I would like to eliminate a possible confusion now.The Mandelbrot set is a special case of a fractal.The reason its name is up there in the co-starring role is Benoit Mandelbrot – invented a mathematics to describe fractals and he also discovered the mouse famous fractal type of them all.

I am going to talk about them because

- it looks like this branch of science makes can make sense of the structure of the natural world.They are ubiquitousThey are beautifulThey appeal to humans's intuitive sense that life doesn’t have an exact answer – How long is the Nile river.

The key word in a discussion of fractals is similarity.

What tools do we have to define similarity – How similar is this to this -

Imagine we go around the table and each of us has to answer this question : How similar are these people. Your answer must be a number. And units. I won't accept percent as your units of similarity because if you think of it % of what ?

We are not going to acquire tools here – too much math

What I hope you will take a away from today's presentations.

Our generations are ill equipped by our education system to use the word similar in a meaningful way – our day to day usage is very very vague.

1.

Despite this – similarity is a profound component of understanding natural processes and natural structures.

2.

Fractals and the Mandelbrot SetMay 30, 2016 9:43 AM

Intro Page 1

http://www.oxforddictionaries.com/definition/english/fractalA curve or geometrical figure, each part of which has the same statistical character as the whole. They are useful in modelling structures (such as snowflakes) in which similar patterns recur at progressively smaller scales, and in describing partly random or chaotic phenomena such as crystal growth and galaxy formation.

http://www.dictionary.com/

a geometrical or physical structure having an irregular or fragmented shape at all

scales of measurement between a greatest and smallest scale such that certain mathematical or physical properties of the structure, as the perimeter of a

curve or the flow rate in a porous medium, behave as if the dimensions of the structure (fractaldimensions) are greater than the spatial dimensions.

Collins English Dictionarya figure or surface generated by successive subdivisions of a simpler polygon or

polyhedron, according to some iterative process

fractalfoundation.orgA fractal is a never-ending pattern. Fractals are infinitely complex patterns that are self-similaracross different scales. They are created by repeating a simple process over and over in an ongoing feedback loop

http://www.mathgoodies.com/A fractal is a figure with repeating patterns containing shapes that are like the

whole but of different sizes throughout.

Gives you an idea that this is not well settled science.

Red Highlight are useful

Purple Highlights hint at some of the power of fractals but have no place in a definition.

Self Similar

An object is said to be self-similar if it looks "roughly" the same on any scale

Fractals - DefinedMay 30, 2016 10:00 AM

Intro Page 2

An object is said to be self-similar if it looks "roughly" the same on any scale

Fractals are types of self similar objects

Getting rid of qualitative words like 'Roughly' requires some very complex math and I will leave that out because I don’t want to appear more ignorant than I do already.

Intro Page 3

Makes you think that fractals were discovered by someone doodling instead of making notes during a lecture.

Fractal Trees (Processing)

How to draw a Paisley Pattern

What do fractals look like ?May 30, 2016 10:30 AM

Intro Page 4

HOW LONG IS THIS ?

ORTHIS

Agreed a stupid question but what I want to make you think about is – wow there are so many subparts to subparts

✐ How To Draw Fractal Tree

Growing Pythagoras tree

Intro Page 5

How to Make a Fractal

Intro Page 6

What else do they look like ?May 30, 2016 10:41 AM

Intro Page 7

Intro Page 8

Intro Page 9

Intro Page 10

Intro Page 11

Intro Page 12

And hence the reason that the Mandelbrot Set – Fractals and Self-Similarity have blended into a confusing set of labels.

These images exploit the properties of the mandelbrot to provide a visual representation.

ZOOMING Mandelbrot

Mandelbrot Zoom 10^227 [1080x1920]

Zooming is a meaningless concept for a Fractal

Because : is a never-ending pattern. that are self-similar across different scales.

And most famously – the Mandelbrot SetMay 30, 2016 11:06 AM

Intro Page 13

Fractal Tree Generator - Thomas Wolfe

Play at homeMay 30, 2016 10:24 AM

Intro Page 14

Beauty

Ubiquitous

They are all over the natural world.

Mandelbrot famously wrote: "Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line."

Take AwayMay 30, 2016 11:11 AM

Intro Page 15

How to calcMandelbr...

With non-complex numbers, such as real, rational or natural numbers, the squaring iteration must always go to infinity unless the star ting number is

one. No matter how many times you square one it will still equal one. But just the slightest bit more or less than one and the iteration of squaring will

attract it to the infinitely large or small. The same behavior holds true for complex numbers: numbers just outside of the ci rcle z = 1 on the complex

plane will jump off into the infinitely large, complex numbers just inside z = 1 will quickly square into zero.

But the magic comes by adding the constant c (a complex number) to the squaring process and starting from z at zero: z -> z^2 + c. Then stable

iterations - a set attracted to neither the infinitely small or infinitely large - become possible. The potentially stable Complex numbers lie both outside

and inside of the circle of z = 1; specifically on the complex plane they lie between -2.4 and .8 on the real number line, the horizontal x grid, and

between -1.2 and +1.2 on the imaginary line, the vertical y grid. These complex numbers in effect stay within the meso-cosmic realm, the world of

Man, even if the z -> z^2 + c iteration process goes on forever. These numbers are contained within the black of the Mandelbrot fractal.

From <http://www.fractalwisdom.com/science-of-chaos/the-mathematics-of-chaos/>

The MandelBrot SetMay 20, 2016 12:11 PM

The Mandelbrot Set Page 16

The Mandelbrot Set Page 17

The Mandelbrot Set Page 18

The Mandelbrot Set Page 19

The Mandelbrot Set Page 20

The set of Natural Numbers. 1,2,3,4 ...The set of Real Numbers. -1......0....all the fractions .....1.....2

These may all be though of as points in a line.One dimensional

Two dimensional Numbers are pairs of numbers.Its most common two take two lines of numbers and make the lines perpendicular to each other and give each line of numbers a name. Lattitude and Longitude for example. X and Y are other names commonly used (in algebra).

We think of them as a location on the plane formed by two lines intersecting each other.Hold your arms stretched forward and cross them at the elbows and satisfy yourself that you are defining a plane.

Now lets say each pair of numbers that define. 2 dimensional number consist of 1 Real Number and 1 imaginary number.

You are fine with the idea of two numbers defining a 2 dimensional point. And giving each of the lines of numbers arbitrary namesSo calling them imaginary is just a name.Kind of

The similarities between real and imaginary numbers is this

As far as the numbers go, the imaginary numbers are just like the real numbers. For every one of the infinite number of real numbers there is amatching imaginary number of the same value.

The difference between real and imaginary numbers is this

They obey different rules of maths.In the world of imaginary numbers when you multiply an imaginary number by itself the result is a negative number.

Said another way - the imaginary component of an imaginary number is give the name i (little letter i). And the size of an imaginary component is the size of the number (R) multiplied by i where i has this special property i^2 = -1

http://www.hiddendimension.com/fractalmath/Divergent_Fractals_Main.html

Complex numbers are numbers with both a real and an imaginary part, and are expressed

in the form a + bi, where i is the square root of -1. Another notation for complex numbers

Imaginary NumbersMay 20, 2016 11:53 AM

The Mandelbrot Set Page 21

in the form a + bi, where i is the square root of -1. Another notation for complex numbers

is (a, b). The following examples illustrate the behavior of complex numbers:addition/subtraction: (a + bi) + (c + di) = (a + c) + (b + d)imultiplication: (a + bi) * (c + di) = (ac - bd) + (ad + bc)idivision: (a + bi)/(c + di) = [(ac + bd) + (bc - ad)i]/(c2 + d2)absolute value of (a + bi): sqrt(a2 + b2)Complex numbers are typically illustrated graphically by using a graph where the real component of the number is graphed on the X axis and the imaginary component is graphed on the Y axis.

From <http://www.hiddendimension.com/fractalmath/Divergent_Fractals_Main.html>

The Mandelbrot Set Page 22

Mathematically, the Mandelbrot set is defined on the plane of complex numbers by picking a starting point cc and

iterating the formula zk+1=z2k+czk+1=zk2+c. The iteration gives you a sequence of numbers that either stays bounded

or spirals out of control further and further from the starting point. The complex number cc belongs to the Mandelbrot

set if the sequence stays within a radius of 2 from the origin.

Plotting the Mandelbrot set is easy: map each pixel on the screen to a complex number, check if it belongs to the set by

iterating the formula, and color the pixel black if it does and white if it doesn’t. Since the iteration may never end we set

a maximum

First of all we need a way to represent complex numbers. Some programming languages like Python include a built-in

complex number type which we could use to implement the iteration using the above formula directly. Other languages

such as Java or JavaScript don’t include complex numbers, but not to worry: we can represent the complex

number z=x+iyz=x+iy as the pair of real numbers (x,y)(x,y). In this representation the Mandelbrot set iteration

becomes:

xk+1yk+1=x2k–y2k+Re c=2xkyk+Im cxk+1=xk2–yk2+Re cyk+1=2xkyk+Im c

All that’s left now is figuring out how to map pixels to complex numbers. That’s an easy task: we want the center of the

image to be mapped to (0,0), so given a pixel we subtract half of the image height from the vertical coordinate, and half

of the width from the horizontal coordinate. Next, the scale: we know that the Mandelbrot set lies within a circle of

radius 2, so the entire width of the image should have length 4. This gives us the following program for plotting the

Mandelbrot set in a C-like language:

From <http://jonisalonen.com/2013/lets-draw-the-mandelbrot-set/>

for (int row = 0; row < height; row++) { for (int col = 0; col < width; col++) { double c_re = (col - width/2.0)*4.0/width; double c_im = (row - height/2.0)*4.0/width; double x = 0, y = 0; int iteration = 0; while (x*x+y*y <= 4 && iteration < max) { double x_new = x*x - y*y + c_re; y = 2*x*y + c_im; x = x_new; iteration++; } if (iteration < max) putpixel(col, row, white); else putpixel(col, row, black); }}

From <http://jonisalonen.com/2013/lets-draw-the-mandelbrot-set/>

Computing the Mandelbrot using a computerMay 20, 2016 12:36 PM

The Mandelbrot Set Page 23

}

From <http://jonisalonen.com/2013/lets-draw-the-mandelbrot-set/>

Where are the colors ?

We only used black and white. In the set or out the set.

Lets enter the non binary world.

The numbers outside the set. Some get very big very fast and some take many iterations to get big.

So now we have 3 colors. Black inside the set. White - outside the set and get big very fast. A third color for numbers that get big more slowly.

Now we can refine and add more colors based on the speed (number of iterations) at which the numbers race towards infinity.

Since the Mandelbrot set has a very fine structure, by plotting only

black or white we don’t get any idea of how close a pixel is to the set.

The number of iterations gives us some idea of that, so let’s use it to

color the pixels that don’t belong to the set. The change needed in the

code is trivial: define a color map and use it instead of white: if (iterations < max) putpixel(col, row, colors[iterations]);

(Implementation in Java.) The image produced by this program is much

prettier and gives a better idea of the structure of the set, with fine

tendrils springing from the bulbs.

The Mandelbrot Set Page 24

tendrils springing from the bulbs.

From <http://jonisalonen.com/2013/lets-draw-the-mandelbrot-set/>

The Mandelbrot Set Page 25

Aa

http://www.hiddendimension.com/fractalmath/Mathematics_Main.html

Divergent Fractals These are the "classic" fractal functions, typified by the Mandelbrot set and the corresponding Julia sets. "Divergent" relates to the regions that are most interesting in creating striking images and art.

Convergent Fractals These are fractals generated using convergent iterative methods such as Newton's Method. You may be surprised - some of the images may look like they came from a "divergent" function!

Hyperbolic Tessellation Fractals Hyperbolic geometry is a non-Euclidean geometry developed independently by Nikolai Lobachevski and Farkas Bolyai. The work of the artist M.C. Escher contains examples of Hyperbolic Tessellation Fractals.

3D Fractals and Higher DimensionsA Mandelbrot set and its corresponding Julia sets comprise the 4-dimensional Juliabrot. Quaternions and Hypercomplex objects are also 4D fractals. 3D fractals are 3-dimensional cuts through these 4-D objects.

Circle and Sphere Inversion Fractals Circle and sphere inversions are closely related to Möbius transformations. The best known

fractal example is the Apollonian Gasket.

Kleinian Group Fractals Mobius transformations, which are also known as fractional linear transformations, can be

used to generate a variety of fractals. This has been popularized by the book "Indra's Pearls" by Mumford, Series and Wright.

Height Field Fractals For a 2-dimensional fractal the 3rd dimension is a function of the number of iterations, the

fractal magnitude, or the orbit trap value.

Strange AttractorsDynamical systems are models with rules that describe the way a quantity changes with time. These systems can behave a strange attractor. It can be graphed but its behavior is complex and unpredictable. Like leaves in the wind, it is impossible to predict where the leaves will end up.

Iterated Function System (IFS) FractalsIFS fractals were developed by Michael Barnsley. The original IFS formulas are called contractive affine transformations, which provide specifications for the self-similarity of the fractal. IFS fractals can also be created using Möbius transformations.

L System FractalsDeveloped by Aristid Lindenmayer to model the morphology of organisms. It is an iterative turtle graphics system.

Divergent Fractals

Mandelbrot fractals are the result of iterating a fractal formula. A fractal formula is a statement

like:

z = z^2 + c

Divergent Fractals

Julia Fractals

A 2nd algorithm begins each orbit by setting z to the complex value associated with the pixel, and c to a fixed complex value,

Types of FractalsMay 30, 2016 11:13 AM

Types of Fractals Page 26

This statement takes 2 complex values found in the variables z and c, and combines them based on the expression to the right of the equal sign; in this case, by squaring z and adding c to the result. The resulting complex value is assigned to the variable z, replacing the previous value of z. This completes the 1st iteration of the formula. A 2nd iteration would evaluate the expression again, this time starting with the new value of z computed in the 1st iteration. This process continues with each step producing a new value for z. The process terminates when the magnitude of z exceeds some threshold value or the specified maximum number of iterations is reached. The magnitude of z is the distance of the point z from the origin of the complex plane (0+0i). The set of all the z values over the entire iteration of the fractal formula is called the orbit and the different z values are called the orbit points.

To produce a fractal image from this process, a window is mapped onto the complex plane composed of a grid of points called pixels. The number of rows and columns of the grid is determined by the size and resolution of the window. For each pixel in the window, we set z to an initial z value, substitute the complex value associated with the pixel into the formula for c, and iterate the formula. If the iteration terminates because the magnitude of z exceeds the threshold, the pixel is said to have escaped and is outside of the Mandelbrot set. The pixel is colored based on the number of iterations it took to escape and/or other characteristics of z. If the magnitude does not exceed the threshold when the maximum number of iterations is reached, the pixel is assumed to be inside the Mandelbrot set and is usually colored black but other colorings are possible based on the value of z when the iteration is terminated. Of course, the pixel may have escaped had we only executed additional iterations of the formula. Fractals generated using this algorithm, are called Mandelbrot fractals.

http://www.fractalsciencekit.com/types/classic.htm

A 2nd algorithm begins each orbit by setting z to the complex value associated with the pixel, and c to a fixed complex value, called the Julia Constant. The rest of the algorithm is identical to the algorithm described above. Fractals generated in this way, are called Julia fractals.

The choice of the Julia Constant in large part controls the character of the resulting Julia fractal. Surprisingly, the Mandelbrot fractal for the same fractal formula provides the best interface for choosing the Julia Constant. It turns out that the best choice for a Julia Constant is a point on the complex plane near the Mandelbrot set boundary. The resulting Julia fractal will have many of the characteristics of the Mandelbrot fractal in the neighborhood of the Julia Constant. The Mandelbrot fractal can be used as a map for choosing the Julia Constant.

http://www.fractalsciencekit.com/types/classic.htm

Types of Fractals Page 27

http://www.bbc.com/news/magazine-11564766

The whole universe is fractal, and so there is something joyfully quintessential about Mandelbrot's insights.…

Animation: Fractals are used in many computer games to render realistic graphics for mountains, landscapes & 3D terrains, especially for flight simulations, computer games, digital artworks & animations.

computer file compression systems

Architecture of the networks that make up the internet

Diagnosing some diseases.

Dealing with complexity - earthquakes

Financial Markets

Antenna and other geometry exploiters

Fluid Mechanics

L-Type are used to model many biological systems

Seismology

Generation of patterns for camouflage, such as MARPAT

Medicine[22]•

http://kluge.in-chemnitz.de/documents/fractal/node2.html

Fractals will maybe revolutionize the way that the universe is seen. Cosmologists usually assume that matter is spread uniformly across space. But observation shows that this is not true. Astronomers agree with that assumption on "small" scales, but most of them think that the universe is smooth at very large scales. However, a dissident group of scientists claims that the structure of the universe is fractal at all scales. If this new theory is proved to be correct, even the big bang models should be adapted. Some years ago we proposed a new approach for the analysis of galaxy and cluster correlations based on the concepts and methods of modern Statistical Physics. This led to the surprising result that galaxy correlations are fractal and not homogeneous up to the limits of the available catalogues. In the meantime many more redshifts have been measured and we have extended our methods also to the analysis of number counts and angular catalogues.The result is that galaxy structures are highly irregular and self-similar. The usual statistical methods, based on the assumption of homogeneity, are therefore inconsistent for all the length scales probed until now. A new, more general, conceptual framework is necessary to identify the real physical properties of these structures. But at present, cosmologists need more data about the matter distribution in the universe to prove (or not) that we are living in a fractal universe.

Astronomy

A fractal heat exchanger designed by Deb Pence at Oregon State University, and etched in silicon. Photo courtesy of Tanner Labs

http://fractalfoundation.org/OFC/OFC-12-2.html

Bio / Chem: Bacteria Cultures Chemical Reactions Human Anatomy Molecules Plants Population Growth Other: Clouds Coastlines and Borderlines Data Compression Diffusion Economy Fractal Art Fractal Music Landscapes Newton's Method Special Effects (Star Trek) Weather

Where we find and use themMay 30, 2016 11:42 AM

Why are Fractals Amazing Page 28

An excerpt from a publication of Amalgamated Research Inc, showing some manufactured fractals used for fluid mixing.

Image courtesy of Amalgamated Research Inc.

The fractals above are developed by Amalgamated Research Inc and are licensed to industrial producers who need to mix fluids together very carefully. The standard way of mixing fluids -stirring - has some important drawbacks. First of all, it produces turbulent mixing, which is unpredictable, so two different batches may not end up identically mixed. Secondly, turbulent mixing is energy intensive and is disruptive to delicate structure.

The engineered fractal fluid mixers provide a different solution to the problem of mixing. Here, there is an actual physical branching fractal network of tubes that can distribute fluid thoroughly into a chamber containing another fluid. This system provides a low -energy reproducible way of mixing chemicals. Fractal networks with different space -filling properties can be custom-made for chemical systems requiring different properties.

Examples of fluid-mixers like these have been used in fields as diverse as high-precision epoxies, the manufacturing of sugar, and chromatography - used to sample and determine small quantities of biological molecules.

Patent drawing for a Sierpinski fractal dipole antenna.

Images courtesy of USPTO.

Cellphone with a Sierpinski Gasket antenna.

Image courtesy of Fractenna Inc..

Fractal patterns can also be found in commercially available antennas, produced for applications such as cellphones and wifi systems by companies such as Fractenna in the US and Fractus in Europe. The self-similar structure of fractal antennas gives them the ability to receive and transmit over a range of frequencies, allowing powerful antennas to be made more compact.

Fractal MedicineModern medicine often involves examining systems in the body to determine if something is malfunctioning. Since the body is full of fractals, we can use fractal math to quantify, describe, diagnose and perhaps soon to help cure diseases.

Why are Fractals Amazing Page 29

A cross-section of a lung showing both emphysema and lung cancer. Image courtesy of the University of Iowa.

With modern imaging equipment such as CT scans and MRI machines, doctors can have access to a huge amount of digital data about a patient. Making sense of all the data can be time-consuming and difficult even for trained experts.

Teaching computers to use mathematical processes to tell the difference between healthy lungs and lungs suffering from emphysema promises to help make faster, more reliable diagnoses. The fractal dimension of the lung appears to vary between healthy and sick lungs, potentially aiding in the automated detection of the disease.REF

A schematic diagram of normal blood vessels, abnormal tumorous blood vessels, 'renormalized' blood vessels, and inadequately space-filling blood vessels.

Image courtesy of Edwin L. Steele Laboratory, Harvard.

Cancer is another disease where fractal analysis may not only help diagnose but also perhaps help treat the condition. It is well known that cancerous tumors - abnormal, rapid growth of cells - often have a characteristic growth of new blood vessels that form a tangled mess instead of the neat, orderly fractal network of healthy blood vessells. Not only can these malfunctioning vessels directly harm the tissue, but they can also make it harder to treat the disease by preventing drugs from reaching into the inner parts of tumors where the drugs are most needed.

Fractal analysis of cancer may be applied in many avenues. To quote Dr. Larry Norton, a cancer researcher at Memorial Sloan Kettering Cancer Center, '"Tumors have a higher fractal dimension than normal tissues indicating their greater internal complexity." Norton noted that fractal dimensions have a tremendous buffering capacity, in that they grow in value even as the tumors themselves change little in terms of their apparent size. But once the dimension reaches a threshold value, the system changes radically, much as a ball traveling across a table drops when it reaches the edge. "That's what happens with cancer," he explained. "People can go out and smoke and not have cancer and then suddenly they do. We're talking here about the power constants of the fractal dimensions - one incremental change gets them in trouble. Imagine if we understood the genes that control that power function. If we could understand those molecular changes, we might have a whole new target for intervention." ' REF

http://fractalfoundation.org/OFC/OFC-12-4.html

Why are Fractals Amazing Page 30

Cables and Bridges

A modern day recreation of an Inca rope suspension bridge. Image courtesy of Rutahsa Adventures, www.Rutahsa.com.

But first, let's look back a bit, at some fractal ideas people have used for centuries in the engineering of super-strong cables. Long before we built the Golden Gate Bridge, the Inca people in South America were building bridges across mountainous ravines that were hundreds of feet wide. The cables in these bridges were woven by hand out of long strands of stiff qoyagrass. Just as in a modern steel cable, a small number of fibers are woven into a larger fiber, several of which are then woven into a larger rope. This repetitive, fractal pattern provides great strength, and it allows a very long cable woven from thousands of individual pieces of grass no longer than a meter or two each.

"Garcilasco de la Vega, in 1604, reported on the cable-making techniques [of the Inca]. The fibers, he wrote, were braided into ropes of the length necessary for the bridge. Three of these ropes were woven together to make a larger rope, and three of them were again braided to make a still larger rope, and so on. The thick cables were pulled across the river with small ropes and attached to stone abutments on each side."REF

Modern engineers still use this same idea today in the construction of high strength cables that make possible such masterpieces as the giant suspension bridges like the Golden Gate Bridge. A steel cable is formed from a bundle of smaller cables which themselves are formed of smaller bundles, etc.

Golden Gate Bridge, San Francisco. Photo courtesy of Aaron Logan.

Cable technology is essential for building suspension bridges. The Inca weavers and the modern steel cable makers both use a repetitive process to create strong, fractal cable patterns.

Why are Fractals Amazing Page 31

Schematic showing a possible configuration of a steel cable bundle in cross section. Image courtesy of Bernard S. Jansen and Jonathan Wolfe.

The repetitive fractal pattern is evident in this cross-section diagram of the parts of a steel cable, but one thing that's not visible above is that the various 'strands,' 'ropes' and 'cables' are all are wrapped to spiral around each other, which greatly increases the total strength of the cable. The image below shows a piece of steel cable from the side, allowing you to see two levels of self-similar braided structure. Note that the bundles at both scales twist in the same direction.

Image courtesy of Bernard S. Jansen.

Why are Fractals Amazing Page 32

https://prezi.com/nwwuoguafu5o/fractals-in-video-games/

Fractal geometry used in games

Landscapes – modelling what the real world looks like -GamesMay 30, 2016 12:16 PM

Why are Fractals Amazing Page 33

Gw2 Fractal 15 - Lava Boss (Imbued Shaman).avi

Fractal Terrain 4. Weather, water and distance field objects. HD

Why are Fractals Amazing Page 34

Why are Fractals Amazing Page 35

https://en.wikipedia.org/wiki/Fractal#cite_note-cerebellum-17

Guy C. Van Orden, PhDA healthy heart beats in an aperiodic rhythm, not too regular or repetitive, and not too random or chaotic. The healthy rhythm lives between those extremes, exhibiting a pattern of fractal variability. Loss of fractal variability signals heart disease, pathological dynamics of the heart

n congestive heart failure, for example, the heartbeat is overly regular, corresponding to a low fractal dimension. And in atrial fibrillation, the heartbeat is overly random, corresponding to a high fractal dimension (Goldberger, 1996; Goldberger, Amaral, et al., 2002; Havlin, Amaral, et al., 1999).

Fractal rhythms appear widely at multiple levels of analysis in healthy physiology (Bassingthwaighte, Liebovitch, & West, 1994) including brain physiology and also in measurements of a healthy person's behavior. As concerns behavior, fractal behavior has been observed in perceptual learning, postural sway, and the timing of perceived reversals of a reversible Necker cube. It is found in motor performances such as spacing and timing of rhythmic movement and the phase relation between rhythmic movements. It is found in tapping, human gait, and repeated measurements of simple reaction time. It appears in controlled cognitive performances including mental rotation, lexical decision, visual search, repeated production of a spatial interval, repeated judgments of an elapsed time, simple classifications, and variation in word naming by skilled readers, an automatic cognitive performance. And finally it is present in variation of ratings of self-esteem and in mood ratings by bipolar patients (Gilden, 2001; Riley & Turvey, 2002; and Van Orden, Holden & Turvey, 2003, are reviews).

Clinicians have traditionally described the normal activity of the heart as ``regular sinus rhythm.'' However, contrary to subjective impression and clinical assumption, cardiac interbeat intervals normally fluctuate in a complex, apparently erratic manner, even in individuals at rest (Fig. 1a) [1, 19]. This highly irregular behavior defies conventional analyses that require ``well-behaved'' (stationary) data sets. Fractal analysis techniques developed above are good candidates for studying this type of time series where fluctuations on multiple time scales appear to occur.www.physionet.org

Fractal Heart

http://www.medicographia.com/2013/01/fractals-and-their-contribution-to-biology-and-medicine/

cellular membrane systems had fractal properties.

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Fractals and their contribution to biology and medicine

Health and Life sciencesMay 30, 2016 12:22 PM

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Fractals and their contribution to biology and medicine

by G. A. Losa, Switzerland

Gabriele A. LOSA, PhDFellow Member of the EuropeanAcademy of Sciences, Institute of Scientific InterdisciplinaryStudies (ISIS), LocarnoSWITZERLAND

The term Fractal coined by Mandelbrot from the Latin adjective fractus (fragmented, irregular) derives from the Latin verb frangere, meaning to break, to create irregular fragments. To be called fractals, biological and/ or natural objects must fulfill a certain number of theoretic and methodological criteria including a high level of organization, shape irregularity, functional morphological and temporal auto-similarity, scale invariance, iterative pathways, and a non-integer peculiar fractal dimension [FD]. Whereas mathematical objects are deterministic invariant and self-similar over an unlimited range of scales, biological components and morphological structures are iterated entities statistically self-similar only within a fractal domain called “scaling window,” ie, only within this scaling window can the scale-invariant (fractal) properties of an irregular object of finite size be observed. The latter needs to be experimentally established for each element, while the scaling range has to account for at least two orders of magnitude. The application of the fractal principle is very valuable for measuring dimensional properties and spatial parameters of irregular biological structures, for understanding the architectural/ morphological organization of living tissues and organs, and for achieving an objective comparison among complex morphogenetic changes occurring through the development of physiological, pathologic, and neoplastic processes. Emphasis will be laid on the fractal contribution to the knowledge of cell membranes, hematological tumors, cell tissue cancers, and brain tissues in healthy and diseased states.

Medicographia. 2012;34:365-374 (see French abstract on page 374)

The Fractal Geometry of Nature, Benoît Mandelbrot’s masterpiece, has provided a novel epistemological framework for interpreting real life and the natural world in a way that avoids any subjective view.1 Founded upon a body of well-defined laws and coherent principles, including those derived from chaos theory,2 fractal geometry allows the recognition and quantitative description of complex shapes, living forms, biologic tissues, and organized patterns of morphologic features correlated through a broad network of functional interactions and metabolic processes that shape adaptive responses and make the process of life possible. Obviously, this is in opposition to the ancient, conventional vision based on Euclidean geometry and widely adopted concepts, such as homeostasis, linearity, smoothness, and thermodynamic reversibility, which stems from a more intuitive—but artificially ideal—view of reality. In the chapter of his work entitled Epilog: The Path to Fractals, Benoit Mandelbrot wrote “The reader knows well that the probability distribution of fractals is hyperbolic, and that the study of fractals is rife with other power law relationships.” Although Mandelbrot’s famous seminal paper on statistical self-similarity and fractal dimension dates back to 1967,3 and the first coherent essay on fractal geometry was published 35 years ago,4 it is worth here recalling exactly how and when the ‘‘heuristic introduction’’ of this innovative discipline occurred or, more vividly expressed, when ‘‘the irruption of fractal geometry’’ into the life sciences such as biology and medicine actually took place.5

Although there no precise date can be given, it is generally agreed that fractal geometry was introduced during the ‘‘golden age’’ of cell biology—that is, between the 1960s and 1990s, under the impulse of Swiss and French groups.6,7 It was discovered that biologic elements, unlike deterministic mathematical structures, express statistical self-similar patterns and fractal properties within a defined interval of scales, termed “scaling window,” in which the relationship between the observation scale and the measured size or length of the object can be established and defined as the fractal dimension (FD).8 The fractal dimension of a biological component remains constant within the scaling window and serves to quantify variations in length, area, or

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remains constant within the scaling window and serves to quantify variations in length, area, or volume with changes in the size of the measuring scale. However, concrete “fractality” exists only when the experimental scaling range encompasses at least two orders of magnitude, namely, spans two decades on the logarithmic scale axis. Data spanning several decades of scale have been previously reported in many other fields: thus, defining a “scaling range” appears an inescapable requisite for assessing the fractality of every biological element. This emphasizes Mandelbrot’s statement “fractals are not a panacea; they are not everywhere.”9

To conclude, the fractal dimension is a statistical measure that correlates the morphological structural complexity of cellular components and biological tissues.10 Fractal dimension is also a numerical descriptor that measures qualitative morphological traits and self-similar properties of biological elements. Recourse to the principles of fractal geometry has revealed that most biological elements, whether at cellular, tissue, or organ level, have self-similar structures within a defined scaling domain that can be characterized by means of the fractal dimension.

Figure 1. Changes of surface density estimates for outer and inner mitochondrial membranes, with increased magnification.

After reference 6: Paumgartner D et al. J Microsc. 1981;121:51-63.© 1981, The Royal Microscopical Society.

Cell membranes and organelles

Application of fractal geometry to cell biology stemmed from the discovery that cellular membrane systems had fractal properties. What started it all was the uncertainty of observations regarding the extent of cell membranes in the liver, as findings from morphometry studies of liver cell membranes by various laboratories failed to match. This triggered much debate as to which of these estimates was correct, and whether liver cells contained 6 or 11 m2

of membranes per cm3, quite a significant difference. This cast doubt on the reliability of stereological methods, since they yielded conflicting results when measurements were made under different magnifications of the electron microscope. Ultimately, it was found that the estimates of surface density of liver cell membranes increased with increased resolution.6

Mandelbrot suggested that these results were attributable to a scaling effect, analogous to the ‘‘Coast of Britain effect.’’3 This explained why measurements of liver cell membranes at higher magnification yielded higher values than at lower magnification.6,11 This scaling effect applies mainly to cellular membranes with a folded surface or an indented profile, such as the inner mitochondrial membrane or the rough endoplasmic reticulum (ER). In fact, the surface density estimate of rough membranes was found to be increased with increasing magnification, while the surface density measure of the smooth outer mitochondrial membrane and of the smooth ER counterpart was only slightly affected by the resolution effect (Figure 1).6

Figure 2. Electron microscopy view of human lymphocytes.

A. Healthy human suppressor T lymphocyte [CD8] haracterized by a wrinkled cell surface. Magnification: 18 400×. B. Human lymphoblastsof acute leukemia (T-ALL), characterized by a smooth cell surface and a low fractal dimension.

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of acute leukemia (T-ALL), characterized by a smooth cell surface and a low fractal dimension. Magnification 18 400×. After reference 23: Losa GA et al. Pathol Res Pract. 1992;188:680-686. © 2011, Elsevier GmbH.

http://www.fractal.org/Life-Science-Technology/Publications/Fractals-and-Human-Biology.pdf

<<Fractals-and-Human-Biology.pdf>>Fractals and Human Biology

We are fractal. Our lungs, our circulatory system, our brains are like trees. They are fractal structures.

Fractal geometry allows bounded curves of infinite length, and closed surfaces with infinite area. It even allows curves with positive volume, and arbitrarily large groups of shapes with exactly the same boundary. This is exactly how our lungs manage to maximize their surface area.

Most natural objects - and that includes us human beings - are composed of many different types of fractals woven into each other, each with parts which have different fractal dimensions. For example, the bronchial tubes in the human lung have one fractal dimension for the first seven generations

of branching, and a different fractal dimension from there on in.

Our lungs cram the area of a tennis court into the area of just a few tennis balls.

The Three-Quarter Power Law

Fractal geometry has revealed some remarkable insights into a ubiquitous and mysterious "three -quarter" law. This particular power law models the way one structure relates to and interacts with another. It is based on the cube of the fourth root. Many three-quarter laws have emerged from the measurement of seemingly unrelated systems, modeling the way that one structure varies with another.

For a long time now, physiologists have had an empirical understanding of how much

blood flows through our circulatory system, and how this relates to the physical size of the vessels that carry it. Research employing fractal rules has revealed a three-quarter power rule law even in the circulatory system.

Our arteries, which account for just 3 per cent of our bodies by volume, can reach every cell in our bodies with nutrients. In the kidneys and lungs, our arteries, veins, and bronchioles all manage to intertwine around a common boundary.

The arteries that deliver the blood, and the veins that take it away, need to share a common interface with the surface of the lungs, in order to aerate the blood. The arteries must provide every cell in our body with nutrients, using the minimum amount of blood.

The kidneys, the liver, the pancreas are all organs constructed along self-similar fractal rules. So too is the most remarkable of all those we know on the planet - the human brain.

The Mysterious Brain

One thing we can say with certainty about the brain is that it is a very fractal piece of kit ! It has an obvious fractal structure. You have only to look at it to see that. It is very crinkled and wrinkled and highly convoluted, as it folds back and back on itself.

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"There is a natural evolutionary route from universal mathematical patterns to the laws of physics to organs as complex as the brain." ... Ian, the English Fractal Guy

It is deeply ironic that this remarkable organ, which is the seat of the mind, and which either created or discovered (we don't know which) the mathematical rules on which it and the entire universe turns, cannot explain or understand its own functioning.

Understanding how our brains function is probably the greatest challenge facing the scientific community at this time. Fractal geometry is at the leading edge of research in this area.

Fractals and Medical Research

All aspects of nature follow mathematical rules and involve some roughness and a lot of irregularity. For example, complex protein surfaces fold up and wrinkle around towards three-dimensional space in a dimension that is around 2.4. Antibodies bind to a virus through their compatibility with the specific fractal dimension of the surface of the cell with which they intend to react.

Consequently, many of the current developments and findings in fractal geometry are in work with surfaces.

Viruses and Bacteria

The receptor molecules on the surfaces of all viruses and bacteria are fractal. Their positioning techniques, the methods they use to determine the chemistry of the body they have invaded and how they will interfere with that body's chemistry, and their binding functions, emerge mathematically by way of the deterministic rules of fractal geometry.

AIDS

The dynamics of the AIDS virus in the human body has been modeled with fractal geometry, which provides the answer to the long-standing puzzle surrounding the unusually long incubation period of the AIDS virus. Many patients remain HIV positive for as long as ten years before the virus decides to kick in, and the onset of the full-blown disease reveals itself in the body.

As the immune system begins to fall apart, the AIDS virus starts to behave chaotically. Studies of the virus at this stage have revealed significant changes in the fractal structure.

Fractal geometry unravels the structural differences that occur at the end of the incubation period of the virus.

Detecting Cancer

The surface structures of cancer cells are crinkly and wrinkly. These convoluted structures display fractal properties which vary markedly during the different stages of the cancer cell's growth.

Fractal geometry I being employed in the initial detection of the presence of cancer cells in the body.

Using computers, mathematical pictures can be obtained, which reveal whether or not cells are going cancerous. The computer is able to measure the fractal structure of cells. If cells are too fractal, it spells trouble. There is something wrong with those cells.

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trouble. There is something wrong with those cells.

Women at Risk

The fractal dimension of cancerous material is higher than that of healthy cells. Alan Penn, who is Adjunct Professor of Mathematic and Engineering at George Washington University, describes his work in this area, "MRI Breast Imaging may improve diagnosis for the 4,000,000 woman at risk for whom mammography isn't effective. Clinical application of MRI has been hampered by difficulty in determining which masses are benign and which are malignant. Research has focused on developing robust fractal dimension estimates which will improve discrimination between benign and malignant

breast masses."

Bubbly Bones and Breaks

Bones contain air bubbles. Bone fractures are fractal. Fractal geometry is being applied particularly and most effectively in the healing of brittle bone fractures.

Fractal Beats

The body structures of all of nature's animals are fractal, and so too is their behavior and even their timing.

Our heartbeats seem regular and rhythmical, but when the structure of the timing is examined in fine detail, it is revealed to be very slightly fractal. And this is very important.

Our heartbeats are not regular. There is an important tiny variation.

This fine variation reduces the wear and tear on the heart drastically. Additionally, heart disease can be detected by extreme and arrhythmic fractal behavior.

"If the beats were regular, the stresses on the heart would be the same on every beat."

... Benoit Mandelbrot

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I need a mathematician to help me and see if my idea is feasible.

Anyone know one ?

Fractals-and-Human-...

My Own InventionMay 31, 2016 9:01 AM

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THIS IS A PROGRAM TO GENERATE A MB SET

fastgraphics

graphsize 384,384refreshkt=319 : m = 4.0xmin=2.1 : xmax=-0.6 : ymin=-1.35 : ymax=1.35dx=(xmax-xmin)/graphwidth : dy=(ymax-ymin)/graphheight

for x=0 to graphwidthjx = xmin+x*dxfor y=0 to graphheight

jy = ymin+y*dyk = 0 : wx = 0.0 : wy = 0.0do

tx = wx*wx-(wy*wy+jx)ty = 2.0*wx*wy+jywx = txwy = tyr = wx*wx+wy*wyk = k+1

until r>m or k>kt

if k>kt thencolor black

else if k<16 then color k*8,k*8,128+k*4if k>=16 and k<64 then color 128+k-16,128+k-16,192+k-16if k>=64 then color kt-k,128+(kt-k)/2,kt-k

end ifplot x,y

next yrefresh

next ximgsave "Mandelbrot_BASIC-256.png", "PNG"Image generated by the script:

Image CompressionMay 31, 2016 9:09 AM

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Count the characters required to write the program

ANSWER = 561

ANY POINT ON THIS IMAGE CAN BE DEFINED By x,y corodinatesAny rectangle can be described by the co-orindates of the opposite corners.

Lets say you want to construct an image that consists of tress and ferns and other flora.

You zoom around the set looking for little segments that you can extract and overlay on a new image to construct the forest. Remember back when we zooned into the MB Set – there were so many different shapes.

You find your rectangle (or polygon) of interest. Do define it fully you need two sets of coordinates and the degree of zoom.

So total bytes = progrsm bytes + SUM OF (all the rectangle definitions to

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So total bytes = progrsm bytes + SUM OF (all the rectangle definitions to make up the set)

To make a simple 1024 requires about 8 Megabytes of data to store.

I think we will come in a lot less.

The Challenge – to automate the search.

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http://www.math.washington.edu/~morrow/mcm/16647.pdfIn order to determine the weight of leaves on an individual tree we set out to develop a model to approximate the number of leaves on any given tree. However, given the large amount of variance between branch structures and the large amount of consistency within a single branch system, we decided to develop a program to model the different major branch systems (the sub-trees with a single large branch as its “trunk” or base) within a tree and the number of leaves that each branch structure develops to approximate the total number of leaves on a tree.

Branch systems have several striking properties that can be quantified and used to describe their physiology. We quantify some of the parameters more pertinent to branch and leaf growth and use them in our model. To best incorporate the inherently fractal nature of tree growth, we decided to use an algorithm that recursively produced auxiliary branch structures from the main branch. We also incorporated a stochastic factor into the algorithms to reflect the inherent randomness of tree growth. To counteract occasional extreme variances that arose from the randomness, we took a Monte Carlo approach: the program creates a large number of sample branches and leaf counts which are then averaged to produce our final estimation.

Our algorithm produced stunningly accurate leaf counts given the innate randomness of the problem. We did small scale testing on several tree branch systems from varying species of trees that we found nearby in different environments and were able to produce not only accurate leaf counts, but also comparable branch structures. We also did one large scale test on an entire oak tree by modeling each of the major branches and summing the results to produce the total leaf count. The results of our simulation produced an average that matches general estimates for leaf counts for mature oaks. Not only did our algorithm produce accurate leaf counts, it also reproduced the shapes and sizes of the branches that we try to model.

The algorithm does have some areas where we would like to improve it. One inherent weakness of our model is that we require the user to choose the parameters themselves, which can be difficult to do on site. However, our model depends a lot more on branch order and lengths, which are more easily estimated. As a result, it is still robust with regards to choices for those parameters which are hard to estimate. We also realize that our model does not take into account several properties related to leaf growth. Still, we feel that our model is a useful predictive tool for scientists who need to estimate the biomass of trees. The accuracy we obtained in small-scale and large-scale testing is rather resounding. Ultimately, we feel that our model provides a very strong starting point and that with some minor work, our model can have significant predictive power.

How Fractal Patterns Perpetuate Through a Tree, and then a Forest - Bonzai Permaculture -

Counting Leaves on a TreeMay 31, 2016 9:24 AM

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Paradox – Low resolution images may contain extra data that allows their conversion to a higher resolution

Using fractals to mine existing dataMay 31, 2016 9:31 AM

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