Coinbase Commerce makes it quick and easy to start accepting cryptocurrency payments on your website or online shop. Here's everything you need to know.
Coinbase Commerce is a digital payment system, designed for use on websites. It lets you accept a variety of cryptocurrencies, paid directly into your crypto wallet. It's easy to get started and crypto exchange Coinbase run it. We've recently started using it to accept donations, and you can too!
Free: No fees to start accepting cryptocurrency.
Global: There's no need to support thousands of different foreign currencies, accept internationally available cryptocurrencies.
Irreversible: Once transferred, customers cannot raise a chargeback or otherwise withdraw their payment.
No middlemen: Transfers happen directly into your wallet. Not even Coinbase sit in on your transactions.
While many of these benefits come from cryptocurrency itself, Coinbase Commerce provides the tooling necessary to quickly and easily accept payments on your website.
How Does Coinbase Commerce Work?
Coinbase Commerce acts as an intermediary between you and your customers, but with a difference. The Coinbase Commerce plugin facilitates transactions between you and your customers. Coinbase Commerce handles the technical details, such as adjusting the price based on the current market rate of your chosen crypto, handling the optional QR code, and sharing your wallet address. It provides a nice interface for users to pay, a configurable time limit (which ensures customers can't hold you to ransom over an old price), and encouragement to users who may change their minds.
In its most basic form, Coinbase Commerce simply shows users your wallet address. Users pay you directly, with no middle party taking a cut. This is a very altruistic service provided by Coinbase, and you don't need to worry about security either. Payments happen on-chain, and you hold the private keys. Coinbase Commerce makes it easy to transfer funds to Coinbase, but you remain in full control of all your funds.
Supporting Bitcoin, Ethereum, Litecoin, or Bitcoin Cash, Coinbase Commerce has plugins available for all of the major e-commerce platforms, including Shopify, Drupal, Magneto, Zen Cart, and more.
Coinbase Commerce supports a huge variety of different payment options. You can provide a donation button, or a single product purchase button. If you're technically minded, you can use the Coinbase Commerce application programming interface (API) to programmatically raise new funding requests, or you can use the webhooks to update any of your own code.
This could be a database storing your stock levels, or a simple email system. The Coinbase Commerce dashboard provides advanced statistics, so you may not need to write any of this additional code. If you're not a coder, the e-commerce plugins discussed above are the easiest way to sell your goods and accept crypto.
You can get started by creating a free account with an email address and password. Once you've verified your email address, you'll need to enable two-factor authentication. Coinbase "take security very seriously", and they won't let you continue without configuring this. Finally, you'll need to confirm your wallet seed. This is a standard security practice, and as Coinbase Commerce doesn't have access to your funds, you'll need to ensure you securely store this in a safe place.
Once registered, you'll gain access to the Coinbase Commerce dashboard. This provides a summary of all your funds and lets you configure all the settings you need to accept payments.
Payment buttons or "checkouts" handle transactions for specific purchases, and you can have as many of them as you like. They consist of a product name, description, and price, along with an optional product image.
Coinbase Commerce is perhaps the quickest and easiest way to start accepting crypto payments on your website. It's fast, free, secure, irreversible, easy to use, and works anywhere in the world.
As Coinbase Commerce acts more like a broker than an exchange, you don't need to worry about the traditional problems with storing coins on an exchange. You own your own private keys, and Coinbase never handles your coins. Coinbase simply provides the tools you need to accept crypto, and that's it. Regardless of how much hate Coinbase receive (and it's a lot: #DeleteCoinbase), Coinbase can never steal your funds. Still, it doesn't hurt to learn how to keep your crypto safe.
If you've started accepting cryptocurrency payments through Coinbase Commerce, then let us know in the comments section below.


We sell 2 different brands, one is "Flower Crafts" that are fabric flowers with no backings and are sold to fabric & crafts stores and to designers. The other brand "Fashion Flowers" is the brand available on our website as well as gift shops & boutiques. These are all great stores; we highly recommend that you pay them a visit!
Deedlebug Designs at "Southern Charm" in the Cary Towne Center mall.
Sisters Jewelry and Gifts Inc.


text:
Home / Press Release / Baton Rouge / Louisiana's First Medical Physics Residency Program Earns National Accreditation
7 years ago POSTED IN: Baton Rouge , Covington , Gonzales , Hammond , Houma
Louisiana's First Medical Physics Residency Program Earns National Accreditation
First in State to Establish Such a Program
(Baton Rouge, La.) The Commission on Medical Physics Education Programs (CAMPEP) has granted full accreditation to the Mary Bird Perkins Cancer Center Medical Physics Residency Program. Mary Bird Perkins is the first center in Louisiana to establish such a program, helping provide highly specialized professionals to the workforce and enhanced care for cancer patients.
"Medical physicists are an essential part of the cancer-fighting team and by training residents, we are contributing to delivery of high quality cancer care for patients," said Todd Stevens, CEO and president, Mary Bird Perkins Cancer Center.
Using a collaborative approach, Mary Bird Perkins formed a medical physics consortium with multiple affiliate sites, including Willis-Knighton Cancer Center in Shreveport, the University of Mississippi Medical Center in Jackson, MS and e+ Louisiana in Lafayette. This allowed for the expansion of residency training opportunities and resources. In conjunction with its affiliates, Mary Bird Perkins now has the largest radiation oncology physics residency training program in the United States.
By establishing the first accredited group model with respected affiliate Centers in Louisiana and Mississippi, the program is creating more openings for graduates of CAMPEP-accredited medical physics graduate programs, like the Mary Bird Perkins-LSU Medical Physics Graduate Program. "We wanted to provide a leading-edge residency program that would provide best in class training and help alleviate the current shortage of physics residency programs in the U.S.," explained Mary Bird Perkins Residency Program Director and Chief of Clinical Physics, John Gibbons, Ph.D. "It has been a great experience to innovatively work with other centers to help meet this important need."
In an effort to address the need for well-trained medical physicists, the American Board of Radiology announced in early 2009 that by 2014 it would require medical physicists to have graduated from or be enrolled in a CAMPEP-accredited residency training program prior to sitting for board certification exams.
A medical physicist brings a unique perspective to the clinical team in a radiation oncology program: that of a scientist trained in physics, including radiological physics; and also in clinical, basic medical, and radiobiological sciences. The physicist performs an important role working along with the radiation oncologist to assure the accurate delivery of all aspects of a treatment prescription.
For additional information about Mary Bird Perkins Cancer Center's medical physics residency program please visit www.marybird.org/medphys.
Mary Bird Perkins Cancer Center is a regional cancer care organization that has been fighting cancer for more than 40 years. With five centers in Baton Rouge, Covington, Hammond, Houma and Gonzales, its service area encompasses 18 parishes across southeast Louisiana. For more information, please visit
www.marybird.org.

redpajama_set_name: RedPajamaCommonCrawl


Our client are one of Europe's leading entertainment brand who are looking for a Talent Acquisition Partner to joing their growing team.
This role will offer a great opportunity to gain European and International exposure, as well as partnering with senior stakeholders across the business. You will have end to end responsibility for the Recruitment function for a number of business functions.
This role will work alongside a strong Recruitment team and Head of Recruitment to focus on delivery but also own and deliver on a number of recruitment projects.
Exposure to technical recrutiment (java developers, data scientists, iOS etc) highly desirable.
An expert in utilising various recruitment tools.
Able to partner with key stakeholders and provide market insights.
This role offers endless benefits with a great working culture including multiple insurances, pension, discounts, bonus schemes, entertainment etc.


JS function to calculate complementary colour?
There is a number of colour picking suites and palette generators on the web but I haven't seen any that calculate the colour live using JS.
A detailed hint or a snippet would be very much appreciated.
I find that taking the bit-wise complement works well, and quickly.
I'm not sure if it's a perfect complement in the sense of "mixes together to form a 70% grey", however a 70% grey is "pure white" in terms of color timing in film. It occurred to me that XORing the RGB hex out of pure white might be a good first approximation. You could also try a darker grey to see how that works for you.
Again, this is a fast approximation and I make no guarantees that it'll be perfectly accurate.
See https://github.com/alfl/textful/blob/master/app.js#L38 for my implementation.
None of the other functions here worked out the box, so I made this one.
Rather than reinventing the wheel, I found a library to work with colors.
This is how you would implement some of the other answers using it.
How can I generate the opposite color according to current color?
Programmatically find complement of colors?
hsl to rgb conversion. Values waaaay off. Compare to solution. What am I doing differently?


text:
Rick Blum
Rick Blum is the Policy Director of the Reporters Committee for Freedom of the Press, where he oversees policy work on a wide range of topics affecting newsgathering and transparency.
For over a decade, Rick coordinated News Media for Open Government, a coalition of major news media associations promoting open government policies and practices. He led the coalition (until 2017 known as the Sunshine in Government Initiative) in its work supporting enactment of the 2007 amendments to the Freedom of Information Act (FOIA) and the FOIA Improvement Act of 2016. His work defending against unnecessary, overbroad or self-serving secrecy leads him to any number of topics, such as tracking space junk, the farm bill, and high-speed rail. In addition, he has helped explain to Congress how the news media keep the public informed based on unauthorized disclosures of classified information ("leaks"), addressing concerns about WikiLeaks and successfully pushing back against anti-leaks proposals in the Intelligence Authorization Act for 2013.
Blum has testified numerous times before Congress on FOIA implementation and exemptions, science and disclosure, and federal legislation to shield journalists from revealing confidential sources. He has appeared in C-SPAN's Washington Journal and has promoted open government through radio show appearances and interviews in newspapers and broadcasts.
He is part of a panelist of reviewers contributing an assessment to the Newseum Institute's First Amendment Report Card and helps organize the Institute's Freedom of Information Day celebration during Sunshine Week. In 2016, he was inducted into the Newseum Institute's Freedom of Information Hall of Fame.
Blum was the founding director of OpenTheGovernment.org, a broad coalition of journalists, labor, and free-speech and environmental advocates. There he launched the Secrecy Report Card, an annual report of quantitative indicators of secrecy and openness in the federal government. Later he served for several years on OTG's steering committee. He also served on the Sunlight Foundation's Advisory Committee on Transparency.
As a policy analyst at OMB Watch from 1997-2001, he worked with environmental groups, librarians, freedom-of-information advocates and others to maintain public access to chemical accident risk management plans. He served for several years on the steering committee of the Working Group on Community Right-to-Know.
He holds a master's degree from Indiana University, where his studies focused on democratization efforts in Russia, and a bachelor's degree from the University of California, Berkeley.


text:
android game mod apk game mod best android game mod best game mod game mod apk android game download android game free
make up wedding jogja make up wedding yogyakarta make up prewedding jogja make up prewedding yogyakarta make up pesta jogja make up pesta yogyakarta make up pernikahan jogja make up pernikahan yogyakarta make up wisuda jogja make up wisuda yogyakarta
Moviepie Musings
Title: E.T. The Extra-Terrestrial
Director: Steven Spielberg
Actors: Dee Wallace Peter Coyote Henry Thomas
MPAA Rating: PG-13
Genre(s): Adventure Science Fiction
Official Site: http://www.imdb.com/title/tt0083866/
The sweetest movie about a lost alien botanist has been restored (yay!) to its original form, and it looks great.
In case you've been living under a rock for a couple of decades, you've most likely seen the classic sci-fi/fantasy film E.T. The Extra-Terrestrial, one of the biggest money-making films of all time. I've probably seen the movie a dozen times, and yes, it makes me cry every time (when E.T. is lying in the river, all gray and sick, getting picked at by raccoons... well, I get choked up just thinking about it).
A sweet film about friendship, love, family, and most of all, home, E.T. for me is a nostalgic love-fest ode to the early 1980s, where at the time the kids on screen reflected the ages of me and my neighborhood friends. E.T. in a way was my generation's Bambi, teaching us values and terrorizing us with a traumatic death scene at the same time (but with Speilberg's touch: he makes us sob, then brings E.T. back to life—whew!).
The film affectionately takes place in generic suburbia, when kids were still allowed to go trick-or-treating alone, and there were woodsy areas to play in only a bike ride away. A little alien is left behind when his ship takes off without him. He wanders into a suburban backyard shed, where he is discovered by a a 10-year-old boy named Elliott (Henry Thomas). Elliott dubs the little guy "E.T.", takes him in, and makes his siblings Michael and Gertie (Robert MacNaughton and Drew Barrymore) make a "most excellent promise" not to tell anyone about the alien.
But in the meantime (of course) the government is on the alien's tracks, with their threatening high-beam flashlights, dark silhouettes, and jangling keys. All the while, poor E.T. just wants to phone home....
The 35th anniversary edition looks great: The original film has been fully restored to 4K (theater quality look and sound), and fans should be pleased to hear that all the extra bells and whistles (introduced at the 20th anniversary, i.e. the new CGI and souped-up scenes), have been stripped back out so the film is in its original form. The bad guys have guns again, and E.T. doesn't "hop" through the woods to the weirdly pulsing spaceship. All is back to the original, which always looked better to start with.
Two deleted scenes are included. One has E.T. and Elliott playing in the bathroom—a scene that I felt didn't add anything to the story, and was a bit distracting. The other new segment makes more sense: a transition scene of the kids' mother (Dee Wallace) driving around in her station wagon on Halloween to find her tardy brood. She comes upon a hilarious riot of suburban-kid-chaos, with flaming pumpkins rolling down the street, toilet paper flying through the air, and an egg smashing on her windshield. Michael and Gertie look guilty when found, and Gertie gets yet another scene-stealing hilarious one-liner (I won't spoil it here).
There are quite a few making-of extras as well as reunions and interviews with cast members through the years. The extras are great, but a little repetitive. There is nothing here newer than 2012, so if you have the 30th (or even the 20th) anniversary editions, you've seen most if not all of this.
Ultimately, the reasons you may want this edition is because it is back in its original form, you get a digital copy, and will look great on your bigger, better TV.
Linda, Webmistress
Enjoys Oscar-bait, foreign films, sci-fi movies, and even movies with talking dogs.
Latest from Linda, Webmistress
A Soldier's Girl
Linda's Best of 2019
More in this category: « Duets Earth (2007) »
Movies Search
MPAA Rating - Select MPAA Rating - G PG PG-13 R NC-17 NR
I get the impression that there are…
Jennifer, Pie Pal
I think I'd flip out if I…
Forget the kid! After watching the trailer,…
Yep! That's about what I would have…
Tom, Pie Pal
It wasn't bad, it just wasn't great.…
Pie Pal Vickie
Oh, there was definitely an after party…
Too bad about 'We Are The Best!'.…
30 years ago, I loved "Big Chill."…
I agree with the review. I thought…
Anime feels all warm and calm.
Derossett
Aliens	Animals	Animation	Christmas	Civil Rights	Disney	Family	Horror	LGBT	Marvel	Musicians	Nature gone wrong	Romance	RomCom	Singing and Singers	Spies	Sports	Superheroes	Teens	World War II
Copyright © 2000-2019 Movie*Pie: All rights reserved.


text:
⟶ Қазақша
⟶ Русский
Поделитесь в соцсетях
Астана: вехи истории
The opening of the photo exhibition of Dmitry Rugis "Kazakhstans"
Today was the official opening of a personal photo exhibition of an active QazaqGeography member and "National Geographic Qazaqstan" magazine photographer Dmitry Rugis.
The exhibition presents a series of photographs by the author under the common name "Kazakhstans". He fills the faces of Kazakhstan people in the history by actively traveling in the country and studying its culture. These are mainly working class people, people from the remote places, from distant villages, who have preserved their ancient traditions.
With all our hearts we congratulate citizens of Kazakhstan with Happy Independence Day of the Republic of Kazakhstan, the main holiday of the country, symbol of our independent government. From the moment our country became independent, our nation managed to build a democratic, legal and economically independent state. We had a great honor to become authors of new history and start our own way that was recognized in the world public community. The republican public association «QazaqGeography» wishes all residents of Kazakhstan peace and conciliation, stability and prosperity, success and good health!
The 4th Asian Conference on Geography (4th ACG)
From December 6 to the 10th the Geographical Society of China and Sun Yat-sen University, supported by the Korean Geographical Society, the Association of Japanese Geographers, the National Association of Geographers of India, the Hong Kong Geographic Association and the Macau Geographic and Education Research Association hosted the 4th Asian Geography Conference. It should be noted that this meeting will leave a mark in the history of Asia of the XXI century.
The Congress of QazaqGeography was held in the new office of the association
On November 17th the Congress of the Republican public association «QazaqGeography» was held under the lead of Askar Mamin, the first deputy of the prime minister of the Republic of Kazakhstan.
XV Forum of the interregional partnership of Kazakstan and Russia
On November 8,9th XV forum of the interregional partnership of Kazakstan and Russia was held in Petropavl, where leaders of both countries Nursultan Nazarbayev and Vladimir Putin participated.
Abylaikit fortress and monastery
In Kazakh people's mind all the notes of Djungars are related to the hostile relationship between two nations and not much. However, thanks to archeological remains and historical findings the picture of Kazakh-Djungar relation in the past becomes clearer. Newly discovered findings might alter stereotyped attitude toward djungars. Tinkay Krevetova portrays the Abylaikit monument, which explores the history of oyrat period.
«Konyr Auliye cave»
In East Kazakhstan region, 200 km away from Semey, an astonishing cave called Konyr Auliye is located. A hidden holy lake can be found inside of it. There are countless cases when spouses would come and pray asking for a child. Eventually their wishes came true. Also, archeologists found a workshop of the Stone Age. The author of the article Tinkay Kretova provides appealing facts about the holy cave in a brief manner.
«Kunanbay mosque»
Do you know that the mosque in Karkaraly region was built by Kunanbay kazhy following tatar artchitecture style only by using solid logs? Despite the purpose of the building have been changed over time it never lost its value from architectural perspective. A cultural monument services as a reminder of Kazakh spiritual tradition.
«Mauseleum of Isabek Ishan Khazret»
There is a mausoleum built in honor of Isabek Ishan Khazret near to the Ekibastuz city 18 km away from Aqkol village, in Kyldykol tract. Who was Isabek Ishan Khazret? And why this particular place was chosen for this purpose? All details you may find in Tinkay Kretova's article.
«Kyzyl Kenish oyrat temple»
Do you know that during the ruling of khoshout's in oyrat confederation on Kent's mountains territory an oyrat temple Kyzyl Kenish was built? The intention behind the construction was to create a town afterward. In her article Tinkay Kretova describes this unique temple.
«Woody mosque in Aqkoltyk»
Have you seen any mosque, which was turned into a library or a club? If not, then you can explore for yourself about ancient mosque preserved till these days in the article by Tinkay Krevetova Woody mosque in Aqkoltyk.
«Chokan in Tezek's village: questions and comments»
In his article Ismailzhan Iminov meticulously describes the story of Chokan's life and his death. Besides, the author vividly depicts the natural beauty of the village, as he personally had been wondering around every corner of it before writing about the scene of actions. In this article, you can also discover Chokan's attitude toward the tsar's power. Most importantly, writing piece explores the mystery of Chokan's death.
Conference «Development of scientific tourism»
On July 19, 2019, the NGO «QazaqQeoqraphy» hosted the conference «Development of Scientific and Informative Tourism».
The opening of the exhibition of the applied art «Geography – a line of cultural perspective»
The opening of the exhibition of the applied art «Geography – a line of cultural perspective» took place on September 14th in the National Museum
Lecture: «Using and drafting topographic and thematic maps for geographical research»
On the 16th of September for the geography teachers and students of the Olympic reserve in the framework of QazaqGeography informative - educational program was held a lecture on the topic «Using and drafting topographic and thematic maps for geographical research».
Horse hiking: mausoleums "Niyaz bi - Kabanbay batyr"
An active member of QazaqGeography and the founder of the "At-travel" Club, Berik Akylbekov, on November 2 organized the horse trip between "Niyaz bi - Kabanbay batyr" Mausoleum.
Master class: "Using Google tools in teaching geography" for the teachers of Nur Sultan city
The master class on the topic "Using Google tools in teaching geography" for the teachers of Nur Sultan city was held on the 30th of October at QazaqGeography platform. organized
Geographical Dictation 2019
On October 27th an international educational campaign «Geographical Dictation» was held at the QazaqGeography Media Center
The V Congress of the QazaqGeography was held in Nur-Sultan
The V Congress of the Republican Public Association "QazaqGeography" was held at the National Museum of the Republic of Kazakhstan. "QazaqGeography carries out systematic work to study the history and culture of Kazakhstan, the development of ecotourism and the implementation of other important projects aimed at maintaining unity and reviving the spiritual values of the Kazakh people," QazaqGeography Chairman A. Mamin said at the opening of the Congress.
Республиканское общественное объединение «QazaqGeography»
Z10G9D2 Казахстан, Нур-Султан
проспект Абая, зд. 22
info@q-g.kz
© 2013—2020 Республиканское общественное объединение «QazaqGeography»
«QazaqGeography»
© 2013—2020 QazaqGeography


text:
(Redirected from Fractals)
A self-similar pattern or set
For other uses, see Fractal (disambiguation).
Mandelbrot set: Self-similarity illustrated by image enlargements. This panel, no magnification.
The same fractal as above, magnified 6-fold. Same patterns reappear, making the exact scale being examined difficult to determine.
The same fractal as above, magnified 100-fold.
The same fractal as above, magnified 2000-fold, where the Mandelbrot set fine detail resembles the detail at low magnification.
In mathematics, a fractal is a subset of a Euclidean space for which the Hausdorff dimension strictly exceeds the topological dimension. Fractals tend to appear nearly the same at different levels, as is illustrated here in the successively small magnifications of the Mandelbrot set;[1][2][3][4] Because of this, fractals are encountered ubiquitously in nature. Fractals exhibit similar patterns at increasingly small scales called self similarity,[5] also known as expanding symmetry or unfolding symmetry; If this replication is exactly the same at every scale, as in the Menger sponge,[6] it is called affine self-similar. Fractal geometry lies within the mathematical branch of topology.
One way that fractals are different from finite geometric figures is the way in which they scale. Doubling the edge lengths of a polygon multiplies its area by four, which is two (the ratio of the new to the old side length) raised to the power of two (the dimension of the space the polygon resides in). Likewise, if the radius of a sphere is doubled, its volume scales by eight, which is two (the ratio of the new to the old radius) to the power of three (the dimension that the sphere resides in). However, if a fractal's one-dimensional lengths are all doubled, the spatial content of the fractal scales by a power that is not necessarily an integer.[1] This power is called the fractal dimension of the fractal, and it usually exceeds the fractal's topological dimension.[7]
Analytically, fractals are usually nowhere differentiable.[1][4][8] An infinite fractal curve can be conceived of as winding through space differently from an ordinary line – although it is still 1-dimensional, its fractal dimension indicates that it also resembles a surface.[1][7]
Sierpinski carpet (to level 6), a fractal with a topological dimension of 1 and a Hausdorff dimension of 1.893
Starting in the 17th century with notions of recursion, fractals have moved through increasingly rigorous mathematical treatment of the concept to the study of continuous but not differentiable functions in the 19th century by the seminal work of Bernard Bolzano, Bernhard Riemann, and Karl Weierstrass,[9] and on to the coining of the word fractal in the 20th century with a subsequent burgeoning of interest in fractals and computer-based modelling in the 20th century.[10][11] The term "fractal" was first used by mathematician Benoit 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.[1][12]
There is some disagreement among mathematicians about how the concept of a fractal should be formally defined. Mandelbrot himself summarized it as "beautiful, damn hard, increasingly useful. That's fractals."[13] More formally, in 1982 Mandelbrot stated that "A fractal is by definition a set for which the Hausdorff–Besicovitch dimension strictly exceeds the topological dimension."[14] Later, seeing this as too restrictive, he simplified and expanded the definition to: "A fractal is a shape made of parts similar to the whole in some way."[15] Still later, Mandelbrot settled on this use of the language: "...to use fractal without a pedantic definition, to use fractal dimension as a generic term applicable to all the variants".[16]
The consensus is that theoretical fractals are infinitely self-similar, iterated, and detailed mathematical constructs having fractal dimensions, of which many examples have been formulated and studied in great depth.[1][2][3] Fractals are not limited to geometric patterns, but can also describe processes in time.[6][4][17][18][19][20] Fractal patterns with various degrees of self-similarity have been rendered or studied in images, structures and sounds[21] and found in nature,[22][23][24][25][26] technology,[27][28][29][30] art,[31][32] architecture[33] and law.[34] Fractals are of particular relevance in the field of chaos theory, since the graphs of most chaotic processes are fractals.[35]
3 Definition and characteristics
4 Common techniques for generating fractals
5 Simulated fractals
6 Natural phenomena with fractal features
7 In creative works
8 Physiological responses
9 Applications in technology
9.1 Ion propulsion
Introduction[edit]
A simple fractal tree created through javascript
The word "fractal" often has different connotations for the lay public as opposed to mathematicians, where the public are more likely to be familiar with fractal art than the mathematical concept. The mathematical concept is difficult to define formally, even for mathematicians, but key features can be understood with little mathematical background.
The feature of "self-similarity", for instance, is easily understood by analogy to zooming in with a lens or other device that zooms in on digital images to uncover finer, previously invisible, new structure. If this is done on fractals, however, no new detail appears; nothing changes and the same pattern repeats over and over, or for some fractals, nearly the same pattern reappears over and over.[5] Self-similarity itself is not necessarily counter-intuitive (e.g., people have pondered self-similarity informally such as in the infinite regress in parallel mirrors or the homunculus, the little man inside the head of the little man inside the head ...). The difference for fractals is that the pattern reproduced must be detailed.[1]:166; 18[2][12]
This idea of being detailed relates to another feature that can be understood without mathematical background: Having a fractal dimension greater than its topological dimension, for instance, refers to how a fractal scales compared to how geometric shapes are usually perceived. A regular line, for instance, is conventionally understood to be one-dimensional; if such a curve is rep-tiled into pieces each 1/3 the length of the original, there are always three equal pieces. A solid square is understood to be two-dimensional; if such a figure is rep-tiled pieces each scaled down by a factor of 1/3 in both dimensions, there are a total of 3² = 9 pieces. We see that for ordinary self-similar objects, being n-dimensional means that when it is rep-tiled into pieces each scaled down by a scale-factor of 1/r, there are a total of rn pieces. Now, consider the Koch curve. It can be rep-tiled into four sub-copies, each scaled down by a scale-factor of 1/3. So, strictly by analogy, we can consider the "dimension" of the Koch curve as being the unique real number D that satisfies 3D = 4, which by no means is an integer! This number is what mathematicians call the fractal dimension of the Koch curve. The fact that the Koch curve has a non-integer fractal dimension is what makes it a fractal.
This also leads to understanding a third feature, that fractals as mathematical equations are "nowhere differentiable". In a concrete sense, this means fractals cannot be measured in traditional ways.[1][4][8] To elaborate, in trying to find the length of a wavy non-fractal curve, one could find straight segments of some measuring tool small enough to lay end to end over the waves, where the pieces could get small enough to be considered to conform to the curve in the normal manner of measuring with a tape measure. But in measuring an infinitely "wiggly" fractal curve such as the Koch snowflake, one would never find a small enough straight segment to conform to the curve, because the jagged pattern would always re-appear, at arbitrarily small scales, essentially pulling a little more of the tape measure into the total length measured each time one attempted to fit it tighter and tighter to the curve. The result is that one must need infinite tape to perfectly cover the entire curve, i.e. the snowflake has infinite perimeter.[1]
A Koch snowflake is a fractal that begins with an equilateral triangle and then replaces the middle third of every line segment with a pair of line segments that form an equilateral bump
The history of fractals traces a path from chiefly theoretical studies to modern applications in computer graphics, with several notable people contributing canonical fractal forms along the way.[10][11] According to Pickover, the mathematics behind fractals began to take shape in the 17th century when the mathematician and philosopher Gottfried Leibniz pondered recursive self-similarity (although he made the mistake of thinking that only the straight line was self-similar in this sense).[36] In his writings, Leibniz used the term "fractional exponents", but lamented that "Geometry" did not yet know of them.[1]:405 Indeed, according to various historical accounts, after that point few mathematicians tackled the issues, and the work of those who did remained obscured largely because of resistance to such unfamiliar emerging concepts, which were sometimes referred to as mathematical "monsters".[8][10][11] Thus, it was not until two centuries had passed that on July 18, 1872 Karl Weierstrass presented the first definition of a function with a graph that would today be considered a fractal, having the non-intuitive property of being everywhere continuous but nowhere differentiable at the Royal Prussian Academy of Sciences.[10]:7[11] In addition, the quotient difference becomes arbitrarily large as the summation index increases.[37] Not long after that, in 1883, Georg Cantor, who attended lectures by Weierstrass,[11] published examples of subsets of the real line known as Cantor sets, which had unusual properties and are now recognized as fractals.[10]:11–24 Also in the last part of that century, Felix Klein and Henri Poincaré introduced a category of fractal that has come to be called "self-inverse" fractals.[1]:166
A Julia set, a fractal related to the Mandelbrot set
A Sierpinski gasket can be generated by a fractal tree.
One of the next milestones came in 1904, when Helge von Koch, extending ideas of Poincaré and dissatisfied with Weierstrass's abstract and analytic definition, gave a more geometric definition including hand-drawn images of a similar function, which is now called the Koch snowflake.[10]:25[11] Another milestone came a decade later in 1915, when Wacław Sierpiński constructed his famous triangle then, one year later, his carpet. By 1918, two French mathematicians, Pierre Fatou and Gaston Julia, though working independently, arrived essentially simultaneously at results describing what are now seen as fractal behaviour associated with mapping complex numbers and iterative functions and leading to further ideas about attractors and repellors (i.e., points that attract or repel other points), which have become very important in the study of fractals.[4][10][11] Very shortly after that work was submitted, by March 1918, Felix Hausdorff expanded the definition of "dimension", significantly for the evolution of the definition of fractals, to allow for sets to have non-integer dimensions.[11] The idea of self-similar curves was taken further by Paul Lévy, who, in his 1938 paper Plane or Space Curves and Surfaces Consisting of Parts Similar to the Whole described a new fractal curve, the Lévy C curve.[notes 1]
A strange attractor that exhibits multifractal scaling
Uniform mass center triangle fractal
2x 120 degrees recursive IFS
Different researchers have postulated that without the aid of modern computer graphics, early investigators were limited to what they could depict in manual drawings, so lacked the means to visualize the beauty and appreciate some of the implications of many of the patterns they had discovered (the Julia set, for instance, could only be visualized through a few iterations as very simple drawings).[1]:179[8][11] That changed, however, in the 1960s, when Benoit Mandelbrot started writing about self-similarity in papers such as How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension,[38][39] which built on earlier work by Lewis Fry Richardson. In 1975[12] Mandelbrot solidified hundreds of years of thought and mathematical development in coining the word "fractal" and illustrated his mathematical definition with striking computer-constructed visualizations. These images, such as of his canonical Mandelbrot set, captured the popular imagination; many of them were based on recursion, leading to the popular meaning of the term "fractal".[40][8][10][36]
In 1980, Loren Carpenter gave a presentation at the SIGGRAPH where he introduced his software for generating and rendering fractally generated landscapes.[41]
Definition and characteristics[edit]
One often cited description that Mandelbrot published to describe geometric fractals is "a rough or fragmented geometric shape that can be split into parts, each of which is (at least approximately) a reduced-size copy of the whole";[1] this is generally helpful but limited. Authors disagree on the exact definition of fractal, but most usually elaborate on the basic ideas of self-similarity and the unusual relationship fractals have with the space they are embedded in.[1][6][2][4][42]
One point agreed on is that fractal patterns are characterized by fractal dimensions, but whereas these numbers quantify complexity (i.e., changing detail with changing scale), they neither uniquely describe nor specify details of how to construct particular fractal patterns.[43] In 1975 when Mandelbrot coined the word "fractal", he did so to denote an object whose Hausdorff–Besicovitch dimension is greater than its topological dimension.[12] However, it has been noted that this requirement is not met by space-filling curves such as the Hilbert curve.[notes 2]
Because of the trouble involved in finding one definition for fractals, some argue that fractals should not be strictly defined at all. According to Falconer, fractals should, in addition to being nowhere differentiable and able to have a fractal dimension, be only generally characterized by a gestalt of the following features;[2]
Self-similarity, which may include:
Exact self-similarity: identical at all scales, such as the Koch snowflake
Quasi self-similarity: approximates the same pattern at different scales; may contain small copies of the entire fractal in distorted and degenerate forms; e.g., the Mandelbrot set's satellites are approximations of the entire set, but not exact copies.
Statistical self-similarity: repeats a pattern stochastically so numerical or statistical measures are preserved across scales; e.g., randomly generated fractals like the well-known example of the coastline of Britain for which one would not expect to find a segment scaled and repeated as neatly as the repeated unit that defines fractals like the Koch snowflake.[4]
Qualitative self-similarity: as in a time series[17]
Multifractal scaling: characterized by more than one fractal dimension or scaling rule
Fine or detailed structure at arbitrarily small scales. A consequence of this structure is fractals may have emergent properties[44] (related to the next criterion in this list).
Irregularity locally and globally that is not easily described in traditional Euclidean geometric language. For images of fractal patterns, this has been expressed by phrases such as "smoothly piling up surfaces" and "swirls upon swirls".[7]
Simple and "perhaps recursive" definitions; see Common techniques for generating fractals
As a group, these criteria form guidelines for excluding certain cases, such as those that may be self-similar without having other typically fractal features. A straight line, for instance, is self-similar but not fractal because it lacks detail, is easily described in Euclidean language, has the same Hausdorff dimension as topological dimension, and is fully defined without a need for recursion.[1][4]
Common techniques for generating fractals[edit]
See also: Fractal-generating software
Self-similar branching pattern modeled in silico using L-systems principles[26]
Images of fractals can be created by fractal generating programs. Because of the butterfly effect, a small change in a single variable can have an unpredictable outcome.
Iterated function systems (IFS) – use fixed geometric replacement rules; may be stochastic or deterministic;[45] e.g., Koch snowflake, Cantor set, Haferman carpet,[46] Sierpinski carpet, Sierpinski gasket, Peano curve, Harter-Heighway dragon curve, T-square, Menger sponge
Strange attractors – use iterations of a map or solutions of a system of initial-value differential or difference equations that exhibit chaos (e.g., see multifractal image, or the logistic map)
L-systems – use string rewriting; may resemble branching patterns, such as in plants, biological cells (e.g., neurons and immune system cells[26]), blood vessels, pulmonary structure,[47] etc. or turtle graphics patterns such as space-filling curves and tilings
Escape-time fractals – use a formula or recurrence relation at each point in a space (such as the complex plane); usually quasi-self-similar; also known as "orbit" fractals; e.g., the Mandelbrot set, Julia set, Burning Ship fractal, Nova fractal and Lyapunov fractal. The 2d vector fields that are generated by one or two iterations of escape-time formulae also give rise to a fractal form when points (or pixel data) are passed through this field repeatedly.
Random fractals – use stochastic rules; e.g., Lévy flight, percolation clusters, self avoiding walks, fractal landscapes, trajectories of Brownian motion and the Brownian tree (i.e., dendritic fractals generated by modeling diffusion-limited aggregation or reaction-limited aggregation clusters).[4]
A fractal generated by a finite subdivision rule for an alternating link
Finite subdivision rules – use a recursive topological algorithm for refining tilings[48] and they are similar to the process of cell division.[49] The iterative processes used in creating the Cantor set and the Sierpinski carpet are examples of finite subdivision rules, as is barycentric subdivision.
Simulated fractals[edit]
Fractal patterns have been modeled extensively, albeit within a range of scales rather than infinitely, owing to the practical limits of physical time and space. Models may simulate theoretical fractals or natural phenomena with fractal features. The outputs of the modelling process may be highly artistic renderings, outputs for investigation, or benchmarks for fractal analysis. Some specific applications of fractals to technology are listed elsewhere. Images and other outputs of modelling are normally referred to as being "fractals" even if they do not have strictly fractal characteristics, such as when it is possible to zoom into a region of the fractal image that does not exhibit any fractal properties. Also, these may include calculation or display artifacts which are not characteristics of true fractals.
Modeled fractals may be sounds,[21] digital images, electrochemical patterns, circadian rhythms,[50] etc. Fractal patterns have been reconstructed in physical 3-dimensional space[29]:10 and virtually, often called "in silico" modeling.[47] Models of fractals are generally created using fractal-generating software that implements techniques such as those outlined above.[4][17][29] As one illustration, trees, ferns, cells of the nervous system,[26] blood and lung vasculature,[47] and other branching patterns in nature can be modeled on a computer by using recursive algorithms and L-systems techniques.[26] The recursive nature of some patterns is obvious in certain examples—a branch from a tree or a frond from a fern is a miniature replica of the whole: not identical, but similar in nature. Similarly, random fractals have been used to describe/create many highly irregular real-world objects. A limitation of modeling fractals is that resemblance of a fractal model to a natural phenomenon does not prove that the phenomenon being modeled is formed by a process similar to the modeling algorithms.
Natural phenomena with fractal features[edit]
Further information: Patterns in nature
Approximate fractals found in nature display self-similarity over extended, but finite, scale ranges. The connection between fractals and leaves, for instance, is currently being used to determine how much carbon is contained in trees.[51] Phenomena known to have fractal features include:
Actin cytoskeleton[52]
Animal coloration patterns
Blood vessels and pulmonary vessels[47]
Crystals[53]
Earthquakes[30][54]
Geometrical optics[55]
Heart rates[22]
Heart sounds[23]
Mountain goat horns
Ocean waves[56]
Psychological subjective perception[57]
Proteins[58]
Rings of Saturn[59][60]
River networks
Romanesco broccoli
Snowflakes[61]
Soil pores[62]
Surfaces in turbulent flows[63][64]
Brownian motion (generated by a one-dimensional Wiener process).[65]
Frost crystals occurring naturally on cold glass form fractal patterns
Fractal basin boundary in a geometrical optical system[55]
A fractal is formed when pulling apart two glue-covered acrylic sheets
High-voltage breakdown within a 4 in (100 mm) block of acrylic glass creates a fractal Lichtenberg figure
Romanesco broccoli, showing self-similar form approximating a natural fractal
Fractal defrosting patterns, polar Mars. The patterns are formed by sublimation of frozen CO2. Width of image is about a kilometer.
Slime mold Brefeldia maxima growing fractally on wood
In creative works[edit]
Further information: Fractal art and Mathematics and art
Since 1999, more than 10 scientific groups have performed fractal analysis on over 50 of Jackson Pollock's (1912–1956) paintings which were created by pouring paint directly onto his horizontal canvases[66][67][68][69][70][71][72][73][74][75][76][77][78] Recently, fractal analysis has been used to achieve a 93% success rate in distinguishing real from imitation Pollocks.[79] Cognitive neuroscientists have shown that Pollock's fractals induce the same stress-reduction in observers as computer-generated fractals and Nature's fractals.[80]
Decalcomania, a technique used by artists such as Max Ernst, can produce fractal-like patterns.[81] It involves pressing paint between two surfaces and pulling them apart.
Cyberneticist Ron Eglash has suggested that fractal geometry and mathematics are prevalent in African art, games, divination, trade, and architecture. Circular houses appear in circles of circles, rectangular houses in rectangles of rectangles, and so on. Such scaling patterns can also be found in African textiles, sculpture, and even cornrow hairstyles.[32][82] Hokky Situngkir also suggested the similar properties in Indonesian traditional art, batik, and ornaments found in traditional houses.[83][84]
Ethnomathematician Ron Eglash has discussed the planned layout of Benin city using fractals as the basis, not only in the city itself and the villages but even in the rooms of houses. He commented that "When Europeans first came to Africa, they considered the architecture very disorganised and thus primitive. It never occurred to them that the Africans might have been using a form of mathematics that they hadn't even discovered yet." [85]
In a 1996 interview with Michael Silverblatt, David Foster Wallace admitted that the structure of the first draft of Infinite Jest he gave to his editor Michael Pietsch was inspired by fractals, specifically the Sierpinski triangle (a.k.a. Sierpinski gasket), but that the edited novel is "more like a lopsided Sierpinsky Gasket".[31]
A fractal that models the surface of a mountain (animation)
3D recursive image
Recursive fractal butterfly image
A fractal flame
Physiological responses[edit]
Humans appear to be especially well-adapted to processing fractal patterns with D values between 1.3 and 1.5.[86] When humans view fractal patterns with D values between 1.3 and 1.5, this tends to reduce physiological stress.[87][88]
Applications in technology[edit]
Main article: Fractal analysis
Fractal antennas[89]
Fractal transistor[90]
Fractal heat exchangers[91]
Architecture[33]
Urban growth[92][93]
Classification of histopathology slides
Fractal landscape or Coastline complexity
Detecting 'life as we don't know it' by fractal analysis[94]
Enzymes (Michaelis-Menten kinetics)
Generation of new music
Signal and image compression
Creation of digital photographic enlargements
Fractal in soil mechanics
Organic environments
Procedural generation
Fractography and fracture mechanics
Small angle scattering theory of fractally rough systems
T-shirts and other fashion
Generation of patterns for camouflage, such as MARPAT
Digital sundial
Technical analysis of price series
Fractals in networks
Medicine[29]
Neuroscience[24][25]
Diagnostic Imaging[28]
Pathology[95][96]
Geology[97]
Geography[98]
Archaeology[99][100]
Soil mechanics[27]
Seismology[30]
Search and rescue[101]
Technical analysis[102]
Morton order space filling curves for GPU cache coherency in texture mapping,[103][104][105] rasterisation[106][107] and indexing of turbulence data.[108][109]
Ion propulsion[edit]
When two-dimensional fractals are iterated many times, the perimeter of the fractal increases up to infinity, but the area may never exceed a certain value. A fractal in three-dimensional space is similar; such a fractal may have an infinite surface area, but never exceed a certain volume.[110] This can be utilized to maximize the efficiency of ion propulsion when choosing electron emitter construction and material. If done correctly, the efficiency of the emission process can be maximized.[111]
Mathematics portal
Banach fixed point theorem
Box counting
Diamond-square algorithm
Feigenbaum function
Form constant
Fractal cosmology
Fractal derivative
Fractalgrid
Fractal string
Fracton
Graftal
Greeble
Lacunarity
List of fractals by Hausdorff dimension
Mandelbulb
Mandelbox
Matryoshka doll
Multifractal system
Multiplicative calculus
Newton fractal
Percolation
Power law
Publications in fractal geometry
Random walk
Self-reference
Strange loop
Wiener process
^ The original paper, Lévy, Paul (1938). "Les Courbes planes ou gauches et les surfaces composées de parties semblables au tout". Journal de l'École Polytechnique: 227–247, 249–291. , is translated in Edgar, pages 181–239.
^ The Hilbert curve map is not a homeomorphism, so it does not preserve topological dimension. The topological dimension and Hausdorff dimension of the image of the Hilbert map in R2 are both 2. Note, however, that the topological dimension of the graph of the Hilbert map (a set in R3) is 1.
^ a b c d e f g h i j k l m n o Mandelbrot, Benoît B. (1983). The fractal geometry of nature. Macmillan. ISBN 978-0-7167-1186-5.
^ a b c d e Falconer, Kenneth (2003). Fractal Geometry: Mathematical Foundations and Applications. John Wiley & Sons. xxv. ISBN 978-0-470-84862-3.
^ a b Briggs, John (1992). Fractals:The Patterns of Chaos. London: Thames and Hudson. p. 148. ISBN 978-0-500-27693-8.
^ a b c d e f g h i j Vicsek, Tamás (1992). Fractal growth phenomena. Singapore/New Jersey: World Scientific. pp. 31, 139–146. ISBN 978-981-02-0668-0.
^ a b Boeing, G. (2016). "Visual Analysis of Nonlinear Dynamical Systems: Chaos, Fractals, Self-Similarity and the Limits of Prediction". Systems. 4 (4): 37. arXiv:1608.04416. doi:10.3390/systems4040037. Retrieved December 2, 2016.
^ a b c Gouyet, Jean-François (1996). Physics and fractal structures. Paris/New York: Masson Springer. ISBN 978-0-387-94153-0.
^ a b c Mandelbrot, Benoît B. (2004). Fractals and Chaos. Berlin: Springer. p. 38. ISBN 978-0-387-20158-0. A fractal set is one for which the fractal (Hausdorff-Besicovitch) dimension strictly exceeds the topological dimension
^ a b c d e Gordon, Nigel (2000). Introducing fractal geometry. Duxford: Icon. p. 71. ISBN 978-1-84046-123-7.
^ Segal, S. L. (June 1978). "Riemann's example of a continuous 'nondifferentiable' function continued". The Mathematical Intelligencer. 1 (2): 81–82. doi:10.1007/BF03023065.
^ a b c d e f g h Edgar, Gerald (2004). Classics on Fractals. Boulder, CO: Westview Press. ISBN 978-0-8133-4153-8.
^ a b c d e f g h i Trochet, Holly (2009). "A History of Fractal Geometry". MacTutor History of Mathematics. Archived from the original on March 12, 2012.
^ a b c d Albers, Donald J.; Alexanderson, Gerald L. (2008). "Benoît Mandelbrot: In his own words". Mathematical people : profiles and interviews. Wellesley, MA: AK Peters. p. 214. ISBN 978-1-56881-340-0.
^ Mandelbrot, Benoit. "24/7 Lecture on Fractals". 2006 Ig Nobel Awards. Improbable Research.
^ Mandelbrot, B. B.: The Fractal Geometry of Nature. W. H. Freeman and Company, New York (1982); p. 15.
^ Jens Feder (2013). Fractals. Springer Science & Business Media. p. 11. ISBN 978-1-4899-2124-6.
^ Gerald Edgar (2007). Measure, Topology, and Fractal Geometry. Springer Science & Business Media. p. 7. ISBN 978-0-387-74749-1.
^ a b c Peters, Edgar (1996). Chaos and order in the capital markets : a new view of cycles, prices, and market volatility. New York: Wiley. ISBN 978-0-471-13938-6.
^ Krapivsky, P. L.; Ben-Naim, E. (1994). "Multiscaling in Stochastic Fractals". Physics Letters A. 196 (3–4): 168. Bibcode:1994PhLA..196..168K. doi:10.1016/0375-9601(94)91220-3.
^ Hassan, M. K.; Rodgers, G. J. (1995). "Models of fragmentation and stochastic fractals". Physics Letters A. 208 (1–2): 95. Bibcode:1995PhLA..208...95H. doi:10.1016/0375-9601(95)00727-k.
^ Hassan, M. K.; Pavel, N. I.; Pandit, R. K.; Kurths, J. (2014). "Dyadic Cantor set and its kinetic and stochastic counterpart". Chaos, Solitons & Fractals. 60: 31–39. arXiv:1401.0249. Bibcode:2014CSF....60...31H. doi:10.1016/j.chaos.2013.12.010.
^ a b Brothers, Harlan J. (2007). "Structural Scaling in Bach's Cello Suite No. 3". Fractals. 15 (1): 89–95. doi:10.1142/S0218348X0700337X.
^ a b Tan, Can Ozan; Cohen, Michael A.; Eckberg, Dwain L.; Taylor, J. Andrew (2009). "Fractal properties of human heart period variability: Physiological and methodological implications". The Journal of Physiology. 587 (15): 3929. doi:10.1113/jphysiol.2009.169219. PMC 2746620. PMID 19528254.
^ a b Buldyrev, Sergey V.; Goldberger, Ary L.; Havlin, Shlomo; Peng, Chung-Kang; Stanley, H. Eugene (1995). "Fractals in Biology and Medicine: From DNA to the Heartbeat". In Bunde, Armin; Havlin, Shlomo (eds.). Fractals in Science. Springer.
^ a b Liu, Jing Z.; Zhang, Lu D.; Yue, Guang H. (2003). "Fractal Dimension in Human Cerebellum Measured by Magnetic Resonance Imaging". Biophysical Journal. 85 (6): 4041–4046. Bibcode:2003BpJ....85.4041L. doi:10.1016/S0006-3495(03)74817-6. PMC 1303704. PMID 14645092.
^ a b Karperien, Audrey L.; Jelinek, Herbert F.; Buchan, Alastair M. (2008). "Box-Counting Analysis of Microglia Form in Schizophrenia, Alzheimer's Disease and Affective Disorder". Fractals. 16 (2): 103. doi:10.1142/S0218348X08003880.
^ a b c d e Jelinek, Herbert F.; Karperien, Audrey; Cornforth, David; Cesar, Roberto; Leandro, Jorge de Jesus Gomes (2002). "MicroMod-an L-systems approach to neural modelling". In Sarker, Ruhul (ed.). Workshop proceedings: the Sixth Australia-Japan Joint Workshop on Intelligent and Evolutionary Systems, University House, ANU. University of New South Wales. ISBN 9780731705054. OCLC 224846454. http://researchoutput.csu.edu.au/R/-?func=dbin-jump-full&object_id=6595&local_base=GEN01-CSU01. Retrieved February 3, 2012. Event location: Canberra, Australia
^ a b Hu, Shougeng; Cheng, Qiuming; Wang, Le; Xie, Shuyun (2012). "Multifractal characterization of urban residential land price in space and time". Applied Geography. 34: 161–170. doi:10.1016/j.apgeog.2011.10.016.
^ a b Karperien, Audrey; Jelinek, Herbert F.; Leandro, Jorge de Jesus Gomes; Soares, João V. B.; Cesar Jr, Roberto M.; Luckie, Alan (2008). "Automated detection of proliferative retinopathy in clinical practice". Clinical Ophthalmology (Auckland, N.Z.). 2 (1): 109–122. doi:10.2147/OPTH.S1579. PMC 2698675. PMID 19668394.
^ a b c d Losa, Gabriele A.; Nonnenmacher, Theo F. (2005). Fractals in biology and medicine. Springer. ISBN 978-3-7643-7172-2.
^ a b c Vannucchi, Paola; Leoni, Lorenzo (2007). "Structural characterization of the Costa Rica décollement: Evidence for seismically-induced fluid pulsing". Earth and Planetary Science Letters. 262 (3–4): 413. Bibcode:2007E&PSL.262..413V. doi:10.1016/j.epsl.2007.07.056.
^ a b Wallace, David Foster. "Bookworm on KCRW". Kcrw.com. Retrieved October 17, 2010.
^ a b Eglash, Ron (1999). "African Fractals: Modern Computing and Indigenous Design". New Brunswick: Rutgers University Press. Retrieved October 17, 2010.
^ a b Ostwald, Michael J., and Vaughan, Josephine (2016) The Fractal Dimension of Architecture. Birhauser, Basel. doi:10.1007/978-3-319-32426-5.
^ Stumpff, Andrew (2013). "The Law is a Fractal: The Attempt to Anticipate Everything". 44. Loyola University Chicago Law Journal: 649. SSRN 2157804.
^ Baranger, Michael. "Chaos, Complexity, and Entropy: A physics talk for non-physicists" (PDF).
^ a b Pickover, Clifford A. (2009). The Math Book: From Pythagoras to the 57th Dimension, 250 Milestones in the History of Mathematics. Sterling. p. 310. ISBN 978-1-4027-5796-9.
^ "Fractal Geometry". www-history.mcs.st-and.ac.uk. Retrieved April 11, 2017.
^ Mandelbrot, B. (1967). "How Long Is the Coast of Britain?". Science. 156 (3775): 636–638. Bibcode:1967Sci...156..636M. doi:10.1126/science.156.3775.636. PMID 17837158.
^ Batty, Michael (April 4, 1985). "Fractals – Geometry Between Dimensions". New Scientist. 105 (1450): 31.
^ Russ, John C. (1994). Fractal surfaces. 1. Springer. p. 1. ISBN 978-0-306-44702-0. Retrieved February 5, 2011.
^ kottke.org. 2009. Vol Libre, an amazing CG film from 1980. [online] Available at: http://kottke.org/09/07/vol-libre-an-amazing-cg-film-from-1980
^ Edgar, Gerald (2008). Measure, topology, and fractal geometry. New York: Springer-Verlag. p. 1. ISBN 978-0-387-74748-4.
^ Karperien, Audrey (2004). Defining microglial morphology: Form, Function, and Fractal Dimension. Charles Sturt University. doi:10.13140/2.1.2815.9048.
^ Spencer, John; Thomas, Michael S. C.; McClelland, James L. (2009). Toward a unified theory of development : connectionism and dynamic systems theory re-considered. Oxford/New York: Oxford University Press. ISBN 978-0-19-530059-8.
^ Frame, Angus (August 3, 1998). "Iterated Function Systems". In Pickover, Clifford A. (ed.). Chaos and fractals: a computer graphical journey : ten year compilation of advanced research. Elsevier. pp. 349–351. ISBN 978-0-444-50002-1. Retrieved February 4, 2012.
^ "Haferman Carpet". WolframAlpha. Retrieved October 18, 2012.
^ a b c d Hahn, Horst K.; Georg, Manfred; Peitgen, Heinz-Otto (2005). "Fractal aspects of three-dimensional vascular constructive optimization". In Losa, Gabriele A.; Nonnenmacher, Theo F. (eds.). Fractals in biology and medicine. Springer. pp. 55–66. ISBN 978-3-7643-7172-2.
^ J. W. Cannon, W. J. Floyd, W. R. Parry. Finite subdivision rules. Conformal Geometry and Dynamics, vol. 5 (2001), pp. 153–196.
^ J. W. Cannon, W. Floyd and W. Parry. Crystal growth, biological cell growth and geometry. Pattern Formation in Biology, Vision and Dynamics, pp. 65–82. World Scientific, 2000. ISBN 981-02-3792-8, ISBN 978-981-02-3792-9.
^ Fathallah-Shaykh, Hassan M. (2011). "Fractal Dimension of the Drosophila Circadian Clock". Fractals. 19 (4): 423–430. doi:10.1142/S0218348X11005476.
^ "Hunting the Hidden Dimension". Nova. PBS. WPMB-Maryland. October 28, 2008.
^ Sadegh, Sanaz (2017). "Plasma Membrane is Compartmentalized by a Self-Similar Cortical Actin Meshwork". Physical Review X. 7 (1). doi:10.1103/PhysRevX.7.011031. PMC 5500227. PMID 28690919.
^ Carbone, Alessandra; Gromov, Mikhael; Prusinkiewicz, Przemyslaw (2000). Pattern formation in biology, vision and dynamics. World Scientific. p. 78. ISBN 978-981-02-3792-9.
^ Sornette, Didier (2004). Critical phenomena in natural sciences: chaos, fractals, selforganization, and disorder: concepts and tools. Springer. pp. 128–140. ISBN 978-3-540-40754-6.
^ a b Sweet, D.; Ott, E.; Yorke, J. A. (1999), "Complex topology in Chaotic scattering: A Laboratory Observation", Nature, 399 (6734): 315, Bibcode:1999Natur.399..315S, doi:10.1038/20573
^ Addison, Paul S. (1997). Fractals and chaos: an illustrated course. CRC Press. pp. 44–46. ISBN 978-0-7503-0400-9. Retrieved February 5, 2011.
^ Pincus, David (September 2009). "The Chaotic Life: Fractal Brains Fractal Thoughts". psychologytoday.com.
^ Enright, Matthew B.; Leitner, David M. (January 27, 2005). "Mass fractal dimension and the compactness of proteins". Physical Review E. 71 (1): 011912. Bibcode:2005PhRvE..71a1912E. doi:10.1103/PhysRevE.71.011912. PMID 15697635.
^ Takayasu, H. (1990). Fractals in the physical sciences. Manchester: Manchester University Press. p. 36. ISBN 9780719034343.
^ Jun, Li; Ostoja-Starzewski, Martin (April 1, 2015). "Edges of Saturn's Rings are Fractal". SpringerPlus. 4,158: 158. doi:10.1186/s40064-015-0926-6. PMC 4392038. PMID 25883885.
^ Meyer, Yves; Roques, Sylvie (1993). Progress in wavelet analysis and applications: proceedings of the International Conference "Wavelets and Applications", Toulouse, France – June 1992. Atlantica Séguier Frontières. p. 25. ISBN 978-2-86332-130-0. Retrieved February 5, 2011.
^ Ozhovan M. I., Dmitriev I. E., Batyukhnova O. G. Fractal structure of pores of clay soil. Atomic Energy, 74, 241–243 (1993).
^ Sreenivasan, K. R.; Meneveau, C. (1986). "The Fractal Facets of Turbulence". Journal of Fluid Mechanics. 173: 357–386. Bibcode:1986JFM...173..357S. doi:10.1017/S0022112086001209.
^ de Silva, C. M.; Philip, J.; Chauhan, K.; Meneveau, C.; Marusic, I. (2013). "Multiscale Geometry and Scaling of the Turbulent–Nonturbulent Interface in High Reynolds Number Boundary Layers". Phys. Rev. Lett. 111 (6039): 192–196. Bibcode:2011Sci...333..192A. doi:10.1126/science.1203223. PMID 21737736.
^ Falconer, Kenneth (2013). Fractals, A Very Short Introduction. Oxford University Press.
^ Taylor, R. P.; et al. (1999). "Fractal Analysis of Pollock's Drip Paintings". Nature. 399 (6735): 422. Bibcode:1999Natur.399..422T. doi:10.1038/20833.
^ Mureika, J. R.; Dyer, C. C.; Cupchik, G. C. (2005). "Multifractal Structure in Nonrepresentational Art". Physical Review E. 72 (4): 046101–1–15. arXiv:physics/0506063. Bibcode:2005PhRvE..72d6101M. doi:10.1103/PhysRevE.72.046101. PMID 16383462.
^ Redies, C.; Hasenstein, J.; Denzler, J. (2007). "Fractal-Like Image Statistics in Visual Art: Similarity to Natural Scenes". Spatial Vision. 21 (1): 137–148. doi:10.1163/156856807782753921. PMID 18073055.
^ Lee, S.; Olsen, S.; Gooch, B. (2007). "Simulating and Analyzing Jackson Pollock's Paintings". Journal of Mathematics and the Arts. 1 (2): 73–83. CiteSeerX 10.1.1.141.7470. doi:10.1080/17513470701451253.
^ Alvarez-Ramirez, J.; Ibarra-Valdez, C.; Rodriguez, E.; Dagdug, L. (2008). "1/f-Noise Structure in Pollock's Drip Paintings". Physica A. 387 (1): 281–295. Bibcode:2008PhyA..387..281A. doi:10.1016/j.physa.2007.08.047.
^ Graham, D. J.; Field, D. J. (2008). "Variations in Intensity for Representative and Abstract Art, and for Art from Eastern and Western Hemispheres" (PDF). Perception. 37 (9): 1341–1352. CiteSeerX 10.1.1.193.4596. doi:10.1068/p5971. PMID 18986061.
^ Alvarez-Ramirez, J.; Echeverria, J. C.; Rodriguez, E. (2008). "Performance of a high-dimensional R/S method for Hurst exponent estimation". Physica A. 387 (26): 6452–6462. Bibcode:2008PhyA..387.6452A. doi:10.1016/j.physa.2008.08.014.
^ Coddington, J.; Elton, J.; Rockmore, D.; Wang, Y. (2008). "Multifractal Analysis and Authentication of Jackson Pollock Paintings". Proceedings of SPIE. 6810 (68100F): 1–12. Bibcode:2008SPIE.6810E..0FC. doi:10.1117/12.765015.
^ Al-Ayyoub, M.; Irfan, M. T.; Stork, D. G. (2009). "Boosting Multi-Feature Visual Texture Classifiers for the Authentification of Jackson Pollock's Drip Paintings". SPIE Proceedings on Computer Vision and Image Analysis of Art II. Computer Vision and Image Analysis of Art II. 7869 (78690H): 78690H. Bibcode:2011SPIE.7869E..0HA. doi:10.1117/12.873142.
^ Mureika, J. R.; Taylor, R. P. (2013). "The Abstract Expressionists and Les Automatistes: multi-fractal depth?". Signal Processing. 93 (3): 573. doi:10.1016/j.sigpro.2012.05.002.
^ Taylor, R. P.; et al. (2005). "Authenticating Pollock Paintings Using Fractal Geometry". Pattern Recognition Letters. 28 (6): 695–702. doi:10.1016/j.patrec.2006.08.012.
^ Jones-Smith, K.; et al. (2006). "Fractal Analysis: Revisiting Pollock's Paintings". Nature. 444 (7119): E9–10. Bibcode:2006Natur.444E...9J. doi:10.1038/nature05398. PMID 17136047.
^ Taylor, R. P.; et al. (2006). "Fractal Analysis: Revisiting Pollock's Paintings (Reply)". Nature. 444 (7119): E10–11. Bibcode:2006Natur.444E..10T. doi:10.1038/nature05399.
^ Shamar, L. (2015). "What Makes a Pollock Pollock: A Machine Vision Approach" (PDF). International Journal of Arts and Technology. 8: 1–10. CiteSeerX 10.1.1.647.365. doi:10.1504/IJART.2015.067389.
^ Taylor, R. P.; Spehar, B.; Van Donkelaar, P.; Hagerhall, C. M. (2011). "Perceptual and Physiological Responses to Jackson Pollock's Fractals". Frontiers in Human Neuroscience. 5: 1–13. doi:10.3389/fnhum.2011.00060. PMC 3124832. PMID 21734876.
^ Frame, Michael; and Mandelbrot, Benoît B.; A Panorama of Fractals and Their Uses
^ Nelson, Bryn; Sophisticated Mathematics Behind African Village Designs Fractal patterns use repetition on large, small scale, San Francisco Chronicle, Wednesday, February 23, 2009
^ Situngkir, Hokky; Dahlan, Rolan (2009). Fisika batik: implementasi kreatif melalui sifat fraktal pada batik secara komputasional. Jakarta: Gramedia Pustaka Utama. ISBN 978-979-22-4484-7
^ Rulistia, Novia D. (October 6, 2015). "Application maps out nation's batik story". The Jakarta Post. Retrieved September 25, 2016.
^ Koutonin, Mawuna (18 March 2016). "Story of cities #5: Benin City, the mighty medieval capital now lost without trace". Retrieved 2 April 2018.
^ Taylor, Richard P. (2016). "Fractal Fluency: An Intimate Relationship Between the Brain and Processing of Fractal Stimuli". In Di Ieva, Antonio (ed.). The Fractal Geometry of the Brain. Springer Series in Computational Neuroscience. Springer. pp. 485–496. ISBN 978-1-4939-3995-4.
^ Taylor, Richard P. (2006). "Reduction of Physiological Stress Using Fractal Art and Architecture". Leonardo. 39 (3): 245–251. doi:10.1162/leon.2006.39.3.245.
^ For further discussion of this effect, see Taylor, Richard P.; Spehar, Branka; Donkelaar, Paul Van; Hagerhall, Caroline M. (2011). "Perceptual and Physiological Responses to Jackson Pollock's Fractals". Frontiers in Human Neuroscience. 5: 60. doi:10.3389/fnhum.2011.00060. PMC 3124832. PMID 21734876.
^ Hohlfeld, Robert G.; Cohen, Nathan (1999). "Self-similarity and the geometric requirements for frequency independence in Antennae". Fractals. 7 (1): 79–84. doi:10.1142/S0218348X99000098.
^ Reiner, Richard; Waltereit, Patrick; Benkhelifa, Fouad; Müller, Stefan; Walcher, Herbert; Wagner, Sandrine; Quay, Rüdiger; Schlechtweg, Michael; Ambacher, Oliver; Ambacher, O. (2012). "Fractal structures for low-resistance large area AlGaN/GaN power transistors". Proceedings of ISPSD: 341–344. doi:10.1109/ISPSD.2012.6229091. ISBN 978-1-4577-1596-9.
^ Zhiwei Huang; Yunho Hwang; Vikrant Aute; Reinhard Radermacher (2016). "Review of Fractal Heat Exchangers" (PDF) International Refrigeration and Air Conditioning Conference. Paper 1725
^ Chen, Yanguang (2011). "Modeling Fractal Structure of City-Size Distributions Using Correlation Functions". PLoS ONE. 6 (9): e24791. arXiv:1104.4682. Bibcode:2011PLoSO...624791C. doi:10.1371/journal.pone.0024791. PMC 3176775. PMID 21949753.
^ "Applications". Archived from the original on October 12, 2007. Retrieved October 21, 2007.
^ "Detecting 'life as we don't know it' by fractal analysis"
^ Smith, Robert F.; Mohr, David N.; Torres, Vicente E.; Offord, Kenneth P.; Melton III, L. Joseph (1989). "Renal insufficiency in community patients with mild asymptomatic microhematuria". Mayo Clinic Proceedings. 64 (4): 409–414. doi:10.1016/s0025-6196(12)65730-9. PMID 2716356.
^ Landini, Gabriel (2011). "Fractals in microscopy". Journal of Microscopy. 241 (1): 1–8. doi:10.1111/j.1365-2818.2010.03454.x. PMID 21118245.
^ Cheng, Qiuming (1997). "Multifractal Modeling and Lacunarity Analysis". Mathematical Geology. 29 (7): 919–932. doi:10.1023/A:1022355723781.
^ Burkle-Elizondo, Gerardo; Valdéz-Cepeda, Ricardo David (2006). "Fractal analysis of Mesoamerican pyramids". Nonlinear Dynamics, Psychology, and Life Sciences. 10 (1): 105–122. PMID 16393505.
^ Brown, Clifford T.; Witschey, Walter R. T.; Liebovitch, Larry S. (2005). "The Broken Past: Fractals in Archaeology". Journal of Archaeological Method and Theory. 12: 37–78. doi:10.1007/s10816-005-2396-6.
^ Saeedi, Panteha; Sorensen, Soren A. "An Algorithmic Approach to Generate After-disaster Test Fields for Search and Rescue Agents" (PDF). Proceedings of the World Congress on Engineering 2009: 93–98. ISBN 978-988-17-0125-1.
^ Bunde, A.; Havlin, S. (2009). "Fractal Geometry, A Brief Introduction to". Encyclopedia of Complexity and Systems Science. p. 3700. doi:10.1007/978-0-387-30440-3_218. ISBN 978-0-387-75888-6.
^ "gpu internals" (PDF).
^ "sony patents".
^ "description of swizzled and hybrid tiled swizzled textures".
^ "nvidia patent".
^ "Johns Hopkins Turbulence Databases".
^ Li, Y.; Perlman, E.; Wang, M.; Yang, y.; Meneveau, C.; Burns, R.; Chen, S.; Szalay, A.; Eyink, G. (2008). "A Public Turbulence Database Cluster and Applications to Study Lagrangian Evolution of Velocity Increments in Turbulence". Journal of Turbulence. 9: N31. arXiv:0804.1703. Bibcode:2008JTurb...9...31L. doi:10.1080/14685240802376389.
^ "Introduction to Fractal Geometry". www.fractal.org. Retrieved April 11, 2017.
^ DeFelice, David (August 18, 2015). "NASA – Ion Propulsion". NASA. Retrieved April 11, 2017.
Barnsley, Michael F.; and Rising, Hawley; Fractals Everywhere. Boston: Academic Press Professional, 1993. ISBN 0-12-079061-0
Duarte, German A.; Fractal Narrative. About the Relationship Between Geometries and Technology and Its Impact on Narrative Spaces. Bielefeld: Transcript, 2014. ISBN 978-3-8376-2829-6
Falconer, Kenneth; Techniques in Fractal Geometry. John Wiley and Sons, 1997. ISBN 0-471-92287-0
Jürgens, Hartmut; Peitgen, Heinz-Otto; and Saupe, Dietmar; Chaos and Fractals: New Frontiers of Science. New York: Springer-Verlag, 1992. ISBN 0-387-97903-4
Mandelbrot, Benoit B.; The Fractal Geometry of Nature. New York: W. H. Freeman and Co., 1982. ISBN 0-7167-1186-9
Peitgen, Heinz-Otto; and Saupe, Dietmar; eds.; The Science of Fractal Images. New York: Springer-Verlag, 1988. ISBN 0-387-96608-0
Pickover, Clifford A.; ed.; Chaos and Fractals: A Computer Graphical Journey – A 10 Year Compilation of Advanced Research. Elsevier, 1998. ISBN 0-444-50002-2
Jones, Jesse; Fractals for the Macintosh, Waite Group Press, Corte Madera, CA, 1993. ISBN 1-878739-46-8.
Lauwerier, Hans; Fractals: Endlessly Repeated Geometrical Figures, Translated by Sophia Gill-Hoffstadt, Princeton University Press, Princeton NJ, 1991. ISBN 0-691-08551-X, cloth. ISBN 0-691-02445-6 paperback. "This book has been written for a wide audience..." Includes sample BASIC programs in an appendix.
Sprott, Julien Clinton (2003). Chaos and Time-Series Analysis. Oxford University Press. ISBN 978-0-19-850839-7.
Wahl, Bernt; Van Roy, Peter; Larsen, Michael; and Kampman, Eric; Exploring Fractals on the Macintosh, Addison Wesley, 1995. ISBN 0-201-62630-6
Lesmoir-Gordon, Nigel; The Colours of Infinity: The Beauty, The Power and the Sense of Fractals. 2004. ISBN 1-904555-05-5 (The book comes with a related DVD of the Arthur C. Clarke documentary introduction to the fractal concept and the Mandelbrot set.)
Liu, Huajie; Fractal Art, Changsha: Hunan Science and Technology Press, 1997, ISBN 9787535722348.
Gouyet, Jean-François; Physics and Fractal Structures (Foreword by B. Mandelbrot); Masson, 1996. ISBN 2-225-85130-1, and New York: Springer-Verlag, 1996. ISBN 978-0-387-94153-0. Out-of-print. Available in PDF version at. "Physics and Fractal Structures" (in French). Jfgouyet.fr. Retrieved October 17, 2010.
Bunde, Armin; Havlin, Shlomo (1996). Fractals and Disordered Systems. Springer.
Bunde, Armin; Havlin, Shlomo (1995). Fractals in Science. Springer.
ben-Avraham, Daniel; Havlin, Shlomo (2000). Diffusion and Reactions in Fractals and Disordered Systems. Cambridge University Press.
Falconer, Kenneth (2013). Fractals, A Very Short Introduction. Oxford University Press.
Wikimedia Commons has media related to Fractal.
Wikibooks has a book on the topic of: Fractals
Fractals at the Library of Congress Web Archives (archived 2001-11-16)
Scaling and Fractals presented by Shlomo Havlin, Bar-Ilan University
Hunting the Hidden Dimension, PBS NOVA, first aired August 24, 2011
Benoit Mandelbrot: Fractals and the Art of Roughness, TED, February 2010
Technical Library on Fractals for controlling fluid
Equations of self-similar fractal measure based on the fractional-order calculus(2007)
Fractal dimensions
Assouad
Box-counting
Hausdorff
Topological
Self-similarity
Iterated function
Barnsley fern
Koch snowflake
Menger sponge
Sierpinski carpet
Sierpinski triangle
Space-filling curve
Blancmange curve
De Rham curve
Dragon curve
Koch curve
Lévy C curve
Peano curve
Sierpiński curve
n-flake
Strange attractor
L-system
Fractal canopy
H tree
Escape-time
Burning Ship fractal
Julia set
Lyapunov fractal
Tricorn
Rendering techniques
Buddhabrot
Orbit trap
Pickover stalk
Random fractals
Brownian motion
Brownian tree
Fractal landscape
Lévy flight
Percolation theory
Self-avoiding walk
Felix Hausdorff
Gaston Julia
Helge von Koch
Paul Lévy
Aleksandr Lyapunov
Wacław Sierpiński
"How Long Is the Coast of Britain?"
Coastline paradox
The Beauty of Fractals (1986 book)
Chaos: Making a New Science (1987 book)
The Fractal Geometry of Nature (1982 book)
Anosov diffeomorphism
Chaos theory in organizational development
Control of chaos
Edge of chaos
Quantum chaos
Santa Fe Institute
Synchronization of chaos
Chaotic maps
Arnold tongue
Arnold's cat map
Baker's map
Complex quadratic map
Complex squaring map
Coupled map lattice
Double pendulum
Double scroll attractor
Duffing equation
Duffing map
Dyadic transformation
Dynamical billiards
Exponential map
Gauss map
Gingerbreadman map
Hénon map
Horseshoe map
Ikeda map
Interval exchange map
Kaplan–Yorke map
Logistic map
Lorenz system
Multiscroll attractor
Rabinovich–Fabrikant equations
Rössler attractor
Swinging Atwood's machine
Tent map
Tinkerbell map
Van der Pol oscillator
Zaslavskii map
Chaos systems
Bouncing ball dynamics
Chua's circuit
Economic bubble
FPUT problem
Tilt-A-Whirl
Chaos theorists
Mary Cartwright
Leon O. Chua
Mitchell Feigenbaum
Celso Grebogi
Martin Gutzwiller
Brosl Hasslacher
Michel Hénon
Svetlana Jitomirskaya
Edward Norton Lorenz
Benoît Mandelbrot
Hee Oh
Edward Ott
Henri Poincaré
Mary Rees
Otto Rössler
David Ruelle
Caroline Series
Oleksandr Mykolayovych Sharkovsky
Nina Snaith
Floris Takens
Audrey Terras
Mary Tsingou
Amie Wilkinson
James A. Yorke
Lai-Sang Young
Catenary
Plastic number
Hyperboloid structure
Minimal surface
Camera lucida
Projective geometry
Tessellation
Wallpaper group
Algorithmic art
Anamorphic art
Computer art
Islamic geometric patterns
Girih
Muqarnas
Croatian Interlace
Interlace (art)
Geodesic dome
List of works designed with the golden ratio
Octacube
Self-portrait in a Convex Mirror
19th–20th Century
The Ancient of Days
Jean Metzinger
Danseuse au café
La condition humaine
The Swallow's Tail
Circle Limit III
Martin and Erik Demaine
Scott Draves
Jan Dibbets
John Ernest
Helaman Ferguson
Peter Forakis
Bathsheba Grossman
George W. Hart
Desmond Paul Henry
John A. Hiigli
Anthony Hill
Charles Jencks
Garden of Cosmic Speculation
Robert Longhurst
Hinke Osinga
Hamid Naderi Yeganeh
A Bird in Flight
Tony Robbin
Oliver Sin
Daina Taimina
Roman Verostko
Polykleitos
De architectura
De prospectiva pingendi
De pictura
De re aedificatoria
Sebastiano Serlio
Regole generali d'architettura
I quattro libri dell'architettura
Vier Bücher von Menschlicher Proportion
Frederik Macody Lund
"Ad Quadratum"
Jay Hambidge
"The Greek Vase"
Samuel Colman
"Nature's Harmonic Unity"
The Grammar of Ornament
Ernest Hanbury Hankin
The Drawing of Geometric Patterns in Saracenic Art
A Mathematician's Apology
George David Birkhoff
Aesthetic Measure
Gödel, Escher, Bach
The 'Life' of a Carpet
Journal of Mathematics and the Arts
Ars Mathematica
The Bridges Organization
European Society for Mathematics and the Arts
Goudreau Museum of Mathematics in Art and Science
Institute For Figuring
Museum of Mathematics
Mathematical beauty
Soap bubble
in crystals
Quasicrystals
in biology
Vortex street
Widmanstätten pattern
Pattern formation
Logarithmic spiral
Plateau's laws
Liber Abaci
Adolf Zeising
Ernst Haeckel
Wilson Bentley
D'Arcy Wentworth Thompson
On Growth and Form
The Chemical Basis of Morphogenesis
Aristid Lindenmayer
How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension
Retrieved from "https://en.wikipedia.org/w/index.php?title=Fractal&oldid=905526071"
Mathematical structures
Computational fields of study
Use mdy dates from May 2018
Articles containing Latin-language text


No matter in which medium I am painting, I like to challenge myself. I love vibrant colors and am very much interested in color combinations and depicting movement. My other area of interest is doing pet portraits in pastels. The challenge here is to capture the animal's personality through their eyes. When I can do that, I am happy.


We want to strengthen excellence, equality and inclusion throughout the education system. The Green Party want to provide easy access to to affordable childcare for all parents, to ease the financial burden many now experience.
Include all sexual identities and gender identities in the Relationships and Sexuality curriculum to avoid excluding LGBTQI people.
The Green Party believes that everyone should be able to benefit fully from education. The education system should empower people to fulfil their potential and improve their prospects no matter what their social background. No child should be denied a school place or be made to feel unwelcome in a school due to their religious background. We believe that religious instruction should take place outside of the classroom.
Introduce a childcare payment, and mandate a price ceiling for childcare services.
Provide for six months of paid parental leave to follow maternity leave and unpaid parental leave up to a child's third birthday.
Limit the financial demands schools makes on parents by subsidising school books and eliminating other charges.
We believe in a new model for childcare in Ireland, one that allows parents to be available to their children in the early stages of their lives and is responsive to the changing childcare needs of parents and children during childhood. We believe in providing an expanded set of childcare options for parents, by providing flexibility in parental leave and childcare subsidies that each family can adapt to their own needs. We want to reduce financial and time pressures on hard working parents by guaranteeing every child access to a high quality, nutritious meal in school.
Create a stronger apprenticeship programme.
Ireland's tertiary education system is chronically underfunded. The Green Party would invest in revitalising our universities. Recognising that a university degree is not useful for all career paths, we would advocate for greatly enhancing our apprenticeship and technical education programmes.


A family run business dating back to 1887. Capezio is a manufacturer of dance shoes, apparel and accessories. They offer hand-crafted customised shoes to meet customers' specifications. Capezio has made shoes for world-class productions such as Chicago, Black Swan, Mama Mia!, The Wizard of Oz, and countless others.
Capezio's global Magento enterprise edition multi-store, wanted a trusted and experienced company to ensure code quality was done at the highest level. They required support to improve the overall cosmetic look and extend functionality to provide customers with an improved user-friendly shopping experience.
Furthermore, we provided our technical skills to extend existing functionality and cosmetic improvements to the overall responsive website. This included creating and integrating new modules for improving the user experience. With quick product previews on the category page, including colour swatches to visually preview configurable product options. To help customers promote products more easily, an extension for referring friends was integrated.
To reduce the bounce rate and create a faster web experience, we identified speed issues and offered recommendations for optimisation.
Capezio was impressed with our technical expertise and ability to meet deadlines. They are extremely happy to see their Magento store now performing faster and at a higher quality. With the newly extended functionality, this had led to increased customer retention and a rise in revenue on their global website.


Seeking assistant manager for busy speciality food store. Previous experience in sales required. Good hospitality and customer skills needed. Grocery store experience necessary.
High paying sales position. Motivated individuals where the sky is the limit, and great benefits.


Rehmannia glutinosa, the Latin name for Rehmanniae root, is just one of the many superior herbs that you will find available from Botanicals, a premiere online dried herbs source. Rehmannia glutinosa is a perennial herb that is native to China. It has another common name of Chinese Foxglove, because the flowers look like foxglove flowers, but hairy. In China the plant is known as Sheng di Huang, and is used in that medical herbal tradition.
Rehamanniae Root is available from Botanicals in a cooked, sulfite-free version, as well as a raw sliced form; both versions are Organic. is proud to offer Rehamanniae Root that is certified Organic by Quality Assurance International. All of the dried herbs that we offer, including this root, undergo stringent testing in order to verify the appearance, flavor, aroma and color of each herb we provide our customers. All of our dried herbs are processed and packaged at our own mill, so that we can ensure the highest standards of best practices go into their manufacture.
Warning: Raw: Not for use in persons with diarrhea.


text:
ORNL News
Past Talks
ORNL Archives
About FORNL
FORNL Board
FORNL Annual reports
Refrigeration — Cooling COVID-19 vaccines
Oak Ridge National Laboratory researchers have retrofitted a commercial refrigeration container designed to ensure COVID-19 vaccines remain at ultra-low temperatures during long transport and while locally stored.
Most COVID vaccines, depending on the manufacturer, are stored at minus 70 or 30 degrees Celsius. Current transport methods use dry ice to maintain desired temperatures. However, longer travel times, particularly to remote locations without supportive infrastructure, require extended refrigeration.
Materials — Radiation-resistant sensors
To advance sensor technologies, Oak Ridge National Laboratory researchers studied piezoelectric materials, which convert mechanical stress into electrical energy, to see how they could handle bombardment with energetic neutrons. This irradiation disturbs the position and behavior of atoms, which can affect the conversion of mechanical stresses into electricity.
Sensors made of piezoelectric materials could help guide the design of prototype nuclear reactors and monitor the health of aging reactors — if the materials can withstand extreme conditions.
Khaleel named associate lab director for national security sciences
Moe Khaleel has been selected to lead the National Security Sciences Directorate, or NSSD, at the Department of Energy's Oak Ridge National Laboratory. In his role as associate lab director, he will oversee efforts in nuclear security, nonproliferation, biosecurity, cybersecurity, grid security and security for manufacturing, autonomous systems and other emerging fields.
"I am excited about working with the NSSD staff at the forefront of addressing national security challenges, and I look forward to the accomplishments we are going to make," Khaleel said.
Top 10 science stories of 2021
Despite challenges posed by the COVID-19 pandemic, research at Oak Ridge National Laboratory in 2021 continued apace, with developments across the lab's broad science and technology missions. Among our most popular news stories in the past year:
Supercomputing: Supercomputers aid scientists studying the smallest particles in the universe
Hubbard named Deputy for Science and Technology at ORNL
Susan Hubbard, an acclaimed scientific leader and researcher, has been named Deputy for Science and Technology at the Department of Energy's Oak Ridge National Laboratory. Her appointment is effective March 1, 2022.
"Dr. Hubbard brings skilled and passionate leadership that will enable world-leading impact across our portfolio," said ORNL Director Thomas Zacharia. "She is a true champion of the DOE mission, the National Laboratory System and our responsibility to leverage powerful user facilities and multidisciplinary teams in the national interest."


text:
Software Survey: Help Needed
December 01, 2013, 9:08 a.m. EST 9 Min Read
As nonprofit organizations struggle with changes in the economy and demands for greater accountability, they are looking to technology and better software to fill the breach.
In spite of the relative importance of nonprofits to American society, these organizations have suffered a continuing decline over the past three decades that left them ill-prepared for the recessionary pressures of the past few years. Both charitable giving and government funding have declined, and nonprofits find themselves struggling with using information to their advantage. In a bid to deal with these forces, nonprofits are increasingly looking to technology and software to help craft effective solutions. This year alone, it's estimated that some 39 percent of NFPs will invest in new technology in a bid to do more with less and to bolster confidence among their constituencies.
NONPROFIT TECHNOLOGIES
Executives of companies that publish accounting and management software for nonprofits generally agree that two trends are making themselves felt in 2013 -- the move to cloud-based computing, and the use of mobile technologies.
"Cloud computing adoption rates are still growing," said Serenic Software president and chief executive officer Randy Keith. "But it definitely has gained momentum and is here to stay as customers become more aware of benefits and more comfortable moving data outside of their firewalls and into the cloud."
"The use of cloud technologies may not yet be a reality," said Paul Lundquist, vice president of sales for Open Systems. "But it has become a discussion point. We have not seen a concerted move to the cloud by the nonprofits themselves, but we have seen a great interest among accountants who want to know that when they are ready to make that move, we will have products to accommodate them."
"Nonprofits lag other sectors in the adoption of new technologies," said Tim Goetz, CEO and co-founder of Aplos Software. "But if you want to know what trends will be major in a year or two in the NFP sector, you need only to look at the small-business marketplace today. There, cloud-based accounting and finance companies are becoming more widely adopted."
"The move to the cloud will also drive changes in how business applications are integrated with accounting," Goetz added. "We see member databases, donations, and other elements moving to become apps that will be integrated. There will be more 'app store' approaches to additional capabilities, rather than stand-alone applications."
"In the last year, the expectation and desire for mobile access to data entry and reporting has been a clear trend," said AccuFund president Peter Stam. "Just walk around any conference and look at what the participants are using. Users are carrying more tablets than PCs."
But mobility and the cloud are not the only technology trends that are emerging. "As nonprofits become more complex, the funder requests and restrictions are becoming more complex, and you can't let anything fall through the cracks," noted Blackbaud senior product marketing manager Jeff Sobers. "This is driving a need to use workflow automation, so that workflow is integrated with document management, and with approval processes."
"Workflow is a major trend," concurred Donald Cassady, president and CEO of Grant Management Systems. "Another and equally important trend is the integration of all of the software within the nonprofit. This trend has emerged strongly in the past 12 months, as executives look for software solutions that play well together."
"The difficulty in fundraising and getting donations are also affecting the technologies coming into use," added Intuit director of product management Gabriel Mendoza. "Gone are the days of pledge drives. Having the ability to accept mobile payments will be crucial to the health of nonprofits."
"Nonprofits will continue to take a hard look at the money and effort expended in past years on accounting system customizations and implementations," said Serenic's Keith. "Not only the initial costs of those systems, but the ongoing year-after-year maintenance costs. They will look for a simpler, effective solutions that still provide the transparency and accountability required by their funders and donors, but solutions that don't compete for resources against fulfilling their mission."
NON-TECH SOLUTIONS
For many nonprofits, the focus is less on a new accounting system, or even the adoption of new technologies. Rather, it is on how the accounting system contributes to their ability to do more with less.
"The principals of fund accounting have not changed," said Joseph Scarano, CEO of Araize. "The issues that are being raised by our customers are related to stricter regulations due to cuts in government funding. Nonprofits need to demonstrate better accountability and program effectiveness to secure funding, whether it be from private or government sources. This puts additional pressure on management to be able to report on their financial activity."
"The theory and practice of fund accounting haven't changed in a thousand years," said Cassady of GMS. "What has changed is the result - the ability to extract the right information to make the nonprofit successful. There is no longer such a thing as a 'standard report' that meets all of the internal and external reporting requirements for fund accounting. Accountability and financial reports have to provide information at two different levels -- to the funding sources and auditor, because you have to keep them happy. And also to program directors, so they can run their programs and manage them within the context of the overall budgets."
"The move to big data is emerging for nonprofits, as it already has for the private business sectors," said Open Systems' Lundquist. "The ability to derive information from prospects and existing customers demands a higher level of customer relationship management, integrated tightly with the ability to [access] internal and external databases to enable nonprofits to create more members and donors."
"The increased competition for funds will require increased performance reporting requirements documenting the effectiveness of programs," agreed AccuFund's Stam. "Requiring systems to better capture and report performance measurements will become a priority. And any nonprofit that relies heavily on spreadsheets for their multiple reporting requirements will mean that they are substituting expensive labor for appropriate accounting/reporting systems."
"Effective record-keeping and document management are critical to the increasing demands for compliance and transparency in NFP operations," said Intuit's Mendoza. "A solid accounting system simplifies this record-keeping, making both audits and compliance reports easier to manage without expending more resources."
"We see three major trends in the management of nonprofits," said Jeff Sobers of Blackbaud. "First, the trends you see in the private sector today, especially big data - data mining and the collection of data. Second is tighter controls over funding from private and corporate foundations. These are today the loosest in terms of requirements, but will become more restrictive and will require more reporting in the year ahead. ... And third is auditor rotation. Mandatory auditor rotation is perhaps three years away, but will become a standard requirement, as it is in other sectors."
"When you have to do more with less, you have to be more accountable to the people who give you money," he added. "We are seeing the permeation of KPI and other metrics tools that enable nonprofits to blend financial and non-financial data to provide a more complete picture of the organization and its operations. Non-accounting staffs are talking more with accountants to tell a better story to their constituents. And accountants are not only more involved in strategic planning, they are involved earlier."
"The less you have to customize software, the less complicated support and implementation are going to be," said Serenic's Keith. "Organizations can simplify fund accounting solutions adoption by carefully considering which internal accounting processes they might tweak to allow out-of-the box solutions to work for them. Solutions can be made-to-fit. With an open mind, new software can be the impetus for evaluating current processes and tweaking practices in ways that are beneficial to the organization's mission."
In the year ahead, the challenges faced by nonprofits -- declining revenues, demands for greater accountability, and the need for better and more integrated management tools -- are not going to go away. Responding to these needs, vendors to the sector are taking a variety of paths to help their accounting clients weather the storm. "Collaboration is a key to the coming year," said Serenic's Keith. "By offering accounting staff, program directors, and even volunteers easy access to the data they need -- via the Web or through organizational portals, such as Microsoft SharePoint -- nonprofits can more easily implement proactive monitoring of expenses against budgets, reduce overhead costs, maintain strong procurement controls, and increase staff in the field, rather than in the accounting office."
"There will be fewer smaller software vendors in the marketplace in the next three years," said Araize's Scarano. "There has been a consolidation in the market with mergers and acquisitions and other vendors going out of business."
"Accountants must enable access to data and be able to report on it," said Lundquist of Open Systems. "The general ledger is the center of any nonprofit, just as it is in any business. It all comes down to the financials. If you can't trust them, there is no accountability, and the nonprofit ultimately suffers from a loss of leadership and credibility."
"Accountants need to understand that their decisions when they are doing their day-to-day jobs and reports, right or wrong, must be defensible," said Cassady of GMS. "Recording transactions in the way it has always been done is no longer acceptable. You have to be able to answer the appropriate questions, and defend the answer. It is better to make a wrong decision and be able to explain why that decision was made, than to make a right decision and not know why. You have to able to, as we used to say in grade school, show your work."
"Accountants are important to the nonprofits they serve," said Sobers. "It is critical that those accountants take a leadership role to create a fiscally responsible nonprofit -- and prove it. I have a phrase that every accountant should take to heart -- 'Stewardship is doing your job well; accountability is being able to prove it.'"
NONPROFIT PLAYERS
AccuFund Accounting Suite and AccuFund Anywhere
www.accufund.com/npo
Aplos Accounting
Aplos Software
www.aplossoftware.com
FastFund Online
Araize
www.araize.com
The Financial Edge V. 7.86
Blackbaud Inc.
www.blackbaud.com/fund-accounting
GMS Nonprofit Accounting and Financial Management Systems
Grant Management Systems
www.gmsactg.com
http://www.intuit.com/
(800) 4INTUIT
Traverse NFP
Open Systems Inc.
www.osas.com
Serenic Navigator 2013
Serenic Software
www.serenic.com


text:
Rutherford Confirmed To Head Warner Music Japan
Warner Music International today (March 5) confirmed that Warner Music Asia Pacific president Lachie Rutherford has added duties as chairman/CEO of its affiliate in Japan.
Hong Kong-based Rutherford replaces Hiroshi Inagaki, who retired as Warner Music Japan chairman/CEO on March 1. All WMJ department heads will now report directly to Rutherford, who will run the company on a day-to-day basis with WMJ president Takashi Yoshida.
"I am confident that under Lachie Rutherford's great leadership, WMJ will rise to the challenges and considerable opportunities in the Japanese marketplace," says WMI chairman/CEO Paul-Rene Albertini, to whom Rutherford continues to report.