Dan Brown’s The Da Vinci code was a book that captured my imagination from the first page. The setting in the art world and academic departments, the murder investigation and the conspiracy theory, was everything that my university life is not, but could be if I were to close my eyes and fantasise a little. He showed the way us academics see ourselves, searching for ultimate truth. And, the bit that really got me, his unifying theme of the golden ratio, phi=1.618…, the divine proportion that, according to the book’s hero Professor Langdon underlies everything from art, to biology, to religion.
While I was inspired, I was never entirely convinced by Brown’s fictional description of phi. The maths I use professionally is very different from the god-like symmetry described as the Code. Certainly, there are some remarkably useful universal constants in maths, including phi, pi (3.14…), Euler’s constant (0.577…), the natural number e (2.71…) and many others. But they each have their limitations and arise from quite specific areas of mathematical study (phi from the solution to Fibonacci’s sequence, pi from circles, Euler’s constant from logarithmic growth, e from exponential growth and so on).
After the publication of Brown’s book, mathematicians pointed out that many of Professor Langdon’s claims about phi don’t hold up to scientific scrutiny. The drawing above, in which golden rectacles are overlaid on Da Vinci’s drawing, are arbitrary, as are those often shown of the Parthenon in Greece. There are many different points and lines on a person or a building which we can measure, and some of them will, by co-indcidence be close to phi. With the exception of occurance of Fibonacci numbers in sunflowers, no concrete role for phi has been established in nature.
The Internet continues, of course, to obsess over the number. They beleive, as Professor Langdon dramatically proclaimed in the book, that “the ancients assumed the number PHI must have been preordained by the Creator of the universe…”
Whether Greeks and people living in the Renaissance did make such an assumption is debatable, but it is easy to find blogs and webpages making similar claims today. Throughout history, some of us have been trying to find preordained order in numbers. Research tends to show that we usually get it wrong
But might modren mathematicians have underestimated the power of phi?
I hadn’t given the Da Vinci code much thought until last year when my PhD student, Yu ‘Ernest’ Liu, came in to my office to discuss the results from his latest computer simulations. Ernest showed me printouts of the graphs from his simulation output, on to which he had scrawled the decimals 1.618 in pen on top of some of the plots.
“This number keeps coming up”, Ernest told me, “as the ratio between chemical growth rates. I looked it up and its called the Golden Ratio. I’m not sure why it is found here though…”
Ernest had already realised the possible significance to this finding. His PhD project was a bit different from most: he is using maths to investigate the origin of life. He has been working for the past three years in order to create a model of chemistry which explains how self-replication arises. His question is how the universe has evolved from a soup of chemical reactions to include the rich variety of biological reproduction that we call life. To find phi in the context of this question was intriguing.
Maybe the naysayers were wrong and phi could be a preordained outcome of simulating reproduction? Could the golden ratio provide the Da Vinci code for life itself?
Ernest and I started by reading through the scientific literature and it turned out that some math biologists had already seen a potential role for phi in self-reproduction. Researchers in Brazil had shown that the frequencies of human nucleotides adenine and cytosine followed the golden ratio, and one indepdendent researcher in France claimed that the human genome was “fine-tuned” to phi. Scientists were still giving credence to the idea that phi is a universal property of biology.
The tone of these articles was intriguing. We needed to investigate exactly how our new results fitted in to this work.
Ernest had found that self-reproduction was a common property of many of the chemical universes he simulated and it was within some of these simulations that the ratio phi would appear. But there were other simulations that produced different ratios. Sometimes he found the important ratio was 1.324… (a number known as the plastic number), other times it was 1.220… (which is the 3rd lower golden ratio) or 1.259… (which is the third root of 2) and in some simulations he couldn’t find any ratio at all. Importantly, phi was no longer the most common ratio, it was one of many interesting numbers that were popping up everywhere.
To solve this mystery Ernest started to write down equations to describe each of the different self-reporducing simulations he had run. Then by solving these as time goes to infinity, Ernest uncovered, what is called, a characteristic polynomial equation for each case. One example of this equation is shown below
and the solution to this equation is none other than lambda = (1+sqrt(5))/2 = phi.
It was now we could see that phi was nothing special. Each characteristic polynomial has its own solution and Ernest could now identify why each of the numbers underlay each of his simulations of chemical life.
And there wasn’t just one of them, or even two. There were many, infinitely many, in fact. Phi, and all the other numbers Ernest found, were algebraic numbers, which are shown in the plot below.
This image is maybe even more stunning than the stacking of golden rectangles on top of each other in Da Vinci’s drawing. And it would be tempting, in showing this picture, to segway back to the imaginative philosophical musings of the fictional Professor Langdon. Maybe a figure like this holds the secrets of life? Each solution to self-reproduction captured in a bright and infinite glow?
This simply isn’t the case. Real science and mathematics doesn’t work this way, even if sometimes we might like it to. Maths throws up a lot of beauty, but that beauty is just a small part of the truth. What Ernest and I found, and what we published in our scientific article, is that the algerbraic numbers are not the key to self-reproduction. What is important in our research is showing that many different chemical universes can produce self-reproduction and life (I’ll write about these findings soon!).
When I was younger I used to dream of something like the Da Vinci code, a unifying equation for life. This is why the book caught my imagination. Some theoreticians, Max Tegnmark and Stephen Wolfram being two of the most notable examples, still seem to beleive that there could be an underlying code to the Universe. I am extremely doubtful. We see glimpses of symmetrical beauty in maths. But the real beauty is that there is no single number or equation for reality. The world we live in is complicated, beautiful, messed up and sometimes impossible to understand. There will never be a single equation to explain why.