# Talk:Fibonacci Numbers

## Contents

### Individuals' statuses

Major strengths of the page

• Great use of pictures, and especially using the text to explain to readers what they should notice in the pictures
• Great connections to the world outside of math and to other branches of math
• In some of the more mathematical parts, great job summarizing results in quasi-plain English and reducing clutter by hiding equations. Also, great visuals in those parts.
• Great job with the equations. You have a *lot* of them, and you do a really good job not only getting the math right, but also organizing the math in a really sensible order.
Abram, 6/4 and 6/15

Typos and the like

• The Fibonacci spirals section now has two sentences that say Fibonacci spirals can be seen in sea shells, snails, and the galaxy.
Iris(7/6) fixed this
Looks good (Abram, 7/6)

• Remove the colon from the beginning of Equation 6
Iris(7/6) fixed this
Looks good (Abram, 7/6)

• There are a couple of places where words should be pluralized. I'll fix those in a final pass-through. I'm also going to very slightly rewrite the section on the origin of the problem.
I couldn't find the singular/plural mistakes, but they aren't significant anyway. See if you are ok with my slight rewrite of the origin of the problem (i cut out a couple of sentences). (Abram, 7/6)
Abram, 7/6

Who cares about the identities and the Mandelbrot set?

You do a great job explaining how the Fibonacci sequence shows up in nature. Could you be more clear about why the abstract mathematical properties of the sequence are interesting? For example, it would be great to know if the identities are just kinda cool, or if they are useful in certain branches of mathematics. Also, even though this page is not about the Mandelbrot set, it's worth one sentence explaining why people are interested in the Mandelbrot set. (Abram, 6/15)

IrisI added a sentence why people are drawn to Mandelbrot set. Professor Maurer and I decided that the part for identities are not necessary (6/23). I think you're right about the identities. Maybe spend 10 minutes seeing if you can find a reason why people care about the Mandelbrot set beyond "it's pretty". (Abram, 6/24)
Iris I added another part saying that it is surprising how a simple formula can generate a complex structure of the mandelbrot set. I hope this is sufficient (6/25)
It seems like there must be something actually useful studying it. Maybe it's a good way for mathematicians to get a perspective on how fractals work in general precisely because the formula is so simple. Unless you happen to know why it's actually useful, though, what you have works great. (Abram, 7/5)
Iris (7/6) I think I'll just leave it as it is. I can't really find exactly why studying mandelbrot set is useful
Sounds good. (Abram, 7/5)

Statements to clarify for the sake of accuracy

• On the sum of squares identities section, somewhere be explicit that each Fibonacci rectangle is composed of squares whose side lengths are every Fibonacci number F_1 through F_n, and that the dimensions of this rectangle are F_n through F_n+1. You have a good example, just not the general statement.

Iris (6/25) I added this part

Looks good. (Abram, 7/5)
• The mouse-over definition of Fibonacci rectangles uses the word "proportion" where "ratio" would probably be a bit better.

Iris (6/25) I fixed this

Hmm, as I think about it, it may not make sense to have a mouse-over here. Mouse-over definitions are a good way of avoiding cluttering a page with an in-text definition, but here, you have to define the Fibonacci sequence in the main text anyway. This information in the mouse-over bubble is important, but maybe move it to the main text. It could become the last sentence of the paragraph. (Abram, 7/5)
Iris(7/6) I fixed this by getting rid of the mouse-over
Looks good. (Abram, 7/6)
• The section on difference tables states that each difference sequence forms a Fibonacci sequence. Actually, each difference sequence *includes* the Fibonacci sequence (it's also true that each difference sequence as a whole follows the recursion rule of the Fibonacci sequence, just with different starting values, but this would take some explaining).

Iris(6/25) I fixed this & added the part in 'future suggestions' that someone might want to explain about the Fiboancci numbers with negative indices.

Cool. Maybe specifically refer to this section of the page in that suggestion. (Abram, 7/5)
Iris(7/6)I referred to it in the suggestion
Looks good. (Abram, 7/6)
• The statement, "The number of rotations and the number of leaves are all consecutive Fibonacci numbers" is a bit vague. Do you mean that the number of clockwise rotations, the number of counter-clockwise rotations, and the number of leaves always form a list of 3 consecutive Fib. numbers, or that some pair of them are always consecutive Fib. numbers, or ...?

Iris(6/25) I fixed this.

Looks good. (Abram, 7/5)
Abram, 6/24

Statements to add or change for the sake of explanation

• For the even and odd indices proofs, maybe point out that all the intermediate terms are disappearing because each one appears once positively and once negatively (my wording here is terrible, but you probably know what I mean). (Abram, 6/24)

Iris(6/25) I fixed this as well, but I'm not sure if my wordings are clear enough

I actually really like that you solved this by adding an intermediate equation. The wording isn't perfect, but it's good, and with the intermediate equation there also, it's very clear. Can you do the same thing for the first identity? (Abram, 7/5)
Looks good. (Abram, 7/6)
• It seems like Equation 1 is mean to show a *result* which is that the left side equals the right side. The "middle side" is part of the *derivation*. Maybe instead of having this middle step in equation 1, add a sentence to the end of the derivation pointing out that F_2 = 1, so that F_n+2 - F_2 = F_n+2 - 1. (Abram, 7/5)
Looks good. (Abram, 7/6)
• In the sum of the squares identity, maybe say, "In fact, any rectangle is composed of every square with side lengths F_1 through F_n, with the value of n depending on the rectangle." Also, when you say, "We can prove this identity by computing the area...", I thought for a moment you were going to prove the fact that you just stated, which is that if you make a Fib. rectangle out of F_1 through F_n squares, then the side lengths are F_n by F_n+1. Rewriting the sentence as, "With this information in mind, we can be prove the identity above by computing the area...", would take care of that possible confusion. (Abram, 7/5)
Looks good. (Abram, 7/6)

Rephrase the golden ratio / art connection

Make it clearer that the connection may or may not be real. (Abram, 6/24) Iris(6/25) I added this & the fact that we don't have any evidence whether Leonardo da Vinci had the golden ratio in mind when he was painting "Vitrivian man"

Well done. (Abram, 7/5).

Maybe move the identities section before the Golden Ratio / Binet sections

The identities are much more accessible than the section on the Golden Ratio and the Binet. With the current order, some not-so-confident readers might cautiously wade into the more mathematical section only to be overwhelmed by the harder sections when they could in fact read the easier sections. I think these two sections are totally independent, so you could just do a cut and paste without rewording anything. (Abram, 6/24)

Iris (6/25) i fixed this

Well done. (Abram, 7/5)

The problem is that when you click on "show more", it now says, "To see the proof, click [Hide]." Instead, maybe an entire sentence like, "Click to show a proof" should be clickable. After being clicked, it should say, "Click to hide the proof". (Abram, 6/24)

Iris(6/30) I fixed this
Well done. (Abram, 7/5)

Don't just tell them facts. Derive them.

Some of the less technical parts state facts about the Fibonacci numbers that could be proven without using equations. For example, in the bee section and in the "origin" section, it is possible to explain why the population progresses in a Fibonacci sequence. If you do this, you help readers think mathematically, instead of just reading facts about math. I know we haven't yet discussed this in person. (Abram, 6/4)

Iris feels that she has dealt with this (reported by Abram, 6/15)
I agree. Nicely done. (Abram, 6/15)

Decide if you like the new approach to the "origins" section.

I tried to reduce the lengths of the paragraphs, refer to the picture more explicitly, and explain one example in detail rather than several examples quickly. See if you like some or all of what I did. (Abram, 6/15)

I actually really like the changes you made. I erased my original section and replaced with yours. (Iris, 6.15)

Keep paragraphs short.

In general, limit paragraphs to 1 - 3 (maybe occasionally 4) not terribly complicated sentences per paragraph.

Iris I made some changes. I don't think I can make the paragraphs any shorter than now (6/23). The text doesn't look too dense anywhere, so I'm happy. (Abram, 6/24)

Integrate the golden ratio and Fibonacci pages better. A proof of why the golden ratio is expressed by fibonacci numbers is included on the Golden Ratio page. I don't know if you'd like to refer to that, but it is something already on the site. It's different than the explanation you provide. (Anna, 6/3)

Iris feels she has dealt with this (reported by Abram, 6/15)
I feel like there is still a lot of redundant material in the golden ratio section in general, but I like your link to the Golden Ratio page and your explanation of how your proof is different from that in the Golden Ratio page. (Abram, 6/15)
I actually got rid of the part where I derive the value of the golden ratio. Instead, I copy-pasted the same section into the Golden ratio page, and made a link to that page. I hope this got rid of the redundancy part. (Iris 6/15)
It looks good to me now. (Abram, 6/24)

Clean up a bit of mathematical language, notation, and rigor.

• Usually, if you use a series of equations to show that x = y, you put x and y at either of the series, rather than putting either one in the middle. For example, to demonstrate that $F_1+F_3+F_5+F_7=F_8$, you should write:
$F_1+F_3+F_5+F_7=1+2+5+13=21=F_8$ instead of
$F_1+F_3+F_5+F_7=1+2+5+13=F_8=21$
I made the change, and a similar problem in 'greatest common divisor' section (Iris, 6/15). Looks good. (Abram, 6/24)
• "A recursive function is one defined in terms of itself" is a bit vague. Let me know if you would like help making this more precise. You also may want to contrast a recursive definition with a closed-form definition.
Iris I talked to pf. Maurer. We made it a bit clearer by adding an example. Looks great. (Abram, 6/24)
• Note that the proofs of some of these identities could be made more rigorous using mathematical induction.
Iris I described the process for Binet's formula, and made a comment about mathematical induction at the beginning of the identities part. Looks great. (Abram, 6/24)
Abram, 6/15, clarified in a conversation on 6/17

#### Anna 6/3

Hi Iris,

First off, in your basic description, I'd actually provide an example of how to get a number from the two previous ones. (eg, saying 1+1=2, 1+2=3, 2+3=5, etc). While this may not seem necessary to you, some people work much better with equations than words, and it would really benefit people like that to see those equations.

Creating a helper page for recussion relationship might be helpful in your more mathematical description--the large mouse over is a bit clunky. Iris I'm not sure about this. I'll leave this comment as suggestions for future

In general, if you are providing a proof on your page, it's best to be explicit about that fact (saying something to the effect of "We will now prove this fact") and to mention the proof method if it's not a direct proof (eg mentioning if it uses induction or is a proof by contradiction). Your proofs are direct proofs, but you don't call always call them proofs.

Iris I added the part about mathematical induction, and mentioned that all the proofs can be made more rigorous by using mathematical induction

I'd like to see some more words explaining your proof of the sum of the first n numbers, though your animations in this section are really awesome.

Iris I added one or two sentences for each identities and properties and explained them in English

That's all for now. Keep up the good work.

Quick comment: Abram, 5/24

Iris, a lot of great progress on this page. Let's discuss it in more detail today around 2, when I arrive, if that's ok.

Smaurer1 05:19, 21 May 2010 (UTC)

I've fixed up the mathematical wording and symbolism in the first few places, including showing the use of \ldots, \quad, and \ (space after the slash). I also showed how to hide comments in the source code (look for < then ! then --, but I can't put them together here or the comment would be hidden). I also showed some better sort of mathematical wording. I also indicated how to get by with fewer math /math pairs; can't put in the brackets here or the symbols disappear. Is there a "verbatim" mode in the wiki language? Also, we all need to learn how to center math expressions in the middle of a displayed line.

Two small issues with the Fibonacci spiral section

• That abstract image next to the picture of the snail shell and the galaxy is kind of cool-looking, but totally unrelated to the point you are making about how these spirals show up in nature.
I agree, I deleted the page. It made sense when it came after the mandelbrot set, but I just deleted it (Iris, 6/15)
• It's great that you introduced a connection by talking about logarithmic spirals, but it's not all that clear why you included them / why logarithmic spirals are relevant to your page.
Abram, 6/15
I took out the logarithmic spiral part. I maybe using a helper page for logarithmic spiral, but I'm not decided yet. (Iris, 6/16)

• A lot of really interesting information about the sequence
• Good balance of mathematical definitions and applications

Things to work on:

• Explaining the spiral-counting in the basic description
• Reformatting the description of the problem's origin
• Tying the Fibonacci sequence to recursion and sequences in general

Iris we addressed all these issues