Stream

Join this community to post or comment
Pinned by moderator

Richard Green sharing

​​​Physical  - 
 
The Fibonacci fractal is a fractal based on the Fibonacci word. The Fibonacci word is an infinite string of zeros and ones, and it has properties similar to those of the Fibonacci sequence. Find out more below.
 
Fractals, Fibonacci, and factorizations

The rule for generating the famous Fibonacci sequence 1, 1, 2, 3, 5, 8, 13, 21, ... is that each number (after the first two) is the sum of the previous two numbers. The Fibonacci word is an infinite string of zeros and ones with properties reminiscent of the Fibonacci sequence, and the Fibonacci fractal, shown in the picture, is a way to represent the Fibonacci word in the form of a fractal.

One way to generate the Fibonacci word is to define strings of zeros and ones by the rules S(0)=0, S(1)=01 and S(n)=S(n–1)S(n–2) when n is at least 2. This gives rise to the sequence of strings 0, 01, 010, 01001, 01001010, 0100101001001, ..., whose limit, as n tends to infinity, is the Fibonacci word. There are other equivalent, but superficially very different, ways to generate this word, including (a) using an explicit formula for each digit given in terms of the golden ratio; (b) using a substitution rule; and (c) using the Zeckendorf representation of integers in terms of Fibonacci numbers.

By suitably interpreting the digits of the Fibonacci word as turtle graphics instructions in a Logo-like programming language, it is possible to represent the word as a fractal. More precisely, if one reads the digits in order, then the n-th digit corresponds to the following sequence of instructions:
1. draw a segment forwards;
2. if the digit is 0, then turn left 90 degrees is n is even, and turn right 90 degrees if n is odd.

The picture shows the result of this procedure after many iterations. The resulting curve has various interesting mathematical properties, some of which concern the square-shaped gaps. By inspection, we count one large square gap (in the middle, at the bottom); five smaller square gaps, and 21 square gaps of the next size down. The numbers of these gaps, sorted by size, turn out to be given by every third Fibonacci number starting with the second 1 (1, 5, 21, 89...) which means that there are 89 squares of the next size down. Furthermore, each square has a side length that is 1+√2 times the side length of the square of the next size down; the number 1+√2 is known as the silver ratio.

The recent paper Factorizations of the Fibonacci Infinite Word by Gabriele Fici (http://arxiv.org/abs/1508.06754) surveys some factorizations of the Fibonacci word and shows how to derive these factorizations using elementary properties of the Fibonacci numbers. In some cases, this gives easier derivations of the results than were previously known. An example of such a factorization involves the sequences S(n) from earlier. Proposition 1 of the paper proves that the Fibonacci word can be factorized as the infinite product 0.1.S(0).S(1).S(2)..., where the symbol . is used to separate the factors.

One of the most surprising factorizations in the paper is Proposition 9, which involves the reversals, T(n), of the strings S(n). The strings T(0), T(1) and so on are then given by the sequence 0, 10, 010, 10010, 01010010, ... Remarkably, the concatenation of the strings T(n) also gives the Fibonacci word, even though the ingredients being used to construct it are backwards and generally not palindromic. Another way to say this is that the Fibonacci word can be factorized as the infinite product T(0).T(1).T(2)...

Relevant links

The 2009 paper The Fibonacci Word fractal by Alexis Monnerot-Dumaine is an excellent guide to the mathematical properties of the fractal, and the picture of the fractal here comes from that paper. You can download the paper for free at https://hal.archives-ouvertes.fr/hal-00367972/document 
Monnerot-Dumaine's paper explains how to construct the Fibonacci word using a substitution rule, and explores what the fractal looks like if one makes turns at angles other than a right angle. 

Fici's paper explains how to construct the word using the Zeckendorf representation of natural numbers. It is a theorem that any positive integer can be expressed uniquely as the sum of one or more distinct non-consecutive Fibonacci numbers. This is called Zeckendorf's Theorem, even though Zeckendorf was not the first to prove it: https://en.wikipedia.org/wiki/Zeckendorf's_theorem

Wikipedia's article on the Fibonacci word gives an explicit formula for the n-th digit of the word and mentions many other interesting properties. For example, the Fibonacci word is often cited as the worst case for algorithms detecting repetitions in a string. https://en.wikipedia.org/wiki/Fibonacci_word

The On-Line Encyclopedia of Integer Sequences on the Fibonacci word: https://oeis.org/A003849

Wikipedia on turtle graphics: https://en.wikipedia.org/wiki/Turtle_graphics

I have posted about the Fibonacci word twice before, although not recently. 
My post from March 2013 discusses the word in the context of self-shuffling words: https://plus.google.com/101584889282878921052/posts/YnUkZ986LMM
My post from December 2012 discusses Fibonacci snowflakes and some generalizations of the Fibonacci word: https://plus.google.com/101584889282878921052/posts/KSuUFJV6tyv

If you're disappointed that I didn't talk about the golden ratio, have a look at the aspect ratio of the accompanying picture.

#mathematics #sciencesunday #spnetwork arXiv:1508.06754
19
4
Shruti Pandey's profile photovictor joel urena pena's profile photo
Add a comment...

HEXAPOLIS

Science News (Pop Sci)  - 
 
A team of researchers (headed by Guihua Yu, an assistant professor) at the University of Texas, have created their version of a self-healing electrical circuit. Demonstrating its intrinsic flexibility, the circuit can not only repair itself, but also restore its conductivity – even when broken into two pieces. Such special attributes are showcased by virtue of the pliant gel material of the circuit – which combines a host of seemingly contrasting properties, including high conductivity and a healing factor that is activated even in room temperatures.

source: http://phys.org/news/2015-08-electrical-circuit-gel.html

abstract: http://pubs.acs.org/doi/abs/10.1021/acs.nanolett.5b03069
A team of researchers at the University of Texas have created a self-healing electrical circuit - made of a conductive hybrid gel.
6
Mike DeSimone's profile photo
 
Can't say I see unintentional circuit board conductor cuts much at all, but I wonder if this could be adapted for component attachment, since cracked solder joints are a big problem, especially if the board is exposed to flexing.

Another place this could be quite useful is wearable, flexible electronics.
Add a comment...

Donna Daugherty

​​​​​​​​​Earth  - 
 
Researchers have found the building blocks of life deep below the seafloor, adding evidence to the theory that life is able to spring up wherever there is water, and of course the right chemistry. The study, published in the Proceedings of the National Academy of Sciences, could have implications for the chances of life existing on other worlds, such as Jupiter’s moon Europa.
5
1
Gary Maccagnone's profile photoWilliham Totland's profile photoMark Johnson's profile photo
4 comments
 
+Gary Maccagnone What do you mean 'line up'?
Add a comment...

Phil Rounds

Applied  - 
 

We've heard that we're running out of oil before...in the 1970s. But we are running out of oil. And climate change is at least partially caused by human activity. We can't pump greenhouse gases into the atmosphere and expect there not to be some consequence sooner or later. Also, our fossil resources are valuable for purposes other than burning.

http://www.techinsider.io/elon-musk-talks-fossil-fuels-with-wait-but-why-2015-8
23
8
Daniel Kirbach's profile photoChris Borre's profile photoNitin Joshi's profile photoValerie Ficca's profile photo
2 comments
 
True
Add a comment...

Simon Green

Science News (Pop Sci)  - 
 
 
An interesting study: " Results support the hypothesis that high socially anxious individuals may demonstrate a unique social-cognitive abilities profile with elevated cognitive empathy tendencies and high accuracy in affective mental state attributions."
Source: www.spiritscienceandmetaphysics.com | Original Post Date: April 25, 2015 - A few years ago, a series of studies came out in an attempt to sort of ‘debunk’ people who practice spirituality.  The study found that people who have a spiritual understanding of life tend to be more suscepti
19
9
Lisa Bonanno (Lunarvapor)'s profile photoAlex Lopez's profile photoSamantha Pearl's profile photoSabine Hiraldo López's profile photo
3 comments
 
It fits the profile?
Add a comment...

Pythagoras Dupree™

Science Bytes (Memes, Cartoons, Images)  - 
 
According the data elicited from the particle collisions at the Large Hadron Collider, it is shown that this higgs boson is an extremely unstable particle that quickly decays through disparate processes, leaving behind delicate fingerprints for the physicists to analyze. The piece of information below is to show four important decay modes that experimenters used to ascertain the higgs existence.
56
16
Cecile Bruxelles's profile photoRoss Hall's profile photoIvan Covarrubias's profile photoChanmilak Watson's profile photo
4 comments
 
Mind blown
Add a comment...

Irene Lobato

Science News (Pop Sci)  - 
 
Praying mantids (more commonly known as mantises) have been beloved and feared by different cultures throughout history. Their ability to camouflage both in color and shape with the environment make them beautiful and terrifying insects at the same time... for other insects. If you want to know more about the biology and diversity of these insects, we invite you to read this article on All you need is Biology blog.

If you enjoyed it, feel free to share it! :)

http://wp.me/p3GsGc-1jc
Praying mantids (more commonly known as mantises) have been beloved and feared by different cultures throughout history. They are agile, strong and specially inconspicuous insects: their great abil...
13
Bunyip Bonsai's profile photo
Add a comment...

Es Einsteinium

Science Bytes (Memes, Cartoons, Images)  - 
 
Science in the Summer 2015

A round up of 151 interesting science related news. Whether that be discoveries, creations, realisations, or just an update of a current situation. 
An interesting watch just to find out how much you actually knew about and whether maybe, you could build upon anything that has been done, might give you a great business idea. 
Knowledge is power. 

If you want the full list, click the link
http://bit.ly/1IDVCco

Video
https://www.youtube.com/watch?v=GvyasI5vgbg

Channel
https://www.youtube.com/EsEinsteinium03

References
http://www.iflscience.com/
http://www.sciencealert.com/
https://www.facebook.com/NotAnExactScienceShow?fref=ts
11
3
Sivakumar Perumal's profile photoMary O'Meara's profile photo
Add a comment...

Understanding Animal Research

Science News (Pop Sci)  - 
 
Leber congenital amaurosis-1 (LCA1) is a disorder that causes severe visual impairment, due to the lack of protein retGC1. Mice lacking this protein have undergone gene therapy to replace it and their vision has been restored successfully. Clinical testing using this approach in LCA1 patients is the next step.

"This study shows the tremendous potential of recombinant (rAAV) gene therapy for the effective treatment of genetic causes of vision loss," says Editor-in-Chief Terence R. Flotte, MD, Celia and Isaac Haidak Professor of Medical Education and Dean, Provost, and Executive Deputy Chancellor, University of Massachusetts Medical School, Worcester, MA.

Original paper: http://online.liebertpub.com/doi/full/10.1089/hum.2015.053
16
2
Daniel Junior's profile photoMukesh DuPont's profile photoAlena Naumova's profile photo
 
Awesome.
Add a comment...

Kristin Milton

Science Outreach  - 
 
Hi everyone - I'm looking for an expert on frogs.
Are you one?
Do you know any plussers who are?

Thanks!
13
1
Nishit Dave's profile photo
Add a comment...
9 comments

Pythagoras Dupree™

Science Bytes (Memes, Cartoons, Images)  - 
 
The infographic below helps elucidate the phenomenon of gravitational lensing. Gravitational lensing happens when light, emitted from a distant source such as a galaxy, is bent when it passes through a myriad of galaxies clustered together as well as dark matter. Since dark matter interacts with baryonic matter (which constitutes 4% of the universe) through gravity. It is the dark matters gravity that bends light and acts like a lens. When the light reaches earth, we see distorted multiple images of the galaxy. I find this fascinating because it is one of the main pieces of evidences endorsing the existence of this dark matter. Based on this knowledge, physicists have launched experiments to try and discern its properties. 
98
32
Joachim Mroß's profile photoJ. Izabella Bardocz's profile photoEmre Duran's profile photodavid masiello's profile photo
7 comments
 
Yep, in certain chain reactions anyway. It sounds like "weird force" acting similar to gravity on light (photon mass?) is allowing us to perceive light in measurable ways, allowing further inference to be made over the vast distances of interstellar space.........weird force, lol, dark matter is being rumored (I haven't heard anything official) to exert a force similar to that of gravity, specific to the dark matter, not to be confused with the rumored anti-positron, dark energy, a product of dark matter(again rumor) not the same thing as the suspected gravity like force.....I find this to be speculation...thus far, yet a think tank worth of people have at least heard of my ideas for "Flotilla the Hun" lol
Add a comment...

About this community

Science on Google+ is a community moderated by scientists, for all people interested in science, both professionals and the general public. The primary goal of this community is to bring real scientists to the public, for science outreach. A secondary and long-term goal is to create an environment that fosters interdisciplinary collaborations; thus, enabling and promoting cloud collaboration between scientists. See Guidelines and Rules section for additional details.

Jiang Ting

Curator's Choice - Mods only  - 
 
,q
1
Annette Chaltain's profile photo
 
I hope this stays peaceful 
Add a comment...

AstroCamp

Science Bytes (Memes, Cartoons, Images)  - 
 
More fun with hydrophobic materials!
 
Water on hydrophobic surface
64
20
Valerie Ficca's profile photoBHARADWAJ Sc's profile photolyncrya's profile photoVarsha Vinnakota's profile photo
4 comments
 
+Dustin Thurston The hydrophobic coating is only around the (wide) perimeter. The green water is adhering to the central area which doesn't have a hydrophobic coating.
Add a comment...

Understanding Animal Research

Science News (Pop Sci)  - 
 
The toxin in the sting of a Brazilian wasp, polybia paulista, appears to kill cancer cells without harming normal cells according to studies in mice. The MP1 toxin is used by the wasp to paralyze its prey, but studies say it could be used to target and destroy cancer cells by targeting the fat molecules found on the outside of cancerous cells.
 
Dr Aine McCarthy, Cancer Research UK, said:
"This early stage research increases our understanding of how the venom of the Brazilian wasp can kill cancer cells in the laboratory. But while these findings are exciting, much more work is needed in the lab and in clinical trials before we will know if drugs based on this research could benefit cancer patients."
 
http://www.bbc.co.uk/news/health-34115112
Original Paper: http://www.cell.com/biophysj/abstract/S0006-3495%2815%2900768-7
The venom of a wasp native to Brazil could be used as a weapon to fight cancer, scientists believe.
25
7
Shawn Saint's profile photoVishnuya Shantharuban's profile photo
Add a comment...

Bryce P-Stevenson

Science Bytes (Memes, Cartoons, Images)  - 
 
Comparisons.
180
39
Peter Monien's profile photoUzair Qaiser's profile photoJoe Biles's profile photoRyan Smith's profile photo
22 comments
 
the caption should say "you" are a speck on the pebble that is here.
Add a comment...
 
What Does Neuroscience Know About Meditation? "The practice has been linked with improved focus and emotional control and there is even indication that it changes a person’s biology." Today in +Smithsonian Magazine Magazine #meditation 
There are still many unknowns, but the practice seems to improves attention and memory
8
3
Elona Porter's profile photoAdelino Andrade's profile photo
Add a comment...

Lemuel Adlawan

Science News (Pop Sci)  - 
 
The Inner Workings Of The Turbocharger (invented 100 years ago) explained in detail in this video animation from YouTube user Thomas Schwenke.

source: http://www.motorauthority.com/news/1099732_the-inner-workings-of-the-turbocharger-explained-video?km
30
10
Rodrigo Casaroto's profile photoEric Harker's profile photo
Add a comment...

Joan Schmelz

Science News (Pop Sci)  - 
 
When Misogyny is a Symptom of Narcissism
Dealing with and responding to other people’s misogynistic and racist behavior is never simple Anonymous tells us that this behavior could be a sign of narcissism. Read more at Women in Astronomy: http://womeninastronomy.blogspot.com/2015/08/when-misogyny-is-symptom-of-narcissism.html
17
4
Romavic Antony's profile photoSue H's profile photosarwar rose's profile photoEamonn Lawler's profile photo
5 comments
 
+Yu-Chi Huang we don't support any hate groups here. Also, claims without evidence. As a woman scientist, I'm certainly a feminist and not a member of a hate group. Speaking out for equality, whether gender, race or class, should not be considered "radical". That's a label to discredit and put down people who speak up for equality. 
Add a comment...

Anders Lorenzen

​​​​​​​​​Earth  - 
 
NOAA declare that July this year was the hottest month ever since records began:
15
1
Felix Albrecht's profile photoPat Burdett's profile photoIvonne Gonzalez's profile photo
3 comments
 
yeep.
Add a comment...
 
Environmental Concerns: Deforestation.

A widespread concern is the Earth’s environment and climate change. Science is making many positive strides to combat this threat with expanding knowledge, understanding and technology. However, there are problems (gaps) that must be faced by other means. The rapid loss of trees (deforestation) is one of these concerns.
 Forests are complex ecosystems and when degraded can exacerbate a devastating chain of events. Although, deforestation has been practiced throughout history, it has increased at an alarming rate within the last fifty years. 
As a result, would making laws to stop the destruction be the answer? It can help to control illegal logging, but the problem is more widespread and diverse. Much is associated with the growing pressure of population growth and the economics of many developing countries.
Many our dependent on their agrarian society. This means that many farmers slash and burn forests to create more usable land. This creates  more intensive agriculture which in turn results in declining levels of soil fertility and crop yield and as a result more arable land is needed. A similar example in American history can be the dust bowl debacle. Although, the great drought hastened the inevitable it actually began years earlier.
So what is the answer? There is no clear formula that works for all but, agroforestry is proving to be very good and has much potential. It has been tested in North America with positive results.  http://www.centerforagroforestry.org/pubs/training/chap1.pdf  and  http://www.usda.gov/wps/portal/usda/usdahome?contentidonly=true&contentid=agroforestry-faqs.html

Now the problem becomes social, trying to overcome mindsets, bias and doubt. How do we convince countries such as Brazil and low income farmers that this is a viable and positive alternative? This is what the research in the Environmental Psychology Journal deals with. Although, the research deals with Malawi, it can reflect the problems associated with other countries. 
13
1
J A Jones-Ford's profile photo
Add a comment...