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

At risk of pointing out the obvious, an entry called 'Phi' really ought to be about Phi. If you want it to be about the golden ratio, make that the title.

The only reason you have the trouble with Phi and phi is that you chose to define them as the roots of a quadratic. Had you used a more natural definition, you could have rejected one of these values. (I mentioned this in posting 11.)

In posting 4, Geggs pointed out the calculator method, starting with an arbitrary value ('Pick any number. Doesn't matter how big it is, just pick it.'). The 'x' in my continued fraction was meant to represent that. Just deleting the 'x' leaves you with a finite continued fraction, which is not equal to phi. You need to show that the continued fraction goes on for ever, or include that x, or replace it with phi.

As to the use of Phi and phi or some other notation, I would tend to disregard Dunlap, Vajda and Koshy. Your primary reference (Knott) does not appear to think much of them either - since of Dunlap's work, Knott says 'wrong and appear to have been copied from Vajda preserving the original errors!', which rubbishes both Dunlap and Vajda, and of Koshy's work he says 'there are typos, errors and missing conditions in far too many of its formulae and some glaring omissions and misattributions'. Knott uses your Phi and phi. His other references are Knuth (phi and phi hat, with mentions of the use of tau in older works), Johnson (available online) uses lambda (with subscripts of + and - when the two roots of the equation are to be distinguished). I have no access to the reference by Rabinowitz. So, no help there. My suggestion was made in posting 21.

The website http://www.friesian.com/golden.htm has a picture of a Penrose tiling, and some useful stuff about powers of phi, which would answer the questions you asked in posting 33, but in a round-about kind of way. In fact, the suggestion is that you can get simple relationships out of any power of phi. Note that this site uses phi (the lower case greek letter), and can be reached from links in your Knott reference.

Although several links have been suggested to you, none appear in the entry. Is there some reason for this? (The golden ratio and Fibonacci numbers are very popular on the net.)

The formulae in the footnotes do not appear correctly if (like me) you hover the mouse over the footnote number to read the footnote. (I do this because h2g2 takes a month of sundays to follow the link.) In particular, all superscripts and subscripts are rendered as normal characters in Internet Explorer. It is a particularly bad idea to have the entire explanation of the continued fraction buried in a footnote, which is itself buried within the continued fraction. The content of footnote 8, that the decimals of 1/phi are the same as those of phi, would be totally obvious if the quadratic had been deduced, and not plucked from the air.

The repeated square root thing is an iteration, not a series. It can only be evaluated working from the right, as it were.

There are no examples of the golden ratio in nature, unless you count the one about fingers, which is dismissed as a myth. But there is a huge literature about the golden ratio in nature, a sound theoretical basis for it, and this largely underpins the subject of phyllotaxis (I hope I spelled that correctly - the theory of leaf formations and seed pods and so forth).

If you find the maths too difficult, it might be better to make this a descriptive entry, and just provide links to the many excellent websites at appropriate places. It's just a thought.

Okay, this has been sitting in our in-box for a really long time. We'd really like to push this through now, and as I'm not a mathematician by any stretch of the imagination (got an O' Level - just) I'm going to need every change outlined really carefully and clearly.

- Where can I find each error?
- What needs changing?
- What does it need changing to?

With regards entities, we'd prefer the &...; format, so stick to that one rather than entitiy numbers... and if you're good we may blob the diagram.

Yikes! I'd forgotten about this, somehow. now I must hurry to get it fixed. I'll probably simplify it a bit, take out some of the hard to explain things... I'm going to try to get this worked on as soon as possible, very sorry...

I was thinking that the title might work if it was changed to 'Phi and the Golden Ratio', incorporating the suggestion from earlier.