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Published on INFORMS OR/MS Today (Joseph Byrum)
Alan Turing’s passion for sunflowers reveals mathematical connection linking the worlds of AI, biology and finance
Alan Turing, the pioneering computer scientist, father of artificial intelligence (AI) and accomplished codebreaker, was also an avid gardener. On its surface, tending to plants at home might seem to be little more than a peaceful retreat for a man whose work played a pivotal role in ending the Second World War [1]. But not in this case. Turing’s garden was an inspiration, not a diversion.
The son of a gardener himself, Turing grew up with an appreciation for nature that stimulated his analytical mind. His interest in plants soon blossomed into a life-long passion for quantitative biology, allowing him to explore the interrelation of subjects through the lens of mathematics. Math is what makes it possible for operations research to work across so many disparate fields and industries. Mathematics is king in AI, finance and biology – and gardens, too.
Turing’s Sunflowers
Turing carefully observed the shape of the sunflowers growing in his garden. He realized that the patterns he saw on their face were not random. Rather, these patterns followed a discernible mathematical formula. This famous observation was celebrated a few years ago through a mass experiment in which members of the British public would grow 3,000 sunflowers so that they could verify Turing’s math [2].
Not surprisingly, the experiment “substantially replicated” [3] findings that were never really in doubt. The idea behind the sunflower event was to inspire the public to take a moment to visualize the power of mathematics in nature, as Turing had done. The results confirmed that the sunflower’s face is comprised of densely packed spirals of seeds that tend to follow a regular pattern, with 89 spirals in one direction and 55 in the other. Eighty-nine and 55 are Fibonacci numbers, so named after the 12th century Italian mathematician who identified the sequence of figures in which each number is the sum of the two preceding numbers.
Fibonacci numbers are interesting, as numbers in this sequence are related to an amount approaching the golden ratio, or 1.618. The golden ratio or golden mean is better known as artists, musicians [4] and architects [5] throughout history have reflected the harmony they found in nature by arranging their work according to the golden ratio. In the case of sunflowers, 55 spirals times 1.618 gives you 89 spirals. The pleasing shape of a sunflower isn’t random, even though the pattern might not be obvious.
Quantitative Biology and the Golden Ratio
The golden ratio and the Fibonacci sequence are important tools for understanding nature. They are found throughout quantitative biology, a science that attempts to understand incredibly complex natural processes using mathematics. Turing was drawn into quantitative biology through his investigation of the source of the fascinating sunflower patterns. In his final published work, he offered a mathematical model to describe the development of embryos as the explanation for the patterns such as the dappled spots on a cow, spots on a leopard or leaves on a stem [6]. These patterns are not random, he showed, but rather developed at the cellular level according to a knowable chemical process that can be described by mathematics. Today, these are called “Turing patterns.” He wanted to go even further with his analysis of the phenomenon.
“It might be possible, however, to treat a few particular cases in detail with the aid of a digital computer,” Turing wrote. “This method has the advantage that it is not so necessary to make simplifying assumptions as it is when doing a more theoretical type of analysis. It might even be possible to take the mechanical aspects of the problem into account as well as the chemical, when applying this type of method.”
This effort continued until his death, as he used the early computing resources he pioneered to further investigate “Fibonacci phyllotaxis” (the arrangement of structures, such as leaves or sunflower spirals according to a Fibonacci sequence). He elaborated on the topic in an unfinished paper [7].
Finance and Fibonacci
Turing understood the power of computers in developing and refining theories to explain order in seemingly random developments, and this methodology lives on even in unlikely places, like the world of finance. The ebbs and flows of the stock market at first seem entirely unexplainable. One day the stock is high, the next it’s down. If people knew why that happened and could replicate the analysis, it would be possible to make a lot of money as a trader.
Traders have discovered that they can gain some advantage by thinking about the market in the way we think about sunflowers. That is, just as there is discernible mathematical pattern hidden in the sunflower’s spirals, there is mathematical order reflected in the moves of the stock market. The Fibonacci sequence can be used to read the trends as the market seeks balance and gain a bit of insight into what future moves might be likely.
The Fibonacci trading ratios are 23.6%, 38.2%, 50%, 61.8%, 78.6% and 100% (50% is not a true Fibonacci number, but it is a favorite of traders). The numbers have a number of uses, but they are commonly consulted to set a target price when making market. The ratios indicate areas of potential support (that is, the price won’t go lower) and resistance (the price won’t go higher) in a stock. In a process known as retracement, traders will look at the stock’s price movement over time compared to each of the Fibonacci price intervals. When a stock that’s on the rise hits each Fibonacci level, the analyst will be prepared for the stock to turn around and decline. Then, when the price hits a lower Fibonacci level, the analyst will be prepared for a correction in the positive direction.
It’s not a precision tool, rather it is an indicator that has proven useful enough to become commonplace in the industry for finding order in seemingly random movements.
Turing isn’t known for contributions to the field of finance, but his enduring work in the field of artificial intelligence is helping reshape the industry. Algorithms find patterns of fraud in credit card transactions; data analytics are used to optimize decision-making and provide personalized banking services. That’s quite a legacy for a curiosity stoked in a sunflower garden.
References
- https://www.cia.gov/news-information/featured-story-archive/2015-featured-story-archive/the-enigma-of-alan-turing.html
- https://webarchive.nationalarchives.gov.uk/20170404144435/http://www.turingsunflowers.com/
- https://royalsocietypublishing.org/doi/full/10.1098/rsos.160091
- https://www.cmuse.org/classical-pieces-with-the-golden-ratio/
- https://www.goldennumber.net/parthenon-phi-golden-ratio/
- http://www.dna.caltech.edu/courses/cs191/paperscs191/turing.pdf
- https://link.springer.com/chapter/10.1007/978-3-662-05642-4_20
