Note: While the relation between the subject and methods used to determine the subject’s conclusion are intended to be humorous, all data in this post is factual.
Dylan here. Did Aang and Katara kiss in the “The Cave of Two Lovers”? It’s one of the most asked questions still in the fandom. In this post, I explore the answer to that question.. scientifically!
Warning: This post is probably hard to understand for some people, skip to the conclusion if all of this makes no sense!
First, we need our data. Jake took screenshots for me of the kiss every .25 seconds, starting from this frame. That frame in my data is the time = 0 seconds. I measured the distance from Aang’s bottom lip to Katara’s bottom lip every .25 seconds of the kiss after that frame. Here is my data from 0 to 1.75 seconds:
Time (s)Distance (mm)0
From 0 to 1.75 seconds, Aang and Katara are moving in at a speed of 9.31 mm/s. Here is a picture of that data represented graphically on my TI-83 Plus.
A linear representation of this data is y=-9.31x+28.96.
Next in the data, Aang and Katara remain at the same place from 1.75 to 3 seconds. After 3 seconds, the torch is out and we cannot see how far away from each other they are. Now, after 3 seconds, it is extremely unlikely that they both would just stand there for any more time. So I assumed they continued to lean in after 3 seconds at the same velocity they did from 0 to 1.75 seconds, 9.31 mm/s. Here is that represented on my calculator:
The points to the left are our original 0 - 1.75 seconds data. The point in the line to the right is the last data point before the lights go out, 13 mm at 3 seconds. The line going through the point is the linear representation of the data from 0 - 1.75 seconds, shifted to the right by 1.25. The equation of the line is y=-9.31(x-1.25)+28.96. The last x-value on the screen is 5.75 seconds, the last frame at which the screen is dark, before the crystals appear. As you can see, the line crosses 0 well before 5.75 seconds. It crosses at about 4.37 seconds, indicating that they did indeed kiss while the screen was dark. In fact, the minimum speed needed for them to go after the screen went dark, before the crystals appeared, for their lips to touch, is 4.73 mm/s (for an explanation for how we got this, hit “Read more..”), well less than the predicted speed of 9.31 mm/s.
Conclusion: Aang and Katara did indeed kiss during “The Cave of Two Lovers”, and that kiss most likely lasted a minimum of 1.38 seconds.
Thanks so much to Jeff for his help with the math side of this post, and Jake for the screenshots!
(Hit “Read more..” for an explanation of how Jeff got to 4.73 mm/s as the minimum speed Aang and Katara would have needed to move in at for their lips to touch.)
When we apply a standard linear regression algorithm to the datasets before the pause and darkness, we come up with this equation: 9.31 * x + 28.96. we then shift the line to the right, to account for the 1.25 second pause, by altering the equation somewhat, to 9.31 * (x - 1.25) + 28.96. According to this equation, they will kiss at 4.36 seconds. This assumes that they begin moving at the same velocity, precisely when the darkness sets in. This gives them 1.39 seconds of kissing time. This means that, keeing velocity constant, they could’ve started up to 1.39 seconds after darkness kicks in. If we wish to determine the minimum velocity that will produce a successful kiss, we take the two datapoints, (3, 13) and (5.75, 0), and find the slope of the line that crosses these two points. Rise over run —> rise is -13 mm, run is 2.75 —> -13 / 2.75 == -4.727272…. So, they could’ve moved at 4.73 mm/s, starting immediately when darkness kicks in, and still kissed. The velocity predicted from their visible movement is 9.31 mm/s, almost twice the minimum.
Omg. Omg. I am blown away. And I thought I was obsessed?