Message boards : Science (non-SETI) : Fibonacci numbers

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Paul D Harris Volunteer tester Send message Joined: 1 Dec 99 Posts: 1122 Credit: 33,600,005 RAC: 0 |
Is there an app that will run on my computer that will calculate Fibonacci numbers similar to a Pi program? |

tullio Volunteer moderator Volunteer tester Send message Joined: 9 Apr 04 Posts: 5989 Credit: 1,382,182 RAC: 3,762 |
I don't know but Enrico Bombieri of the Princeton Institute for Advanced Study mentions a "Fibonacci quarterly" in an article on Fibonacci included in a book "Fibonacci tra arte e scienza" published by Cassa Di Risparmio di Pisa with Luigi A.Radicati di Brozolo (ed). They should know. Tullio |

Lint trap Send message Joined: 30 May 03 Posts: 871 Credit: 28,092,319 RAC: 0 |
I think Fibonacci sequences are also involved in some Fractals, particularly the Mandelbrot Set. see http://www.sunflowerblog.ch/2007/06/03/the-fibonacci-numbers-and-mandelbrots-fractals/. Lt |

Convergence Send message Joined: 23 Jun 08 Posts: 117 Credit: 2,841,589 RAC: 0 |
Is there an app that will run on my computer that will calculate Fibonacci numbers similar to a Pi program? Sure. It's pretty easy to make even on your own. |

JLConawayII Send message Joined: 2 Apr 02 Posts: 188 Credit: 2,838,179 RAC: 0 |
I think Fibonacci sequences are also involved in some Fractals, particularly the Mandelbrot Set. Yeah they pop up all over in mathematics. Phi (the golden ratio) comes immediately to mind. |

Chris S Volunteer tester Send message Joined: 19 Nov 00 Posts: 39121 Credit: 26,521,531 RAC: 52,680 |
Another number, a ratio kept popping into my mind as I wrote the previous post that featured the golden ratio of 1.618. The number was 1.4. It turns out that there is another “golden” ration based on 1.414, which is the square root of 2, which also happens to be the length of the diagonal of a square with sides of length 1. |

William Rothamel Send message Joined: 25 Oct 06 Posts: 3271 Credit: 1,317,089 RAC: 45 |
There is a "Binet" formula and also a generating function where you round to get the answer. Do a Google and you can get to other generating sites as well. If you are interested in Fibonacci numbers, the golden ration and hidden structure in Mathematics, contact me via private message and I will send you a paper that I wrote a few years back on the subject. |

Michael Watson Send message Joined: 7 Feb 08 Posts: 866 Credit: 807,379 RAC: 1,524 |
If you average the ratios of adjacent Fibonacci numbers: 1,1,2,3,5,8,13,21... with Lucas number ratios: 2,1,3,4.7,11,18,29... you can approach Phi, to any desired level of accuracy, in fewer steps than using either alone. Michael |

Sarge Volunteer tester Send message Joined: 25 Aug 99 Posts: 11083 Credit: 3,649,116 RAC: 4,294 |
(1 + sqrt(5)) / 2 |

Sarge Volunteer tester Send message Joined: 25 Aug 99 Posts: 11083 Credit: 3,649,116 RAC: 4,294 |
a:=1; > b:=1; > > for k from 1 to 100 do > c:=a+b: > print(c); > a:=b: > b:=c: > od: a := 1 b := 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987 1597 2584 4181 6765 10946 17711 28657 46368 75025 121393 196418 317811 514229 832040 1346269 2178309 3524578 5702887 9227465 14930352 24157817 39088169 63245986 102334155 165580141 267914296 433494437 701408733 1134903170 1836311903 2971215073 4807526976 7778742049 12586269025 20365011074 32951280099 53316291173 86267571272 139583862445 225851433717 365435296162 591286729879 956722026041 1548008755920 2504730781961 4052739537881 6557470319842 10610209857723 17167680177565 27777890035288 44945570212853 72723460248141 117669030460994 190392490709135 308061521170129 498454011879264 806515533049393 1304969544928657 2111485077978050 3416454622906707 5527939700884757 8944394323791464 14472334024676221 23416728348467685 37889062373143906 61305790721611591 99194853094755497 160500643816367088 259695496911122585 420196140727489673 679891637638612258 1100087778366101931 1779979416004714189 2880067194370816120 4660046610375530309 7540113804746346429 12200160415121876738 19740274219868223167 31940434634990099905 51680708854858323072 83621143489848422977 135301852344706746049 218922995834555169026 354224848179261915075 573147844013817084101 927372692193078999176 > |

Sarge Volunteer tester Send message Joined: 25 Aug 99 Posts: 11083 Credit: 3,649,116 RAC: 4,294 |
a:=1; > > for k from 1 to 10 do > b:=a*(1+sqrt(5))/2: > print(evalf(b)); > a:=b: > od: a := 1 1.618033989 2.618033990 4.236067980 6.854101971 11.09016995 17.94427193 29.03444188 46.97871382 76.01315574 122.9918696 > |

William Rothamel Send message Joined: 25 Oct 06 Posts: 3271 Credit: 1,317,089 RAC: 45 |
If you average the ratios of adjacent Fibonacci numbers: 1,1,2,3,5,8,13,21... with === actually you can start with any numbers--say two arbitrary three digit numbers. Add them together and then add this sum to the 2nd number and then continue on by adding this sum to the previous one and so on. You will soon approach PHI at the tenth number if you take the ratio of the tenth to the ninth. This ratio will be at least 1.61 and possibly you could round it to 1.618. Try it for yourself--makes a good magic trick. |

Paul D Harris Volunteer tester Send message Joined: 1 Dec 99 Posts: 1122 Credit: 33,600,005 RAC: 0 |
Nice respondence to Fibonacci numbers thanks. I am not a math wize I took my math in the old days before computers and we use slide rules so what does a:=1 mean As a part of the symbol sometimes used to mean "A is defined as 1" or is a=1? And what does === mean? |

Chris S Volunteer tester Send message Joined: 19 Nov 00 Posts: 39121 Credit: 26,521,531 RAC: 52,680 |
the : means a variable in computer languages. |

jason_gee Volunteer developer Volunteer tester Send message Joined: 24 Nov 06 Posts: 7388 Credit: 88,632,154 RAC: 10,430 |
They're 'organic' ratios, rather than absolute numbers. just use the golden ratio ;) See any Fibonacci expansion contours or Golden Ratios in the speakers I made when I was 17? and here's some fibonacci sequences to play on them :D 'BT's Fibonacci Sequence' "Living by the wisdom of computer science doesn't sound so bad after all. And unlike most advice, it's backed up by proofs." -- Algorithms to live by: The computer science of human decisions. |

Sarge Volunteer tester Send message Joined: 25 Aug 99 Posts: 11083 Credit: 3,649,116 RAC: 4,294 |
Nice respondence to Fibonacci numbers thanks. On the software I was using, well known constants such as Pi cannot be redefined. If I type "evalf(Pi);", it returns 10 digits (or whatever I set it for) of 3.141592654... . If I want to define Sarge's constant as 2.176218, i could use SC:=2.176218. So, indeed, the colon before the equals sign indicates I am defining something the software package does not already know to be equal. |

William Rothamel Send message Joined: 25 Oct 06 Posts: 3271 Credit: 1,317,089 RAC: 45 |
Pi is an infinite series. The nth Fibonacci number has a closed form equation. What's interesting is that the equation has irrational numbers in it yet gives a whole number (rational) |

Paul D Harris Volunteer tester Send message Joined: 1 Dec 99 Posts: 1122 Credit: 33,600,005 RAC: 0 |
I was watching the Science channel on TV and the show was about alien contact and they were discussing seti and the aliens would communicate through math and it would probally be Fibonacci numbers. So I had a program that will run Pi numers to when ever I stop the program so I was wondering about Fibonacci numbers. So I thought I would make this posting and see what the seti crunchers would say. 1+1=2 2+1=3 3+2=5 5+3=8 etc. |

Sarge Volunteer tester Send message Joined: 25 Aug 99 Posts: 11083 Credit: 3,649,116 RAC: 4,294 |
Pi is an infinite series. The nth Fibonacci number has a closed form equation. What's interesting is that the equation has irrational numbers in it yet gives a whole number (rational) That is one definition of Pi. One could also define it as 4 * integral(sqrt(1 - x^2),0 .. 1,dx). So many ways to start. |

Nick: ID 666 Volunteer tester Send message Joined: 18 May 99 Posts: 10632 Credit: 30,862,042 RAC: 2,495 |
You also need to look at Lucas Numbers, 2,1,3,4,7,11,18,....... and Benford's Law which deals with the initial number. |

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