White Album 03-04: Isosceles Love Triangles

With both love interests now fully integrated into the plot the battle is finally joined. The question on everyone’s mind now is: who will win? Well fortunately for you guys we can mathematically determine this.

Early returns for Setsuna look good

Early returns for Setsuna look good

I could write about the events of White Album 2 episodes 3 and 4 but that’s boring and I’m sure you’d all rather do some math instead. I won’t completely stray from the show in question since today we will be covering the mathematics of love triangles. In order to do this it is useful to start with the basics of the love triangle. The most basic form of the love triangle is the equilateral love triangle. Under an equilateral love triangle each potential love interest has an equal chance of winning:

Equal chances at least for a while anyway

Equal chances at least for a while anyway

At the other extreme the love triangle becomes a line at which point one character has a 100% chance of winning and the other has a 0% chance of winning:

Sometimes glasses aren’t all you need

Sometimes glasses aren’t all you need

Other examples include:

Obtuse Love Triangle

Wherein affection for one member of a love triangle forces the other members apart

Actually the bottom left might just hate everybody now that I think about it

Actually the bottom left might just hate everybody now that I think about it

Isosceles Love Triangle

Wherein one character shows a distinct but not absolute preference for one of the other two members

Preferences not final

Preferences not final

The movement of points on the love triangle is governed by the triangular area stasis theorem which states that, all else equal, any movement of one point on a love triangle will cause movement by the other points to maintain the area of the triangle constant. When combined with the vertex gravitational constant which governs attraction of nearby triangle vertices it follows two close points will naturally move together and this action will naturally force the third point away.

It has been theorized that it is possible for a love triangle to ultimately progress not into a line as seen above but rather into a point or “love singularity” as all three points on the triangle come together. This requires an extraordinary amount of energy in order to overcome the triangle’s natural desire to maintain constant area. Some scientist claim to have created such a singularity in the laboratory but there is no record of anybody successfully accomplishing such a feat in the wild.

Misfortune has befallen all who try

Misfortune has befallen all who try

Given we know the properties of love triangles we should be able to determine the probability of a given character winning given the geometry of the love triangle in question. Through observation it has been determined that the probability of the farthest triangle member winning a given love triangle is a Bernoulli random variable with probability p given by the square of the ratio of [short side]/(sqrt(2)*[long side]).

The distribution is not purely linear because love triangles exhibit some degree of stickiness in practice. Thus when the geometry is very much in one person’s favor it will tend to stay that way. This is the basis for the “First Girl Wins” theorem which states that, all else equal, the first member of a love triangle to appear will be more likely to win than the subsequent members.

So where does this leave us with White Album? If you observe this screenshot we can calculate the relative distances from Haruki to Setsuna and Kazusa:

Super accurate distances (without units) written here

Super accurate distances (without units) written here

As the distances shown here indicate, the ratio of Setsuna/Kazusa is 0.75 which based on the above definition would indicate a probability of 0.71875 that Setsuna wins compared to a probability of 0.28125 that Kazusa wins (and a 0.0 probability that Iizuka wins). Given these probabilities and the fact that Setsuna has first girl status going for her it should come as no surprise that I will predict the winner to be:

Kazusa

That’s right. The reasoning behind this is that although love triangles display stickiness in the short run they also tend to regress to equilibrium in the long run provided two vertices do not move too close to each other. Since we are only on episode 4 and the love triangle ratio for Setsuna is only a modest 1.333 (repeating of course) it is actually statistically more likely that Kazusa will win based on the long run love triangle distribution which has a peak probability of 0.85 for [short side]/[long side] ratios of ~0.8. Ergo, Setsuna is screwed and I’m bummed out for the next 8 weeks. Damnit math, why do you have to ruin everything?

(Note: all probabilities and equations generated here are not back-fitted values based on the author’s observations of the interactions of the show referenced here but rather the result of many hours of in depth calculation and experimentation performed in the Love Lab. Question their validity at your own peril)
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This entry was posted in Fall 2013, White Album 2 and tagged , , , , , , . Bookmark the permalink.

2 Responses to White Album 03-04: Isosceles Love Triangles

  1. BrandonR says:

    Math is tough but fair.

  2. Pingback: White Album 2 06: Being Setsuna is Suffering | Pedantic Perspective

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