Thank you ! :-)))
Wake up call....good morning from very hot Bulgaria :-D
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)...
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
.....and live a courageous life.... ♥
People with Alzheimer’s disease have fat deposits in the brain. For the first time since the disease was described 109 years ago, researchers have discovered accumulations of fat droplets in the brain of patients who died from the disease and have identified the nature of the fat.
Image credit finecooking.com
Is among the 20 most beautiful lakes in the world to 17th place. The park covers an area of 33,000 hectares and includes 16 lakes in succession, connected by waterfalls.Plitvice is the oldest national park in Southeast Europa.All'interno the park there are also many caves of which only a small part is agibile.I lakes are formed by two rivers: the White River and the Black River, which flow in the river Korana. The waters of these rivers are rich in calcareous salts (mostly calcium carbonate and magnesium carbonate), from the dissolution of carbonate rocks forming the geological structure of sito.Questi salts are precipitated by vegetation, forming layers of travertine , a sedimentary rock recently. Over time, these deposits forming real natural dams that act as barriers to water, growing by about a centimeter per year. At one point the water pressure breaks these natural levees, opening new paths in the ground.
This mechanism, in fact common to all the calcareous water, in Plitvice has assumed a particular importance.
The beauty of the National Park Plitvice, Croatia, is increased in the second round of the New Seven Wonders of Nature. ~ DK
#waterfall #waterfallphotography #nature #naturephotography #amazing #awesomeness #colorful #landscape #beautiful #view #photooftheday #summer #whatshot #whatshotnow #whatshotongoogleplus
- Rhysician, present
- University of Sofia - MedicineMedicine
Уважаеми съграждани, честит празник на светите братя Кирил и Методий, - ...
Уважаеми съграждани, честит празник на светите братя Кирил и Методий, - вестник Борба, областен всекидневник за новините от региона Велико Т
Discovered a disease? WHO has new rules for avoiding offensive names
To avoid stigma, names shouldn’t mention people, places, jobs, food, or animals
Великден - традиции и обичаи | LifeStyle Framar.bg
Великден е най-святият празник за християните по целия свят. На този ден Християнската църква отбелязва Възкресението на Божия син Исус Хрис
Double Rainbow Image, Vancouver - National Geographic Photo of the Day
A double rainbow stretches over Horseshoe Bay in Vancouver, Canada, in this National Geographic Photo of the Day from our Your Shot communit
«Ciao Virna!», il mondo dello spettacolo piange la regina del cinema
Il mondo dello spettacolo piange la scomparsa di Virna Lisi, l'attrice italiana amata in tutto il mondo morta a 78 anni per un tumore scoper