*H. R. Currie and G. M. Currie, Open Access Journal*

This one was brought to our attention by the Evil Mathologre. It is a tricky one, since it involves the work of a school student, and the student is in no way a target for our criticism. Out of such concerns, we haven’t made this post a WitCH; it should be considered in the same vein as this Maths Masters column.

As reported in Wagga’s *Daily Advertiser* a couple weeks ago, and as picked up by *The Canberra Times*, IB student Hugo Currie was given a “mathematics assignment” (presumably an Internal Assessment) on the golden ratio:

“… *we had to investigate an element of the golden ratio in the built or natural environment so I decided to look at atomic structure …*“.

Hugo considered the atomic mass number* A* (protons plus neutrons) of nuclides (isotopes), comparing *A* to the number *N* of neutrons and the number *Z* of protons. Of course, *A* = *N* + *Z*. Hugo then looked for “fibonacci nuclides”, nuclides for which the ratios *A*/*N* and *N*/*Z* are very good approximations to the golden ratio. He found a bunch, and suggested his results as a guide to hunting for new elements and nuclides. Hugo’s graphic above is a good illustrative summary of his investigation; the horizontal axis is *N*, the vertical axis is *Z*, and the black line indicates known stable nuclides.

OK, no big deal. From our perspective, having a class sent off to hunt for the golden ratio is asking for trouble, but it’s just an IA, and Hugo’s work seems interestingly exploratory-ish, in the manner the IB foolishly demands. But why did Hugo make the news, and what’s the problem?

In May, Hugo published a paper, co-authored with his father Professor Geoffrey Currie, in the peer-reviewed *Open Science Journal*. And, yes, of course that made the news. And yes, that’s the problem.

Unsurprisingly but unfortunately, we can see little if anything research-worthy in the Curries’ paper, and we noticed a number of “Uh-oh”s. A fine IA, sure, but not a research paper, and not news.

We’ll leave it at that. Readers are free to hunt for the uh-ohs.

Kind of a crappy assignment though, innit?

As I wrote, it’s asking for trouble.

how?

Hi, Liam. The problem is that Fibonacci/GR, while mathematically cool, is also a faddish thing, with a long associated history of pseudoscience. It means one has to tread carefully, reading claims and techniques with concerted skepticism. The article by Markowsky is not well-written, but it indicates clearly enough the standard and common pitfalls.

A lot of the associated modelling, including in serious research journals, amounts to juggling numbers until something around 1.6 emerges, and then pronouncing success. But, as I wrote in this post, unless you have a model, a suggested explanation, OR you have a *lot* of decimals, 1.6-ish is not an argument for the golden ratio: it’s just an argument for 1.6-ish.

Is it really correct to conclude a line of best fit which may have a gradient between 1.6 and 1.8 (I don’t have the data, so can’t do my own analysis) is the magical 1.618… ?!?

I do like Phi as a number, in so much as it is an unexpected solution to a few recreational puzzles, but to conclude that this gradient is

equalto Phi is… baseless? Wrong word, but I can’t think of the right one.RF, it seems to me more that the GR line best fits the nuclides that best fit that line.

The whole shaded area seems to be something other than linear as well…

Well, you have to read the article to see what they’re getting at.

Doesn’t Phi equal the definition of it not an approximation of the number? So you would need to prove what ever you are looking at to exhibit the definition (and not the approximation)? Otherwise you’re just finding patterns that may or may not actually exist.

Hi, Potii. Pretty much. One could reasonably look for numerical evidence of phi. But, without some model somehow connected to the definition of phi, it’s not clear why one should bother, and it can easily stray into reading tea leaves.

Hi,

If we start with an assumption of a Fibonacci Series we will get Phi in the limit of the ratio of successive terms but there are many other varieties of tea to be considered

eg https://en.wikipedia.org/wiki/Metallic_mean

Steve R

As a mathematical physicist, I can only sigh at the content of the nuclides article that Marty referred to. It is a classic example of what happens when word processors and graphics software come together in our modern world: people are easily led to believe that whatever they write is correct, because it has lots of colours and a few equations. It gets published in a journal of clearly dubious worth, and is then picked up by an ignorant media that equates “published” with “worth publishing”. We can only wonder who the reviewers were. Perhaps they were others who had been “published” in a similar way?

Rather than comment on the imagined fitting of a line of the sought-after slope to the data, I’ll make one comment that has nothing to do with nuclides. On page 4, the article says “Indeed, Phi experts have been critical of false claims of the golden ratio in art and architecture (including pyramids)”. Let’s look at that, because there’s more to it than meets the eye.

Note that , which is, of course, very close to 4. It follows that if is present in a given scenario, then if we can play around with some natural numbers to produce a 4 somewhere, we might be led to believe that is present too, to an accuracy of 1 part in 1000.

This seems to be the case with the Great Pyramid. Its angle of slope is extremely close to . Whether that is a coincidence or the result of wheels being used in the surveying, no one knows; but note that another classic pyramid slope is approximately , and so it might well be the case that this appearance of is no coincidence, even though might never have been inserted into pyramid geometry by conscious design.

It follows from the above coincidence of that the cosine of the Great Pyramid’s slope is very close to . This is often presented as undeniable evidence by golden-ratio enthusiasts that was purposely built in to the Great Pyramid. But this presence of might be a coincidence that has resulted from wheels being used to survey the pyramids.

(Edited by Marty, 14/07, at Don’s request)

In general, the way to think about this is WHY would you ever have any atoms greater than hydrogen. The atomic number (element) is determined by the number of protons. And protons repel each other. At least the circling electrons get to avoid each other a fair amount. But the protons are pushed together. How can that be? Well, it turns out that neutrons act as a sort of glue to allow this to happen. Too many (or too few) neutrons gives instability. There are also sort of “magic numbers” (shell filling) in which certain amounts are preferred. I have no clue how the Golden Ratio comes in though.

P.s. I do like me some the Mae West Curve: https://teachingcalculus.com/2015/10/19/are-there-any-extrema/