# What's up with Fibonacci numbers? **URL:** \ **Category:** Cecil's Columns/Staff Reports\ **Created:** [June 22, 2004, 11:25am UTC](https://boards.straightdope.com/t/whats-up-with-fibonacci-numbers/251606 "2004-06-22T11:25:19Z")\ **Posts on this page:** 18\ **Page:** 1
**Author:** ![jester21](https://avatars.discourse-cdn.com/v4/letter/j/ed655f/32.png) [@jester21](https://boards.straightdope.com/u/jester21)\ **Post date:** [June 22, 2004, 11:25am UTC](https://boards.straightdope.com/t/whats-up-with-fibonacci-numbers/251606/1 "2004-06-22T11:25:19Z")
What the hell? I only understood the part about the bunnies… 😕 * * * Link to Staff Report: [What’s up with Fibonacci numbers?](http://www.straightdope.com/mailbag/mfibonacci.html) – CKDH ---
**Author:** ![li\_l\_Dickie\_Dirtz](https://avatars.discourse-cdn.com/v4/letter/l/2bfe46/32.png) [@li\_l\_Dickie\_Dirtz](https://boards.straightdope.com/u/li_l_Dickie_Dirtz)\ **Post date:** [June 22, 2004, 12:29pm UTC](https://boards.straightdope.com/t/whats-up-with-fibonacci-numbers/251606/2 "2004-06-22T12:29:24Z")
There’s a typo in the Staff Report, I believe. > [@](#): > > Whether the exact Fibonacci sequence is part of that structure **of** just a consequence of the simplicity of this model example is pretty open, but most mathematicians would chalk it up to chance. That second **of** should be an **or**. ---
**Author:** ![C\_K\_Dexter\_Haven](https://avatars.discourse-cdn.com/v4/letter/c/b2d939/32.png) [@C\_K\_Dexter\_Haven](https://boards.straightdope.com/u/C_K_Dexter_Haven)\ **Post date:** [June 22, 2004, 12:54pm UTC](https://boards.straightdope.com/t/whats-up-with-fibonacci-numbers/251606/3 "2004-06-22T12:54:06Z")
Thanks for spotting that, **li’l Dickie** , we’ll fix it. **jester21** , when you start a thread about a Staff Report, it’s helpful to other readers if you provide a link so they’ll know what you’re talking about. OK, it’s on the front page now, but in a week or so it will have vanished into the depths, and yet folks might still be reading this thread. No biggie, you’ll know for next time, and I have edited a link into your post. **Mathochist** , congrats on a terrific Staff Report! ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [June 22, 2004, 2:45pm UTC](https://boards.straightdope.com/t/whats-up-with-fibonacci-numbers/251606/4 "2004-06-22T14:45:45Z")
> [@C K Dexter Haven](#): > > **Mathochist** , congrats on a terrific Staff Report! I-a have used up all my English! Kösönöm! Kösönöm! ---
**Author:** ![Exapno\_Mapcase](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/exapno_mapcase/32/1051_2.png) [@Exapno\_Mapcase](https://boards.straightdope.com/u/Exapno_Mapcase)\ **Post date:** [June 22, 2004, 5:36pm UTC](https://boards.straightdope.com/t/whats-up-with-fibonacci-numbers/251606/5 "2004-06-22T17:36:27Z")
While you do mention that there are sequences plural, I wish you had gone on to mention specifically that you can start with any two numbers and start adding them in this way and come up with fascinating results. It makes the 1,1 sequence more of a special case and less magic to non-mathematicians. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [June 22, 2004, 6:10pm UTC](https://boards.straightdope.com/t/whats-up-with-fibonacci-numbers/251606/6 "2004-06-22T18:10:07Z")
> [@Exapno Mapcase](#): > > While you do mention that there are sequences plural, I wish you had gone on to mention specifically that you can start with any two numbers and start adding them in this way and come up with fascinating results. It makes the 1,1 sequence more of a special case and less magic to non-mathematicians. In particular, the fact that f[sub]n+1[/sub]/f[sub]n[/sub] converges to [symbol]f[/symbol] does not depend on the starting values. ---
**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\ **Post date:** [June 22, 2004, 6:28pm UTC](https://boards.straightdope.com/t/whats-up-with-fibonacci-numbers/251606/7 "2004-06-22T18:28:30Z")
It may be worth noting that there is a magazine about the Fibonacci numbers ( **The Fibonacci Quarterly** ) that has been published since, I think, 1963! ---
**Author:** ![Siddhartha\_Vicious](https://avatars.discourse-cdn.com/v4/letter/s/919ad9/32.png) [@Siddhartha\_Vicious](https://boards.straightdope.com/u/Siddhartha_Vicious)\ **Post date:** [June 22, 2004, 9:58pm UTC](https://boards.straightdope.com/t/whats-up-with-fibonacci-numbers/251606/8 "2004-06-22T21:58:54Z")
> [@Thudlow Boink](#): > > It may be worth noting that there is a magazine about the Fibonacci numbers ( **The Fibonacci Quarterly** ) that has been published since, I think, 1963! What can there be to say about Fibonacci numbers that could fill up a magazine four times a year? Honestly. I’m not being a smart ass. By the way, I love your username, **Mathochist**. ---
**Author:** ![wolf\_meister](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/wolf_meister/32/15202_2.png) [@wolf\_meister](https://boards.straightdope.com/u/wolf_meister)\ **Post date:** [June 22, 2004, 11:25pm UTC](https://boards.straightdope.com/t/whats-up-with-fibonacci-numbers/251606/9 "2004-06-22T23:25:29Z")
**Siddharta** Okay, here’s something that was probably in a Fibonacci Quarterly at one time or another. It’s not a well-known math trick so you can impress some folks with this. Start with any 2 numbers (they do NOT have to be Fibonacci numbers). Now add these up as you would a Fibonacci Sequence. Let’s start with 7 and 9 and carry these out to a tenth term. 7 + 9 + 16 + 25 + 41 + 66 + 107 + 173 + 280 + 453 When you (or someone else) has written these out, you can surprise your “audience” by announcing the total of all 10 numbers (which in this case is 1,177). And what is the trick? Whenever you start with 2 numbers and add them as such out to a 10th term, the total will always be the 7th term times 11. (It is easy to multiply by 11 in your head.) ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [June 23, 2004, 1:26am UTC](https://boards.straightdope.com/t/whats-up-with-fibonacci-numbers/251606/10 "2004-06-23T01:26:52Z")
> [@Thudlow Boink](#): > > It may be worth noting that there is a magazine about the Fibonacci numbers ( **The Fibonacci Quarterly** ) that has been published since, I think, 1963! [Here’s](http://www.mscs.dal.ca/Fibonacci/) the homepage for the Fibonacci Association, with links to the Quarterly, indices, etc. There really is a lot of fascinating stuff about the general idea behind such sequences. While it’s not a thick or polished journal (the last time I read it regularly), it’s a fun read sometimes. It is most notable for the problems and solutions sections, which too many Scientific journals avoid. Like I’ve said before, the Fibonacci numbers have many useful purposes in Computer Science alone. ---
**Author:** ![whirlwind](https://avatars.discourse-cdn.com/v4/letter/w/5f8ce5/32.png) [@whirlwind](https://boards.straightdope.com/u/whirlwind)\ **Post date:** [June 23, 2004, 1:43am UTC](https://boards.straightdope.com/t/whats-up-with-fibonacci-numbers/251606/11 "2004-06-23T01:43:41Z")
> [@Article Text](#): > > since they’re a good way to pack a lot of volume in a small space After hearing this phrase on a Volkswagon beetle commercial, by some British dude, it’s been annoying me ever since. Spirals, and more specifically spheres, do not allow maximum volume in a small space. Volume and space are the same thing. If I had one liter of volume, then I’d also possess 1 liter of space. What people mean to say is that spheres provide “maximum volume with the least amount of surface area” – or in the case of spirals, maximum volume with the most protection and least surface area (the surface area of the spiral cone surrounds vital parts of organisms). ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [June 23, 2004, 2:34am UTC](https://boards.straightdope.com/t/whats-up-with-fibonacci-numbers/251606/12 "2004-06-23T02:34:08Z")
> [@Siddhartha Vicious](#): > > What can there be to say about Fibonacci numbers that could fill up a magazine four times a year? Well, they could talk about the Lucas numbers, or the Fibonacci polynomials, or the generalized Fibonacci numbers… There’s no shortage of things to say. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [June 23, 2004, 7:38am UTC](https://boards.straightdope.com/t/whats-up-with-fibonacci-numbers/251606/13 "2004-06-23T07:38:32Z")
> [@whirlwind](#): > > After hearing this phrase on a Volkswagon beetle commercial, by some British dude, it’s been annoying me ever since. > > Spirals, and more specifically spheres, do not allow maximum volume in a small space. Volume and space are the same thing. > > If I had one liter of volume, then I’d also possess 1 liter of space. > > What people mean to say is that spheres provide “maximum volume with the least amount of surface area” – or in the case of spirals, maximum volume with the most protection and least surface area (the surface area of the spiral cone surrounds vital parts of organisms). Okay, to be a little more specific: If you have some number of, say, sunflower seeds to present, a circular formation of some kind will be the most efficient way to do so. further, the easiest way from a developmental point of view to form a circle is to grow out in a spiral formation from the center. I think the usual use of the phrase (mine, in particular) is rather use with the term “space”. In this case, I mean diameter: the longest distance between two points in the set. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [June 23, 2004, 7:50am UTC](https://boards.straightdope.com/t/whats-up-with-fibonacci-numbers/251606/14 "2004-06-23T07:50:53Z")
> [@Siddhartha Vicious](#): > > What can there be to say about Fibonacci numbers that could fill up a magazine four times a year? > > Honestly. I’m not being a smart ass. > > By the way, I love your username, **Mathochist**. As elsewhere noted, the Fibonacci sequence has a particularly rich structure based on a recurrence relation (which I didn’t bother going into in the article). Pick two different starting numbers and get a new sequence. I wasn’t aware that this journal existed (IANACombinatoricist), but I’ll bet they also get into other recurrence relations, which are very useful for, say, generating pseudo-random bit strings in computers. There’s also the notion of a “generating function”, which I thought would overwhelm the average reader. Define a power series (remember Calc 2) F(x) = Sum[sub]k=0[/sub][sup]infinity[/sup] F[sub]k[/sub]/k! x[sup]k[/sup] Then F[sub]k[/sub] = Fsup/sup. _This_ function is now holomorphic in some region of the complex plane, and thus is subject to analytic techniques. For instance, it’s uniquely defined by the differential equation with boundary-value conditions F’’(x) - F’(x) - F(x) = 0 F(0) = F[sub]0[/sub] F’(0) = F[sub]1[/sub] And so there are a lot of properties that simply follow from the theory of complex-valued linear ODEs. ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [June 24, 2004, 6:12am UTC](https://boards.straightdope.com/t/whats-up-with-fibonacci-numbers/251606/15 "2004-06-24T06:12:27Z")
> [@jester21](#): > > What the hell? I only understood the part about the bunnies… 😕 Evidently other people got caught up even on that. From an email I recieved just now: > [@CarlFink](#): > > I noticed one odd (not to say, implausible) thing in your essay on > the Fibonnacci sequence. You wrote: > > ``` > # Pairs of bunnies become sexually mature at 2 months. > # Each month, each sexually mature pair begets a new pair. > > ``` > > And later, you wrote, “Next month we’ll have all of these bunnies > (since they never die) plus one new pair for each pair that was born > at least a month ago (from all the pairs that will be mature by next > month).” > > The implication here is that a doe rabbit’s first pregnancy takes > zero time, or that she’s immediately impregnated at birth (or > something in between). > > I would have used the “Comment on this answer” link, but it requires > paying money. Carl, of _course_ it’s odd and improbable. I think the comment that “bunnies never die” is even odder, personally. Allow me a parable. Three scientists – a biologist, a chemist, and a theoretical physicist – are all given a grant to determine how to improve the milk yield of cows. At the conference held at the end of the grant period, each of them gives a lecture. The biologist stands up and says, “I have devised a system of exercises by which any cow can be made to improve its milk yield by 5%.” He proceeds to explain his technique, returns to the audience, and confers with his biologist friends. The chemist stands up and says, “I have devised a system of injections by which any cow can be made to improve its milk yield by 10%.” He also proceeds to explain his technique, returns to the audience, and confers with his chemist friends. The theoretical physicist stands up and says, “I have devised a method by which any cow can be made to _double_ its milk yield.” The audience, needless to say, is shocked and clamoring for more information on this miraculous discovery. She continues to explain: “Consider a spherical cow…” ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [July 5, 2004, 6:48pm UTC](https://boards.straightdope.com/t/whats-up-with-fibonacci-numbers/251606/16 "2004-07-05T18:48:37Z")
> [@Siddhartha Vicious](#): > > What can there be to say about Fibonacci numbers that could fill up a magazine four times a year? One cute thing is an explicit formula for each fibonacci number. Instead of defining them recursively, it is fairly easy to show that f(n) = phi^n / sqrt(5) - (1-phi)^n / sqrt(5) where phi is the golden ratio (1+sqrt(5))/2 of course > [@](#): > > Honestly. I’m not being a smart ass. Said Siddhartha Vicious. 🙂 It is your nature. Embrace it. ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [July 6, 2004, 2:00am UTC](https://boards.straightdope.com/t/whats-up-with-fibonacci-numbers/251606/17 "2004-07-06T02:00:43Z")
> [@Mathochist](#): > > I-a have used up all my English! Kösönöm! Kösönöm! Well, since we like being accuratearound here, it’s Kös **z** önöm! ---
**Author:** ![Mathochist](https://avatars.discourse-cdn.com/v4/letter/m/c89c15/32.png) [@Mathochist](https://boards.straightdope.com/u/Mathochist)\ **Post date:** [July 6, 2004, 2:09am UTC](https://boards.straightdope.com/t/whats-up-with-fibonacci-numbers/251606/18 "2004-07-06T02:09:50Z")
> [@pulykamell](#): > > Well, since we like being accuratearound here, it’s > > Kös **z** önöm! Feh. Me dum American