovery of fractals have allowed us to decrease the size of our cell phones every year and at the same time they have helped doctors anticipate heart problems in our bodies way before they happen. Without the discovery of fractals, our technology, entertainment, our health, etc.

Would not be where it is today. References Bourke, P. Julia Set Fractal (2D)(2001). http://local.wasp.uwa.edu.au/~pbourke/fractals/juliaset/ Hoffman,J. Benoit Mandelbrot, Novel Mathematician, Dies at 85. The New York Times (2010). http://www.nytimes.com/2010/10/17/us/17mandelbrot.html?_r=1 Jersey,B.,And Shwarz,M.Hunting the Hidden Dimension.(2008).

http://www.pbs.org/wgbh/nova/fractals/ Laubender, P. “What is a Fractal?” Fractaline (1999). http://www.peter-laubender.de/fractaline/what_is_a_fractal.htm Lauwerier, H.Fractals Endlessly Repeated Geometrical Figures.Princeton, New Jersey. pgs 13,15-16, 32-33,(2010).

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#### What was the first fractal?

Unobtainable The item(s) or effects described on this page exist as functional game items, but cannot be acquired through normal gameplay. First Fractal Projectile created

First Fractal

The First Fractal is an unobtainable melee weapon that was originally supposed to be the new endgame sword for 1.4.0.1, but the concept was discarded during development. Its place is now taken by Zenith, though this item is still in Terraria ‘s code. It strongly resembles the Terra Blade, A player using the First Fractal against Target Dummies, Using the weapon summons projectiles that resemble copies of the player (with all dyes removed) holding one of 15 different sprite variations of the sword in a flying barrage of attacks that follow the cursor.

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## What are 3 well known fractals?

Generating fractals – Three common techniques for generating fractals are:

Escape-time fractals — These are defined by a recurrence relation at each point in a space (such as the complex plane). Examples of this type are the Mandelbrot set, Julia set, the Burning Ship fractal and the Lyapunov fractal. Iterated function systems — These have a fixed geometric replacement rule. Cantor set, Sierpinski carpet, Sierpinski gasket, Peano curve, Koch snowflake, Harter-Heighway dragon curve, T-Square, Menger sponge, are some examples of such fractals. Random fractals — Generated by stochastic rather than deterministic processes, for example, trajectories of the Brownian motion, Lévy flight, fractal landscapes and the Brownian tree. The latter yields so-called mass- or dendritic fractals, for example, diffusion-limited aggregation or reaction-limited aggregation clusters.