h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Entry: Phi - A2194120
Author: AK (a maniac) - U236276

This is my first guide entry... I really didn't know much about Phi, this is the absolute limit of my understanding of it. I had never heard of phi(lower-case) at all.
Parts of this entry were copied from an old, unedited entry on the same topic, and I credited this to the author of that entry also.

This is nicely laid out and well written. For a first entry it is very well done!

There are a few things that could be added and a few things that can be improved.

1. You mention properties of phi and give the following two:

phi = 1 / Phi

Phi * phi = 1

These are not really two separate properties.

2. You should put a zero before the point in .618 when you give this value for Phi as it is easy to miss the point unless the zero is there.

3. There's a very neat formula for the Fibonacci numbers which involves Phi. I can't remember exactly, but I'll look it up.

4. I wrote a short entry on Phi which has not been submit to Peer Review. (It was done offline - I only put it onto h2g2 a minute ago). It's at A2200384. Feel free to copy anything out of this without asking and I don't need any credit for it.

Fn=(Phi^n-phi^n)/sqrt(5)

Thanks, Old Hairy!

A thousand apologies. The correct formula is Fn=(Phi^n-(-phi)^n)/sqrt(5). Are you aware of A471151? A detailed review will follow later (honest!).

Interesting. Good entry

Pimms

Thank you,
Yes I had seen the page of Fibonacci, but I didn't really understnad the equation... however I think I'll go put it in anyway.
btu first, is the formula
"F(n) = (Phi^n + Phi^-n)/sqrt(5) if n is odd
F(n) = (Phi^n - Phi^-n)/sqrt(5) if n is even"
like it is on the entry, or
"Fn=(Phi^n-(-phi)^n)/sqrt(5)?"
I'm going to use the last two parts in your entry, Gnomon.
*makes changes*

Those two versions of the formula say exactly the same thing, just in different ways. I'd use whichever you find easiest to understand

One feature of the golden mean that I particularly like, (cus I'm like, sad) is that is relatively easy to generate on a calcualtor.

Pick any number. Doesn't matter how big it is, just pick it. Then invert it. Add 1 to the answer. Then invert that answer. Then add 1 to that answer. Then invert that answer. Then add 1 to that answer.... and so on. Keep doing this for long enough, and your calc will home in on two numbers, namely, Phi and phi. Yes, boys and girls, numerical methods can be fun.

You mentioned the relation of the golden mean to the human body as well, and I thought you would like this. I have a book (Bluff your way in Maths) that states, but obviously doesn't try to prove, that the difference between a woman's height and the height of her naval is... surprise, surprise... Phi. Quite why it specifically refers to women but not men is beyond me. I can't say I've tested it myself.


Geggs

...or just take the square root of 5, add one, and divide by two. tada.
how will it hone in one two numbers? one if its positive, one if its negative?
I've heard of the navel thing too, while looking for info for this entry...
I heard somewhere a whiel back that there was a similiar ratio between the forearm and the upper part of the arm... I think that was talking about Phi...
You mean the difference or the ratio, though?

Hello AK (a maniac). Sorry for the delay in responding properly.

I am very pleased to see someone sufficiently interested in maths to write an entry. I also note that this is your first attempt at a guide entry. So well done on both counts.

I hope you will not be discouraged by the criticisms which follow. I am sure you will end up with an entry which is picked for the guide, provided you persevere. Considerable reworking will be required to achieve the goal of selection, but I'll help you all I can.

The entry has two main problems at the moment. Firstly, it seems to lack any structure, so seems to be a collection of miscellaneous facts about the golden ratio (hereafter GR). For example, it alternates between geometry, algebra and other matters in a rather unpredictable way. I suggest that you gather up the geometric facts, the algebraic facts, and the other facts (some of which may be disputed). The other major problem is the notation you have used, which made the entry very difficult for me to follow. It is at odds with the literature on GR, usually (in recreational mathematics and modern serious work) denoted by the lower case Greek letter phi. In older serious works in was denoted by a lower case Greek tau. There is little standardisation for what you call phi, but in 'The Art of Computer Programming' by D.E. Knuth, a phi with an overhead caret (^) is used, and I think that would be called phi hat.

Now for the details, in the order they appear in the entry at the moment.

You begin with it being known to the Greeks (presumably the ancient ones). Euclid called it the extreme and mean ratio. In fact, if (A+B)/A=A/B, then A/B is the GR. Something to do with the ratio of the sum to the whole being equal to the ratio of the whole to the part. It would serve the entry well to use this definition, then you can later deduce that x=A/B satisfies the quadratic you give, which leads to the value 1.618 which you give. (At the moment, you define its value, so it does, so to speak, appear from nowhere.) The aesthetic properties can be discussed without any maths if the GR is to do with the whole and a part, which would help enormously to keep a non-mathematically inclined reader interested.

Having introduced it in geometric and aesthetic terms, you go straight to a number and some mathematical properties. But the definition as above leads directly to the quadratic, whose solution gives the value, and can also give the continued fraction (N.B. continued, not continuous). The other so called properties are just rearrangements of the quadratic.

Then you return to properties other than those of the Golden Rectangle. Unfortunately, previous use of the term Golden Rectangle is hidden in a footnote. Again, the first you mention are geometrical, but the first you expand on are Fibonacci sequences.

You say that the ratio of successive Fibonacci numbers converges to phi, which should be Phi in your own notation. That is either due to the general expression for Fibonacci numbers (given in an earlier posting), or is a consequence of the continued fraction expansion of the golden ratio, whose succesive convergents are ratios of consecutive Fibonacci numbers.

The convergence property for generalised Fibonacci sequences, that is with arbitrary starting values, is just not true. If the first term is the GR, and the next term is -1, the ratio of consecutive terms is absolutely fixed, as -1/GR, so one does not need to wait for convergence. It is also not true of all integer values, just start with 0, 0 to see that. However, it is true for all non-zero integer starting values, a fact which is easily shown from the general solution of the Fibonacci recurrence.

The thing about the nearest points to the line through the origin whose gradient is the golden ratio is not very well expressed. The phrase 'the points it crosses closest to' is noxious after 'the line does not cross any points'. You need something more like 'the points nearest the line'. By the way, this particular feature was discovered by Felix Klein. He put it that if you imagine the line as an infinite piece of string, nailed down at infinity, your points as pins in a board, and move the end of the string at the origin, keeping the string taut, then the pins touched are all the convergents of the continued fraction less than the golden ratio, or all the convergents above the golden ratio. Klein discovered this in 1897, and published it in 'Ausgewahlte Kapitel der Zahlentheorie' in 1907.

The phraseology is wrong in 'This process can be continued infinitely', and I would suggest '... continued indefinitely'. I would also suggest that 'This spiral is special as the nautilus shell is curved in this exact same way' be reworded along the lines 'This spiral is known as the nautilus curve, as the shell of the nautilus organism is similarly shaped', but with a different word for organism (I don't know what sort of organism it is, a crustacean maybe).

I do not understand golden triangles. If the sides are in the ratio 1:phi:phi^2 then the triangle collaspes to a line, since phi^2=phi+1. Did you mean either of 1:1:phi or 1:phi:phi? If so, then does it matter which angle you bisect? (it must do, otherwise 1:1:phi could become two 1:1:phi/2 triangles, which are not golden.) My 1:phi:phi^2 example is one interpretation of your 'a triangle in which either one or two sides being phi times longer than its other sides', which ought to be a little more precise. 'the points of each triangle will make a curve like the rectangle did' is not quite true. Pedantically, the points of each triangle will lie on a curve, and I do not know whether or not that is also the nautilus curve (it might be).

I tend to avoid examples involving pentagrams if possible, since they have a certain cult status. However, there is a close association of pentagrams with GR, so perhaps you should include them here. I suggest you define them as the result of joining every non-adjacent pair of vertices of a regular pentagon with a straight line, and thereby avoid any reference to a surrounding circle. In what you say about pentagrams, I think there is a typo. Should 'you draw a pentagram inside of it' be 'you draw a pentagon inside of it'? If not, then there are a bewildering number of points (10 on each pentagram) that could be joined, and it is not quite clear how the inscribed pentagram is obtained (unless by my method from the inner pentagon of the outer pentagram).

You have the geometric examples, but the GR can easily be constructed, straight line and compass style. That could do with mentioning.

The facts about the finger are thin, and shaky. There may be lots of connections of human body proportions to GR, since the human body is often superimposed on a pentagram. But just to state the ratios of a finger, without connecting this with natural growth and Fibonacci, is open to challenge. For example, given the low precision possible for such measurements, why should a number which is about 8/5=1.6 be GR=1.618 rather than pi/2=1.57? The example of the geometry of the Parthenon would fit must better in the introduction, with the stuff about ancient Greeks. The music example is very shaky. At the moment, in Peer review, is an entry about maths in music. Some of the numbers in that entry are/were being hotly disputed (I suggest taking a look).

pi, e and the rest are irrational, but that is not because 'the decimal value does not terminate'. For example, 1/3 is certainly rational, but 0.3333....

The number phi, written out a lot a bit, is available from many web sites. Are there any which are worth a link.

Enough of criticsm! Time for a compliment! I thought the idea of running Fibonacci backwards to get -1/GR was quite novel, and interesting. Never come across that before, so far as I remember. I hope that you stick with this entry, and bring it up to standard. Writing maths stuff was getting a bit lonely.

Sorry, but I went away to do my last long post, so missed what was going on meanwhile.

Re posting 7
The formulae are the same. Having an nth power of -phi makes the sign change according to whether n is odd or even. Different ways of saying the same thing, as it says in posting 8.

Re posting 9
Geggs method evaluates the continued fraction from the bottom up, if you stop to think about it. It fails if you ask the calculator to divide by zero (a magic calculator which divides by zero to get infinity, and divides by infinity to get zero, gets the right answer anyway). Otherwise, the starting value is buried deeper and deeper in the continued fraction, so matters less and less.

Well, Old Hairy...

The first thing I've got to say (in defense of myself), is that several of the passages were copied drectly from an old entry on Phi, and I credited it to that author. A643844
Discouraging? nah...
I don't quite understand the golden triangles but it was on a website I found from google. Apparently though you have to bisect one of the base angles.
I don't know how to put a caret over the Phi symbol...

I'll be working on what you just listed, a bit at a time, for a little while... (I don't have time now...)
Which part would you suggest putting first?

Hello AK (a maniac)

I took a quick look around the net to see if I could find something at an appropriate level which covers the material of your entry. The following is excellent. It covers everything you and I mention, and much, much more, but is fairly easy to read. It also has (from my point of view) the great merit that all the maths results are proved. The proof the phi is irrational is quite interesting. The link is:-
http://www.mcs.surrey.ac.uk/Personal/R.Knott/Fibonacci/phi.html#euclidlinks

Just one more thing. I have prepared an entry which may be of some help, which you are free to poach. It is called Non-PR edit A2079885 and contains a continued fraction all formatted as a multi-deck fraction, an approximation to phi. (It is an adaptation of part of my incomplete entry A2147401 about continued fractions.)

I trust you know how to lift the GuideML of an entry, but if not let me know. With all this, I reckon I can leave you to your own devices until you say otherwise.

I hope I haven't overwhelmed you. How is it going?

I've been working, slowly,... but i haven't been on in the last few days. And I've been doing most of it offline. Which reminds me I need to get back to it, certainly...

Your change of name worried me a bit. Do you need A2079885 or can I recycle that entry now?

oh, the name... I jsut change it randomly. I started regretting calling myself AK a long time ago but if I change it all the rosters are wrong. So I just change the the part at the end.
I'll stick the fraction somewhere on the entry for now, so you can "recycle" that entry...
There, go ahead and recycle it.

I have some changes to make but I need to know: What symbol should I use for Phi, and what for phi?
phi, tau, tau with a triangle thingy, or whatever...?