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Explainer: what are fractals?

Explainer: what are fractals? | anti dogmanti | Scoop.it
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…...
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Fractals can be found everywhere in the world around you. 

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 CommonsClick to enlarge

 

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. Though cylinders and rectangles may be great for modelling the shapes of technology, there are precious few regular shapes to be found in the natural world.

 

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 CommonsClick to enlarge

 

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.

1. Infinite Intricacy

Many patterns of nature are so irregular and fragmented that, compared with Euclid … Nature exhibits not simply a higher degree but an altogether different level of complexity. – Benoît Mandelbröt, The Fractal Geometry of Nature

In 1861, the discovery of the world’s first fractal sent shockwaves through the mathematical community.

If you pick up a pen and doodle a zig-zag, you should end up with a number of sharp corners connected by smooth lines. To show it could be done, the German mathematician Karl Weierstrass constructed a zig-zag that was so jagged, it was nothing but corners – the ultimate mathematical staccato.

No matter how many times the shape was magnified, any glimmer of a smooth line would invariably dissolve into a never-ending cascade of corners, packed ever-more tightly together. Weierstrass' shape had irregular details at every possible scale – the first key feature of a fractal shape.

 

The first fractal shape, discovered by Weierstrass. Zooming in reveals more and more corner points – and no smoothness at all. Wikimedia CommonsClick to enlarge

 

Mathematicians labelled Weierstrass' shape as “pathological,” as it stood in defiance of the tried-and-tested tools of calculus that had been so painstakingly assembled over the previous few hundred years. It remained just a tantalising glimpse of a completely new kind of shape until modern computing power gave mathematicians the keys to the promised land.

2. Zoom Symmetry

I found myself, in other words, constructing a geometry … of things which had no geometry. – Benoît Mandelbröt, 1924-2010

The blossoming of fractal geometry into a new branch of mathematics is largely thanks to the Polish-born mathematicianBenoît Mandelbröt and his seminal 1977 essay The Fractal Geometry of Nature.

 

Mandelbröt’s talk, Fractals and the Art of Roughness at TED2010.

 

Mandelbröt worked for IBM New York in the 1960s. With the company’s immense computing power at his disposal, he was able to explore the strange new world of fractals for the first time.

Perhaps the most famous fractal today is the Mandelbröt set (as shown below), named after its discoverer. To draw it exactly is impossible, but it can be approximated by painstakingly colouring each point in the plane separately.

 

The Mandelbröt set, a famous fractal that can only be drawn by computers. Note how smaller pieces of the set closely resemble the whole. Wikimedia CommonsClick to enlarge

 

To choose the right colour for a specific point, we apply a simple movement rule to the point over and over again and watch how long it takes for the point to “escape” off the page. Though practically impossible to create by hand, modern interactive applets (such as this one, created by British designer Paul Neave) allow you to create and explore these sets in real time.

These computer programs allow you to spot a new kind of symmetry associated with fractals. To mathematicians, a symmetry is an action that when applied to a shape will leave it looking (more or less) the same.

For instance, we say that a square has rotational symmetry because there’s no way to tell if a square has been spun around by 90 degrees when you weren’t looking.

The infinite intricacy of fractals permits them a completely new type of symmetry that isn’t found in ordinary shapes. Incredibly, zooming in on a small region of a fractal leaves you looking at the same shape you started with. Tiny bits of the fractal can look exactly the same as the whole.

Far from being a mathematical curiosity, this zoom symmetry can be found everywhere in nature – once you know to look for it.

 

A bolt of lightning reveals its zoom symmetry for a split second – each branch resembles a small copy of the whole shape. Wikimedia CommonsClick to enlarge

 

3. Complexity from simplicity

Bottomless wonders spring from simple rules which are repeated without end. – Benoît Mandelbröt, 1924-2010

As Mandelbröt was putting fractals under the microscope, the British mathematician Michael Barnsley (currently of the Australian National University) was approaching the same objects from a different angle.

Though the geometry of fractal shapes is infinitely complex, a third trait of fractals is that their complexity arises from very simple core definitions. The shape of a fractal can be completely captured by a small list of mathematical mappings that describe exactly how the smaller copies are arranged to form the whole fractal.

Barnsley’s influential 1988 book Fractals Everywhere contained an algorithm, known as the Chaos Game, that allowed computers to quickly generate any fractal shape from its known mappings.

The Chaos Game took a starting point in space and tracked its motion as it hopped around. Each hop was determined by selecting one of the mappings at random.

Remarkably, no matter the starting point and the order in which the mappings were traversed, the point would quickly be sucked onto a “strange attractor” – the fractal shape – and once there, it would dance around on it forever.

These fractal attractors lie at the heart of Chaos Theory. Since the behaviour of a chaotic system also dances around a fractal attractor, the infinite intricacy of fractal shapes means the slightest nudge to the system can move the point off the attractor entirely.

Crucially, Barnsley found a way to take any desired shape and calculate its list of fractal mappings. Since the complex shape could be completely reconstructed from the simple maps, Barnsley’s algorithms were instrumental in the new field of image compression – allowing the original edition of Microsoft Encarta to pack tens of thousands of images onto a single CD.

 

The Barnsley Fern. No, it’s not a real fern – it’s a mathematical image generated by playing the Chaos Game with four particular maps. Using fractal geometry, complex natural shapes can be encoded with simple mathematical rules Michael RoseClick to enlarge

 

4. Fractional dimensions

Nature has played a joke on the mathematicians. The 19th-Century mathematicians may have been lacking in imagination, but Nature was not. – F J Dyson, as quoted by Benoît Mandelbröt, The Fractal Nature of Geometry

The last and most striking feature of fractals is that are not one-, two- or three-dimensional, but somewhere in-between. Nature seems perfectly happy to use fractional dimensions, so we should be too. In order to do so, we must first clarify what we mean by “dimension”.

The idea of “dimension” has many different (but consistent) mathematical definitions. Intuitively, we can think of a shape’s dimension as a measure of how rough the shape is, or a score that reflects how well the shape fills up its surrounding space.

These intuitive ideas can be made mathematically precise. To illustrate a fractional dimension, think about a piece of paper, which is (practically) 2-dimensional. A solid sphere is 3-dimensional and fills up more space than the piece of paper.

Now crumple the paper into a ball. You now have a fractal-like shape that fills up more space than the paper, but not as much space as the solid sphere. It scores approximately 2.5 for its dimension.

Similarly, your lungs are about 2.97 dimensional – their fractal geometry allows them to pack lots of surface area (a few tennis courts) into a small volume (a few tennis balls). Packing such a huge surface area into your body provides you with the ability to extract enough oxygen to keep you alive.

 

The rings of Saturn arranged in a fractal structure, as seen by the Cassini spacecraft. (Earth is the small dot at the upper-left of the rings) Wikimedia CommonsClick to enlarge

 

Fractals can be found everywhere in the world around you, from a humble fern to the structure of the universe on the largest of scales.

Even certain parts of your anatomy are fractal, including your brain. If you are mindful of fractals, you will be struck by the sheer variety of places you can find them as you go about your daily routine – from clouds, plants and the landscape to church windows and laboratories …

Fractal mathematics not only allows us to begin modelling the shapes of nature, it can also reawaken our childlike wonder at the world around us.


With thanks to Jon Borwein.

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Juan Enriquez: Will our kids be a different species? | Video on TED.com

TED Talks Throughout human evolution, multiple versions of humans co-existed. Could we be mid-upgrade now?
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Thanks. Great video
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‘Hyperbolic metamaterials’ closer to reality | KurzweilAI

‘Hyperbolic metamaterials’ closer to reality | KurzweilAI | anti dogmanti | Scoop.it
Hyperbolic metamaterials could bring optical advances including powerful microscopes, quantum computers and high-performance solar cells. The graphic at left
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Means so much - we will be able to "see" more.

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Researchers, Startups Hope One Drop of Blood Could Diagnose All Types of Cancer

Researchers, Startups Hope One Drop of Blood Could Diagnose All Types of Cancer | anti dogmanti | Scoop.it

As genetics reveals the incredible diversity among cancer cells, researchers have largely given up pursuing a silver bullet to cure all types of cancer. Instead, many have begun searching for the next-best thing: a silver bullet test to diagnose all cancers. The test would look for markers of cancer in the patient’s blood, where the process of tumor-making leaves a trail that can often be picked up before tumors are big enough to spot.

 

And early diagnosis makes a big difference in survival rates. When cancer is found in Stage 0, as it’s just getting started, or in Stage 1, it kills only 10 percent of patients, regardless of what type of cancer it is, for the most part. Many of the cancers we know as the deadliest are so known because they are rarely found in earlier stages.

 

 


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6 Wild Quotes From Christopher Hitchens That Will Remind You Why You're An Atheist

6 Wild Quotes From Christopher Hitchens That Will Remind You Why You're An Atheist | anti dogmanti | Scoop.it
Hitch may have passed on, but his words still ring loud and clear.
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Tony Abbott incorrect on the history of marriage

Tony Abbott incorrect on the history of marriage | anti dogmanti | Scoop.it
Prime Minister Tony Abbott says marriage has always been between a man and a woman, but that's not the case.

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add your insight...

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Infographic: Article Summary Writing Tips

How and Why to add an Article Summary to each Post.Have a look next time you do a search. In addition to seeing the heading, you can also see a few sentences about what is next.This is where you can expand on the heading by giving reasons why people should read on, by enticing readers with some juicy anticipation of what is to come.Can
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Parkwood Golf Club

Parkwood Golf Club - revampedMy husband started playing golf in 2003 when we were living in Fiji for 12 months.When we returned to Australia in October 2004, he joined the Parkwood International Golf Club, not far from where we lived.Over an area of 26 km Gold Coast coastline, there are more than 50 golf clubs of varying length, standard
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A ‘universal smart window’ for instant control of lighting and heat | KurzweilAI

A ‘universal smart window’ for instant control of lighting and heat | KurzweilAI | anti dogmanti | Scoop.it
Smart-window glass that can be switched to block heat or light (credit: Lawrence Berkeley National Laboratory) Researchers at the U.S. Department of
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Researchers track facial expressions to improve teaching software | KurzweilAI

Researchers track facial expressions to improve teaching software | KurzweilAI | anti dogmanti | Scoop.it
Student workstation with depth camera, skin conductance bracelet, and computer with webcam (credit: Joseph F. Grafsgaard et al.) Research from North
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Potential new target to thwart antibiotic resistance: Viruses in gut confer antibiotic resistance to bacteria

Potential new target to thwart antibiotic resistance: Viruses in gut confer antibiotic resistance to bacteria | anti dogmanti | Scoop.it
Bacteria in the gut that are under attack by antibiotics have allies no one had anticipated, scientists have found.
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Islam and the Misuses of Ecstasy : Sam Harris

Islam and the Misuses of Ecstasy : Sam Harris | anti dogmanti | Scoop.it
Sam Harris, neuroscientist and author of the New York Times bestsellers, The End of Faith, Letter to a Christian Nation, and The Moral Landscape.
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Man gets 3D-printed face

Man gets 3D-printed face | anti dogmanti | Scoop.it
When restaurant manager Eric Moger surprised his girlfriend by proposing over Christmas dinner, he could have no idea that less than a year later his life and appearance would be changed beyond recognition.
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Response to Controversy : Sam Harris

Response to Controversy : Sam Harris | anti dogmanti | Scoop.it
Sam Harris, neuroscientist and author of the New York Times bestsellers, The End of Faith, Letter to a Christian Nation, and The Moral Landscape.
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He analyses with such clarity.

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Who is Narendra Modi - India's New Prime Minister?

Who is Narendra Modi - India's New Prime Minister? | anti dogmanti | Scoop.it
Who is Narendra Modi?

India's election, which was won convincingly by the Bharatiya Janata Party (BJP), revolved around only one male: India's next head of sta
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An interesting man

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Replacing a defective gene with a correct sequence to treat genetic disorders | KurzweilAI

Replacing a defective gene with a correct sequence to treat genetic disorders | KurzweilAI | anti dogmanti | Scoop.it
(Credit: Christine Daniloff/MIT) Using a new gene-editing system based on bacterial proteins, MIT researchers have cured mice of a rare liver disorder
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Just amazing!

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Multivitamin and mineral use and breast cancer mortality in older women with invasive breast cancer in the women’s health initiative - Springer

Multivitamin and mineral use and breast cancer mortality in older women with invasive breast cancer in the women’s health initiative - Springer | anti dogmanti | Scoop.it

Postmenopausal women with invasive breast cancer using MVM (multi-vitamins with minerals) had lower breast cancer mortality than non-users. The results suggest a possible role for daily MVM use in attenuating breast cancer mortality in women with invasive breast cancer but the findings require confirmation.


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I love anything from Ray and Terry - they are in the forefront of anti aging research.

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Bees use 'biological autopilot' to land › News in Science (ABC Science)

Bees use 'biological autopilot' to land › News in Science (ABC Science) | anti dogmanti | Scoop.it
Bees get a perfect touchdown by detecting how fast their landing site 'zooms in' as they approach, new research has found.
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Rise and Shine: 5 Things Uber Successful People Do First Thing

Rise and Shine! Morning time simply became your new best buddy.Love it or despise it, making use of the morning hours prior to work may be the secret to a successful, and healthy, lifestyle. That's right, early rising is a typical quality discovered in numerous CEOs, government officials, and other influential individuals who have the rise and
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Content Marketing the New SEO – Infographic

Another great Content Marketing infographic from Berrie Pelser of Wordpress Hosting SEO.More and more today, business uses great content marketing to attract more readers, shares and likes.Makes sense really.Good content marketing works better in the long term, instead of unreliable black hat SEO which can often be affected by changes
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The Most Astounding Fact – Neil deGrasse Tyson

The Most Astounding Fact – Neil deGrasse Tyson | anti dogmanti | Scoop.it

Time magazine once asked astropysicist Neil deGrasse Tyson 10 questions.

One of those questions asked by a Time reader was “What is the most astounding fact you can share with us about the Universe?”

Neil’s response is very awe-inspiring, especially when brought to life in this video which is a compilation from various sources by Max Schlickenmeyer.

Our knowledge of the universe and where we come from is known more today than at any time in history.

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I get goosebumps whenever I watch this video!

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Any Day Now, Malaria, TB and AIDS will be Dodos.

Any Day Now, Malaria, TB and AIDS will be Dodos.
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Watch the inspiring video and then contact your local member to spread the word.

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BigBrain: an ultra-high-resolution 3D roadmap of the human brain | KurzweilAI

BigBrain: an ultra-high-resolution 3D roadmap of the human brain | KurzweilAI | anti dogmanti | Scoop.it
BigBrain (credit: Montreal Neurological Institute and Forschungszentrum Jülich) A landmark three-dimensional (3-D) digital reconstruction of a complete
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Unfrozen mystery: Water reveals a new secret

Unfrozen mystery: Water reveals a new secret | anti dogmanti | Scoop.it
Using revolutionary new techniques, a team led by Carnegie's Malcolm Guthrie has made a striking discovery about how ice behaves under pressure, changing ideas that date back almost 50 years.
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Using Thorium for Energy

Using Thorium for Energy

Liquid Fluoride Thorium Reactors LFTRs were invented 50 years ago by an American named Alvin Weinberg.

LFTRs are revolutionary liquid reactors that run not on uranium, but thorium. These work and have been built before.

The main reason this technology is not in widespread use today is our irrational fear of nuclear energy, despite the fact that more people have died from fossil fuels and even hydroelectric power than nuclear power.

That plus the multinational companies and governments invested in fossil fuel use such as oil, coal and gas.

How much Thorium for Energy do we Have?

Latest research says we have at least 2.6 million tonnes of it on earth, distributed over all the continents.

For every kilogram of thorium, LFTRs can produce 3.5 million Kwh of energy.

This is 70 times greater than uranium and 10,000 times greater than oil.

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Huge online attack exposes internet's vulnerability - tech - 29 March 2013 - New Scientist

Huge online attack exposes internet's vulnerability - tech - 29 March 2013 - New Scientist | anti dogmanti | Scoop.it
The largest online attack ever reported – which may have slowed down the internet itself – is over, but the next battleground is already
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China’s next-generation Internet is a world-beater | KurzweilAI

China’s next-generation Internet is a world-beater | KurzweilAI | anti dogmanti | Scoop.it
Artist rendering of city-sized cloud computing and office complex being built in China (IBM) An open-access report published in the Philosophical
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can't wait for it!

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