Lots of these photos on the internet today, for obvious reasons (this one from
The Atlantic). Am I correct in assuming that the lengths of the icicles on the lower power line would form a normal distribution (a bell-shaped curve without any skew?) Eyeballing it seems to suggest this, but is it fair to assume the shape of the distribution? I obviously don't have time to make the measurements...
Addendum: A tip of the blogging cap to reader Kniffler, who had ChatGPT analyse the image:
Normal distribution? Probably not. The cable is going to be hanging in a catenary, which means there will be a tendency for water to flow towards the middle, so you'd get slightly shorter icicles on the ends, longer in the middle. My hunch is that you'd get a better match with a catenary plus a noise function. And the noise function would give an average length plus a random variation, but adjacent icicles would probably also 'feed' off one another, smoothing things out. There's an academic paper's worth of analysis lurking here, for sure.
ReplyDeleteI see no evidence in the photo of a catenary curve. Those typically develop in power lines in hot weather when the metal expands. In winter cables typically tighten up, as shown here. My impression of a normal distribution stands.
DeleteI also had the feeling there was a papers worth of analysis available in this. Although I certainly lacked the insight about the curve of the cable.
ReplyDeleteAll suspended cables, chains etc. follow a catenary curve. Even a metal rod would. Shallowness of the curve varies depending on stiffness of the cable and the tension. And once the surface it's one is wetted, water will flow down even the shallowest of curves, so even though it's basicaly not perceptible I'd still expect a slight bias towards the centre. Icicle formation isn't fully random - even in the picture, while the lengths vary, the spacing between icicles is very uniform - so I'm dubious that a normal distribution is a good baseline model to use. I think it you measured the icicle lengths on the bottom wire and compared them to a normal distribution, the match would be poor. They don't look random enough to me. A lot of natural processes (esp. watery ones, such as, famously, wave sizes) tend to follow power laws, which give a different curve.
ReplyDeleteI suppose another approach would be to ask, if you wanted a bunch of icicles whose heights did follow a normal distribution, what conditions should you try to arrange?
I have considered actually measuring the icicles with my calipers. There are 114 of them. But life is too short. I'm moving on...
ReplyDeleteI got ChatGPT to do some image analysis, which was probably not quicker than getting a ruler out, but was kinda fun. You can see its final output below, which needed a few corrections to produce the graph.
DeleteThere aren’t really enough data points, but it looks like the curve might not even be smooth. There may be some variation in how stable the different lengths are.
https://imgur.com/a/tusJasp
Thank you, Kniffler. I'll interrupt my blogcation to post this for the readers to poder and argue about.
DeleteLooks like a nice noisy normal distribution to me. And it might not even be noise, just a small data set.
DeleteI think if it were graphed in intervals of 2 units rather than one, the curve would be smoother, have a taller peak, but still with a small skew to the left. But I agree that more icicles would be better.
DeleteScientific data is always imperfect. The data matters. Not the shape you can graph it in.....
DeleteOne of the biggest misconceptions is that scientific data taking is like taking out a tape measure and measuring the size of your door frame. It is not. Generally, you have a bowl of hot soup and are asked to measure the distance to Pluto. And then after having sorted out how you use a bowl of hot soup to measure a vast distance, you realize that the distance that you're trying to measure is rather dependent on the position of both earth and Pluto in time. A problem clearly not presumed in the original question. But remember, you only have a bowl of hot soup to solve this problem, no calendar or clock. Yet.
What I find even more interesting than the distribution of icicle lengths hanging from the wire is this: Why are the icicles hanging from the twigs on the tree normal to the tree twig rather than the ground? Seems like they should be hanging straight down.
ReplyDeleteI would bet the twig icicles began by descending vertically to the ground, but that over time the accumulation of weight on the end of the branch caused it to fall downward.
DeleteA THOUGHT EXPERIMENT....
ReplyDeleteLet's assume that each part of the wire gets the same amount of water. After all, the rain is not going to purposely distribute itself to fit a bell curve.
If that is true, and if also assume all other things are even (like temperature and wind), the would it not seem to follow that all the icicles would be of equal length. Because, again, as with the water, the, um, chief icicle is not going to be saying things like "Hey, Serge, you need to tighten up by about 40 millimeters so that we can do this fabulous thing I have in mind regarding bell curve distribution!" SMILE
So it seems to follow that, in theory, all of them would be of identical length. Yet clearly they are not. So it now would appear that all things are NOT equal. Something, somewhere and somehow, is playing upon/influencing these icicles to amuse us.
This is just my limited notion of the matter. But there is one more thing: It just might be that the bell curve is somehow a part of physics itself. That is, things adhere to the bell curve distribution because it is somehow part and parcel of the universal fabric. If so, we could say that the bell curve was not invented...but discovered.
An interesting thought. And perhaps related, when I wrote my comment about perhaps a smoother curve were it graphed in units of 2 instead of 1, I also realized that if the distribution were graphed in units of 50 (1-50, 50-100, 100-150...), that would "prove" that all the icicles are in fact the same size!!
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