Talk:Golden Ratio

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Contents 1 Abram 11/4 2 Abram 10/29 3 Abram 7/7 4 Abram 7/1 5 Steve Maurer 6/23 5.1 Abram 6/16 6 Alan 6/15 6.1 Abram 6/15 6.2 Gene 6/11

Abram 11/4

I very slightly modified the quadratic derivation of phi, making steps a bit more explicit. See if you're happy with it.

Looks good .

A mouse-over for Fibonacci could read: "An infinite list of numbers beginning 1, 1, 2, 3, 5, 8,... The pattern is that after the first two 1's, each number is the sum of the two entries immediately preceding it." I think having the link to the page is good, but I think people might also appreciate this two sentence mouse-over.

I would add the mouse-over myself, but as I recall, there may be something tricky about having a text be both a link and moused-over, that you may know how to deal with.

Yeah, um, I'm not going to be any help here. My thoughts are to either ask someone else, or leave it as is .

Re the subscript typo, I completely agree that it was a small typo. It's just that some people have preferences for using, say, f_n = f_n-1 + f_n-2 and others prefer f_n+2 = f_n + f_n+1, so I wasn't sure if you felt that one was more appropriate than the other.

Abram 10/29

The clean-up of the irrationality proof and of the page in general are great, and moving the history section to the beginning was great.

It would be nice for this page to be as accessible as possible to not-very-mathy people because the algebra is pretty simple. Really making this page really accessible requires a few things:

• Think-alouds reveal that people have trouble with the definition of . They aren't likely to really grasp that the described proportion isn't something that's generally true, but rather something that only works if you divide a line segment in the right place.
• Think-alouds similarly reveal that when people get down to the section deriving phi using the quadratic formula, they have already forgotten that the starting proportion was definitional for , rather than some kind of algebraic identity. In fact, I think each step in this algebra actually needs annotation. It's a pain, but lots of readers need it.

I've tried to address these issues by changing the phrasing and adding words to each step of the derivation. Let me know if that's enough

Small things:

• has a subscript error, but I'm not changing it because I'm not sure what indexing you think would be most consistent with the rest of the proof.

that was one short typo, and it didn't impact the rest of the proof

• Fibonacci sequence should get a mouse-over definition.

Where and why? I decided that a definition didn't fit in a mouse over, which is why I created the whole new page. The only place where I don't see the link is in the section title, which I could add in if you think it's neccessary. .

• implies that . I don't think this is totally self-evident. Can you explain briefly.

Yeah, that was super opaque, and it actually took me a while to figure out where that came from. It should be clear now .

• The golden triangle and golden gnomon should be labeled.

done.

• I'm cleaning up a couple typos and subscript errors, and also the somewhat confusing mouse-over definition of contradiction.

Abram 7/7

Yeah, the more I think about it, the more I think that your opening is really not a basic description, but is in fact a more mathematical explanation, and that opening with the historic mythic section will make the page much more accessible to the average reader (see comments below).

I'm on it. This page's structure requires heavy revision.

Abram 7/1

As always, there's a lot of interesting math and images in here. I know Chris just gave you a bunch of suggestions, but I want to add one suggestion that dovetails with what he is talking about.

Believe it or not, your basic geometric description is going to be fairly inaccessible to a lot of people. One big reason for this is that many people don't get that a ratio can be thought of as a number, not just a comparison between two numbers. So, for instance, people have no trouble with the idea of a ratio of 3 to 1 or 3 to 10, because these are obviously comparison statements, but have lots of trouble with the idea of a ratio of 3 or .3 because these are single numbers. Also, people will struggle to understand, based on the opening paragraph, that the golden ratio is constant.

The reason this dovetails with Chris's comments is because one way to warm people up to the idea of a ratio as a number is to *begin* with a historical/mythical section. This gives you an opportunity to explain that there is this number that people used to believe was related to beauty and the human body: if you take the ratio of height to navel-whatever distance, you get this number (or so people believed); if you take the ratio of the right segments in a pentagram, you get approximately this navel, etc.

This way, by the time people get to any equations, they will be really likely to have figured out that the golden ratio is a specific number. It would be especially great if your historical/mythical section could give some clue as to why people believed that this ratio that appeared in beautiful objects and the human body is specifically the ratio that satisfies the geometric definition you later provide.

Steve Maurer 6/23

First, let me second all the remarks that this page already has many fine features, not the least of which is that it has attractive images and good, accessible mathematics, with depth and variety. We are spending so much time correcting it because it is worth the effort.

Intro. The number itself is never mentioned, nor all its wonderful mathematical properties. How about:

The golden ratio, approximately 1.618, is called golden because many geometric figures involving this ratio are often said to possess special beauty. Be that true or not, the ratio has many beautiful and surprising mathematical properties.

Maybe I should go on and say

The golden ratio is the ratio of any two numbers a and b that satisfy a/b = b/(a+b)

but since you say that immediately afterwards, in the basics section, I see no reason to say it here. What is the most important general point you want to make about a concept? That's what should go at the top of the page.

Basic Section. There are still many math errors and I have edited them on the page itself. Also, I think you ought to stick to one order; if you do a/b to start with, don't invert and do b/a later. Nothing logically wrong with this, but it adds one more level of difficulty for readers.

Under Geometric, in the first gif, it would be "a+b is to b as b is to a" not a is to b.

Algebraic: Your substitution of a * phi = b should result in expressions involving a, not b. Specifically, your first line after the substitution should be

{a\phi \over a} = {a + a\phi \over a\phi},

which simplifies to

\phi = {\phi + 1\over \phi}

The next line is ok, except that it should be labeled or archored, because you need it later. I will call it (1) for now.

Note the small change at the end of the Algebraic section.

Triangle section.

 The second triangle is not a Gnomon as you have defined it, which I see is the definition in the Mac dictionary widget.  I think this definition is wrong.  See the Mathworld page on Golden Gnomon.  I would just call this a second golden triangle.

Note the small editing change for the sentence that starts "Phi". Symbols should never be replaced by their names - in math writing two slightly different things always suggest slightly different meanings. But also sentences generally cannot start with a symbol.

Color ratio gif: Can you give a reference? These facts are hardly obvious.

History. As I said today at your presentation, you have gone overboard in debunking these beliefs and calling them myths. See my rewrite.

Infinite fractions. I've given this its standard name - the value of the fraction is not infinite, only its representation, so using "infinite" is ambiguous. I agree with Abram's explanation as to why you cannot say that the sequence gets infinite, but I have used less formal wording than "as n goes to infinity", even though Abram is correct that his wording is standard mathematical terminology.

Proof of irrationality. All the right components are here, but not in the best order. The proof needs to start: Suppose \phi is rational. Then it can be written as b/a, where a and b are integers. But if \phi can be expressed as a ratio of integers, then it can be expressed as a ratio of integers in lowest terms. However, we show below that whenever \phi = b/a (integers or not) then also \phi = a/(b-a). Thus b/a cannot be integers in lowest terms. Therefore, \phi cannot be expressed as the ratio of two integers at all.

Now you need to show that \phi = b/a implies that \phi = a/(b-a). You can do that using the idea in the proof you have tried, but be careful because you or your reader will get mixed up between a, b, c! I suggest you work out a proof on paper and show it to Abram, and only write it up in the wiki when you agree on it.

The key about my rewrite above is that it puts the key conceptual ideas of the proof first.

Thanks for the comments. I've hopefully incorporated them all, with the exception of history. I may need to revise it in content and its position within the article (see Abram's comments).

-Alan

Abram 6/16

I found in my readings that the golden ratio is actually not aesthetically pleasing in any quantifiable sense, so I can't use that in my introduction now. Previous drafts of this page had the opening sentence trying to introduce some aspect of the golden ratio (pentagram etc) but it got too complicated.

Huh, interesting, that's news to me. There still may be a way of making the intro spicier, by saying something about why it's an object of interest. For instance, "The golden ratio, so named because it was believed in ___ times to ____, is the ratio of two numbers a and b...." or, "The golden ratio, which has such and such an interesting property, is the ratio of two numbers and b...".

My phrasing is excessively wordy, so feel free to change it.

OK, I'll do that.

It should be clear from the algebra that \varphi is unique in that it is the only number with these properties.

How about, "This algebra shows that if two numbers a and b satisfy the proportion described earlier, then this is the only value the ratio can have." This spells out exactly which property you are taking about.

A few other things:

• I really like the material you've chosen to write about for this topic and the overall way you are organizing the material. This is going to be a great page.
• The history section should include citations.
• The mathematical section still needs some cleaning up. For example:
  • In the phi = ... = 3/2, phi = ... = 5/3, etc, each equal sign after the phi should be an approx. equals. Also, it would be nice to state that each truncated function is a better approximation of phi than was the previous.
  • In phi = 1.6....9, the = should be an approx. equals.
  • It doesn't technically make sense to say "As the Fibonacci sequence approaches infinite length" because the sequence is itself the infinite list of terms. What you could say, that is correct and avoids too much heavy notation is, "as n goes to , the ratio of the nth term of the Fibonacci sequence to the previous term approaches phi."
  • I'm actually having trouble following your irrationality proof. In the parts I do understand, there are a few places where you basically say A implies B where what you mean is B implies A. I can talk to you about these sections, and give a couple quick pointers on writing proofs by contradiction in general, tomorrow (Wednesday).
  • You should not feel bad that I'm noticing so many problems in this section. Writing proofs about a topic you don't know too well, when your proof-writing experience is probably limited, and trying to make such writing meet a high enough standard that it's ready for the general public, is a huge task.

Alan 6/15

Abram-

"Re. Gene's suggestion of spicing things up, you could have your opening caption not describe phi mathematically, but rather have it describe something about how the image may draw aesthetic appeal from the fact that various dimensions have a certain ratio, which is called the golden ratio. "

I found in my readings that the golden ratio is actually not aesthetically pleasing in any quantifiable sense, so I can't use that in my introduction now. Previous drafts of this page had the opening sentence trying to introduce some aspect of the golden ratio (pentagram etc) but it got too complicated. "- Your geometric description doesn't make it clear that a and b can't be anything you want them to be - that only certain (a,b) pairs will allow the proportion you describe to hold true. Similarly, you should point out right at the beginning that when this proportion does hold true, the ratio in question always comes out to the same value, which is called ."

My phrasing is excessively wordy, so feel free to change it. Professor Klotz-

"Alan, your beginning might not capture hearts. Want to add something like "we'll see lotsa things this means below" or something."

I'm not sure where that should be or if that tone would be too elementary.

"Infinite Fractal: what's the "it" you begin with? I think we need something else to explain the mouseover def for recursion, not sure what. " I'll put it as a red link helper page. "Where do the infinite fraction expressions come from? Your ... in the 4th phi should be descending, not horizontal."

Not sure what you mean here.

Hopefully I addressed everything else.

Abram 6/15

Hi Alan,

- I really like the geometric interpretations and applications you include, and the way you use colors to dramatically reduce the number of symbols needed to write relationships (especially in the star).

- Your algebraic derivation of is quite clear.

- Re. Gene's suggestion of spicing things up, you could have your opening caption not describe phi mathematically, but rather have it describe something about how the image may draw aesthetic appeal from the fact that various dimensions have a certain ratio, which is called the golden ratio.

- Your geometric description doesn't make it clear that a and b can't be anything you want them to be - that only certain (a,b) pairs will allow the proportion you describe to hold true. Similarly, you should point out right at the beginning that when this proportion does hold true, the ratio in question always comes out to the same value, which is called .

- In this section, it should be made clear that this ratio is a specific value which can exist even in contexts that do not include line segments divided the way you have divided them. This is another big issue for non-math people: when you derive a mathematical statement in the context of a specific problem, they will need explicit explanation about exactly what way the statement applies/has truth outside the context of that problem. This would be an excellent opportunity to refer back to the page's main image.

- The more mathematical suggestion could use an overhaul on how to use mathematical language to articulate your ideas more precisely. That's something you and I or Anna or Steve or Gene could do in person.

- I have various other suggestions, but this is a good starting place.

Gene 6/11

Alan, your beginning might not capture hearts. Want to add something like "we'll see lotsa things this means below" or something.

Your organization is good and clear, and nicely broken up with pictures.

Under Algebraic, might you want to give away the phi notation right at the beginning of the page? "We'll see it has a special numeric value" or something. (I'm not sure).

Need an image illustrating your definition of golden gnomon.

Infinite Fractal: what's the "it" you begin with? I think we need something else to explain the mouseover def for recursion, not sure what.

Where do the infinite fraction expressions come from? Your ... in the 4th phi should be descending, not horizontal.

"If we evaluate truncations...": we don't set phi = 1, it's a fixed number! We could replace it by 1 and say we get the results on the right hand side.

Thus we discover that the golden ratio is the limit of the ratio of Fib nos, or something.

"As the Fibonacci sequence approaches infinity": I think this could be slightly rephrased.

Proof of Irrationality that the Golden Ratio is an Irrational Number. (Maybe have mouseover here?)

Your proof could be clearer at the end. "equivalent fractions?" And stress the contradiction.

But this is good stuff, Alan.

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