# Petrarch and the mathematics of love

#### However, we are all well aware of the fact that love, despite its powerful strength, is also subject to change over time and, often, it can even end. Why and how does this happen? What are the main factors that affect a love story? How long is it likely to last? Despite love being so important, we have to admit how little we know about its dynamics.

Several mathematicians have tried to describe the dynamics of love through the language they know best, maths. Mathematics is ultimately the study of patterns in natural phenomena and love is full of patterns. This might sound to most readers as a very dry approach to the debate, a way to kill the romance and make love another dull subject to be studied in schools and universities. However, you might be surprised. This is probably one of the few topics that can make maths fun and reconcile most of us with maths. Moreover, the application of math to the dynamics of love might create business opportunities and dramatically improve the quality of relationships. Just think about all those dating apps and websites looking for the perfect system of matching two people according to their profiles (assuming that everyone provides true information about him/herself). To the best of our knowledge, many attempts have been made so far, but with little success.

Business Insider published an article on a mathematical formula that predicts how long love will last. This formula was created by surveying 2,000 women and men on a number of factors that are likely to affect, on a long-term basis, a love relationship. The result of this study is a deterministic, quite lengthy equation based on factors such as number of previous partners, the importance both attach to humor, good looks, sex and children. Each variable is multiplied by a constant factor, that weighs the importance of the variable over the total result; however, the article does not say how you should estimate the value of these variables. We guess that the only way to go is to write this equation in an Excel spreadsheet, play a little bit with the numbers and try to find the best combination that maximises the result.

The mathematician Dr Hannah Fry published a book titled “The mathematics of love” and also gave a Ted talk about it (which we strongly recommend to watch, you will not be disappointed). She takes a more pragmatic approach and, instead of creating a magic formula, she uses a well-established mathematical tool, called Optimal Stopping Theory, to describe the best process to maximise the chances of finding the right partner. According to Wikipedia, the optimal stopping theory is concerned with finding the best strategy to choose the time to take a particular action, in order to maximise an expected reward or minimise an expected cost. According to this theory, if you would like to settle down before you are forty years old, you should start dating when you are 15 years old, reject everybody you are dating within the first 37% of your dating window and then pick up the next candidate who is better than everybody else you have gone out with up to that moment. The model does not assure that you will meet that person, but it says that you are maximising the probability of finding the best match and have a satisfying relationship.

Another branch of maths that deals extensively with optimizing dating strategies is game theory; actually, according to the movie “A Beautiful Mind”, the creator of this branch, John Nash, had his first intuition thanks to a dating problem. This happened one night when he was in a bar with a group of friends watching a group of beautiful women and wondering what was the best strategy to court the most attractive of them.

In conclusion, you have a wealth of options to maximise your probability of finding a good partner. However, not very many models have been proposed about the dynamics of love in established couples. We know of only one model that tries to describe this phase of love stories. Dr Sergio Rinaldi, an Italian professor, teaching in the Polytechnic of Milan, proposed in a peer-reviewed paper a system of ordinary differential equations to study the dynamics of love, once you have found a partner. He calibrated his system on an extensively-described love story, the one between Petrarch, a celebrated Italian poet of the 14th century, and Laura, a beautiful but married lady. Petrarch wrote 366 poems, collected in a book called Canzoniere, that reports all the phases of Petrarch’s feelings towards Laura, that lasted approximately 20 years.

On the basis of a linguistic and lyrical analysis of the poems, Dr Rinaldi could assign a grade to each poem, ranging from -1 to +1. The maximum grade (+1) is assigned to poems describing a period of ecstatic love, while negative grades are assigned to the different shades of despair. When sorted in chronological order, these poems show that Petrarch’s love followed a quite regular cyclical pattern, alternating periods of ecstasy to periods of despair. Dr Rinaldi then developed a system of three ordinary differential equations that fitted these cycles. Two of the three equations describe the reactions of Petrarch and Laura to the love of the partner; they are based the appeal (physical and intellectual) of Petrarch and Laura for each other. The third equation describes Petrarch’s poetical inspiration sustained by Laura’s love. The following image shows the results obtained from the model, with the original parameters.

It is interesting to see that Laura’s feelings follow a wavy patterns, alternating periods where she is nicer towards Petrarch and periods where she behaves in a colder manner, however, the plot of her stays in the negative area, meaning that Petrarch never managed to warm her up.This is a consequence of how the model is parametrized. Laura’s appeal to Petrarch is a constant and it is set equal to -1. If we change this parameter to a positive number, say +1, the plot changes as in the following picture.

Laura’s feeling remain colder than Petrarch’s one, but over the years her love goes in the positive area and reaches an equilibrium value. So does Petrarch, but his feelings are stronger as he is more attracted by Laura (his appeal parameter is set equal to 2).

This model can be labelled as a predictive one because it studies the development of a long-term natural phenomenon that depends on the behaviour and interactions of different people. This is the type of model that we build on a daily basis here in Creme Global, thanks to our data scientist experts and cutting-edge technologies. To show the power of a predictive model, and for fun, we are creating a version of Rinaldi’s love model so you can try to simulate the pattern of your own relationship. Coming soon.

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