Why are Fibonacci numbers significant in computer science?

Question

Fibonacci numbers have become a popular introduction to recursion for Computer Science students and there\'s a strong argument that they persist within nature. For these reasons

Answer 1

Fibonacci sequences are indeed found everywhere in nature/life. They're useful at modeling growth of animal populations, plant cell growth, snowflake shape, plant shape, cryptography, and of course computer science. I've heard it being referred to as the DNA pattern of nature.

Fibonacci heap's have already been mentioned; the number of children of each node in the heap is at most log(n). Also the subtree starting a node with m children is at least (m+2)th fibonacci number.

Torrent like protocols which use a system of nodes and supernodes use a fibonacci to decide when a new super node is needed and how many subnodes it will manage. They do node management based on the fibonacci spiral (golden ratio). See the photo below how nodes are split/merged (partitioned from one large square into smaller ones and vice versa). See photo: http://smartpei.typepad.com/.a/6a00d83451db7969e20115704556bd970b-pi

Some occurences in nature

http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/sneezewort.GIF

http://img.blogster.com/view/anacoana/post-uploads/finger.gif

http://jwilson.coe.uga.edu/EMAT6680/Simmons/6690Pictures/pinecone3yellow.gif

http://2.bp.blogspot.com/-X5II-IhjXuU/TVbHrpmRnLI/AAAAAAAAABU/nv73Y9Ylkkw/s320/amazing_fun_featured_2561778790105101600S600x600Q85_200907231856306879.jpg

Answer 2

Their computation as a power of [[0,1],[1,1]] matrix can be considered as the most primitive problem of Operational Research (sort of like Prisoner's Dilemma is the most primitive problem of Game Theory).

Answer 3

Symbols with frequencies that are successive fibonacci numbers create maximum depth huffman trees, which trees correspond to source symbols being encoded with maximum length binary codes. Non-fibonacci source symbol frequencies create more balanced trees, with shorter codes. The code length has direct implications in the description complexity of the finite state machine that is responsible for decoding a given huffman code.

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Conjecture: The 1st(fib) image will be compressed to 38bits, while the 2nd(uniform) with 50bits. It seems that the closer your source symbol frequencies are to fibonacci numbers the shorter the final binary sequence, the better the compression, maybe optimal in the huffman model.

Further Reading:

Buro, M. (1993). On the maximum length of Huffman codes. Information Processing Letters, 45(5), 219-223. doi:10.1016/0020-0190(93)90207-p

Answer 4

If you have an algorithm that can be successfully explained in a simple and concise mannor with understandable examples in CS and nature, what better teaching tool could someone come up with?

Answer 5

Greatest Common Divisor is another magic; see this for too many magics. But Fibonacci numbers are easy to calculate; also it has a specific name. For example, natural numbers 1,2,3,4,5 have too many logic; all primes are within them; sum of 1..n is computable, each one can produce with other ones, ... but no one take care about them :)

One important thing I forgot about it is Golden Ratio, which has very important impact in real life (for example you like wide monitors :)

Answer 6

Let me add another data structure to yours: Fibonacci trees. They are interesting because the calculation of the next position in the tree can be done by mere addition of the previous nodes:

http://xw2k.nist.gov/dads/html/fibonacciTree.html

It ties well in with the discussion by templatetypedef on AVL-trees (an AVL tree can at worst have fibonacci structure). I've also seen buffers extended in fibonacci-steps rather than powers of two in some cases.