Shakespeare's famous tragedy Romeo and Juliet tells the story of two young lovers who want to be together despite the protestations of their families. Although they know that they shouldn't be together, they can't help the way they feel. In fact, in the famous balcony scene ("Romeo, Romeo! wherefore art though Romeo?"), Juliet describes her love in the following way:
My bounty is as boundless as the sea,
My love as deep; the more I give to thee,
The more I have, for both are infinite.
Juliet's feelings only cause Romeo to feel even stronger for her, and this feedback loop of strong emotion, as we all know, does not end well. But with a bit of mathematics, could anyone have predicted their fate?
Let's suppose, for example, that each week, Romeo's love is 10% stronger than Juliet's love during the previous week. Meanwhile, Juliet's love is 30% stronger than Romeo's the previous week. In other words, if R(w) and J(w) indicate Romeo and Juliet's feelings in week w, we have R(w) = 1.1 × J(w - 1), and J(w) = 1.3 × R(w - 1). As long as Romeo and Juliet start out liking each other — that is, as long as R(0) and J(0) are positive — their love is destined to grow quickly and without bound. In fact, this will happen whenever their multipliers (1.1 and 1.3 in this case) are greater than 1, and their initial values are positive. But is this such a good thing? After all, if your love for someone else grows so quickly, at some point wouldn't it be difficult to do anything but obsess over the object of your affection?
Maybe it would've been better if Juliet hadn't liked Romeo initially; that is, if J(0) had been negative instead of positive. Or, what if Romeo was afraid of commitment, so that his multiplier were negative instead of positive? In other words, what if Juliet's strong positive feelings caused his feelings to sour, but her negative feelings made his feelings positive? By playing around with these parameters, you can try to come up with a more stable model for their love.
In all of this, of course, there are other factors we're neglecting. For example, Romeo's feelings probably don't just depend on Juliet's; they likely depend on his previous feelings as well. And the same is true for Juliet. Instead of each individual having only one multiplier, then, it's probably more realistic for them to each have two:
R(w) = a × J(w - 1) + b × R(w - 1),
J(w) = m × R(w - 1) + n × J(w - 1).
(In our initial example, a = 1.1 and m = 1.3.)
This opens the door to all kinds of personality types. For example, in the actual story, Romeo is pretty eager to love, and his feelings seem to improve as both Juliet's feelings and his own feelings grow (a > 0, b > 0). But perhaps a more reasonable approach is to be cautious, so that when your partner feels positive about the relationship, you do too, but you sometimes doubt your own feelings (a > 0, b < 0). There are several other personality types to consider as well.
If we incorporate these other factors, we can model even more types of relationships. You can create a few yourself using the interactive below (created with Desmos). Based on what you learn, would you offer up any advice to potential Romeos and Juliets? Which relationships do you think are the most/least appealing? And do any of them correspond to what you view as an "ideal" relationship?
Teachers: interested in having this conversation with your students? Then check out our latest lesson, Romeo & Juliet.
(Thanks as well to Steven Strogatz, whose 1988 paper first explored the intersection between Romeo, Juliet, and math.)