The Golden Ratio

The Golden Ratio

Recently, I finished a book by Dr. Mario Livio, a distinguished Israeli-American astrophysicist, entitled The Golden Ratio. If you divide a line into two segments so that the length of the whole line* to the larger segment is the same ratio as the length of the larger to the smaller segment, that ratio comes out to 1.61803398… on ad infinitum: the Golden Ratio, or phi (not to be confused with pi).

The Golden Ratio** is intimately connected with Fibonacci numbers and the logarithmic spiral, which appear in nature in everything from galaxies to daisy petals. And besides popping up in unexpected places, the Golden Ratio as a mathematical object has some fascinating properties.

For instance, while the Golden Ratio is “the most irrational of irrational numbers” – that is, the number least expressible as a fraction, and thus the most unique of the known irrationals – it is, in fact, a ratio. Even in its infinite uniqueness, it does not stand alone; it is most easily and exactly expressed in the way I have described it above, as a relationship between three line segments which form a larger unity together.

To clarify, every time you have a larger and a smaller line segment in this ratio, their sum compared to the larger segment will also produce this ratio.  Where you have one relationship, you always have the others; they cannot be separated.

These three mathematical objects, whose length cannot be defined except in relation to each other, exist outside of time in eternal, infinite, and unique relationship, unable to be neatly categorized by the human mind. Does this begin to sound familiar? This is why Pacioli in 1509 popularized Phi by calling it “the Divine Proportion” – a picture of the Trinity.

Just the Facts

I found this fascinating, but Dr. Livio is not interested in mystic mumbo-jumbo added onto the hard math; the greatest passion he can offer is a mild surprise when the same pattern rears its head over and over in all aspects of nature. By his own admission, he finds people’s interest in the mystic allure of mathematics or the hidden patterns of the universe “inexplicable” (p 168).

Now, one might think this difference between Dr. Livio and me means our interests only slightly overlap, but in fact it causes a great deal of friction. Because Dr. Livio spends most of his book trying to dispel any mystical connections numbers might have. After a while, you wonder why he bothered writing it at all, since he spends most of his time discussing all the cool things the Golden Ratio is not.

By the conclusion of the book, it is clear why this tension occurs. You see, Dr. Livio is bound hand and foot to his naturalistic mindset. He cannot bring himself to consider that there might be a beyond to ground objective truth, beauty, or mystery; everything must be a subjective illusion of the human mind. This conclusion, though, was clearly decided on before the evidence was jammed awkwardly into place, because Dr. Livio can clearly see that 2+2 will go on equaling 4 whether anyone thinks so or not. Which leaves him with a problem.

Two Solutions

Dr. Livio’s solution to this quandary is to invoke the holy grail of academic tie-ins: evolution. Apparently, depending on biology, 1+1 might have evolved to equal 1 instead of 2 (p 251); in addition, it’s possible that “physics has selected” the fittest of the mathematical theories to follow (p 245). How physics existed without math, I’m not sure. Dr. Livio then confusedly asserts that after being selected – not by physics, now, but by humans – math “took on a life of its own” (p 250).

As someone who’s studied formal logic, physics, and math at Harvard, let me assure you that this makes no sense to me, either. Clearly, Dr. Livio hopes that just mentioning what he diplomatically calls “a… less common interpretation of natural selection” (p 252) will be enough to assuage our doubts.

There is order in the universe, Dr. Livio asserts. But, he cautions, without the divine, there should be no order in the universe. The conclusion would seem to be obvious, but Dr. Livio makes a tremendous effort not to see it. In the process, his argument degenerates into pseudo-scientific hand-waving and appeals to authority, with a few fractals thrown in for good measure.

In short, Dr. Livio knows, and has taught me, a great deal about the Golden Ratio – but he has not understood any of it.

*I understand that, technically, the ‘whole line’ is in fact a line segment as well, but this is hard enough to explain without using the same term for all three objects.

**In the interests of full disclosure, Dr. Livio explains in great detail (and I suspect he’s right) why many of the allusions to art, architecture, and coffee tables, etc. in this video are probably not real instances of the Golden Ratio being consciously used. However, the instances we find in nature are true instances of phi.

Photo credit: wiki commons “Golden Ratio line2”