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Side View of World Record Bubble - courtesy: Shaun Li
Recently I had a bit of fun, helping Graeme Denton measure his Guinness World Record (GWR) Largest (Indoor) Soap Bubble.
It was a lot harder than I initially thought, so a few words seem in order.
Measuring tape was out of the question. Bubbles are far too fragile and transient for us to get up close and personal with one, so a bit of maths seemed the way to go. That’s why Graeme contacted me, his friendly, neighbourhood maths professor.
My first question was, "What does 'largest' actually mean?"
The GWR specifies it as the bubble enclosing the largest volume. I thought that estimating the volume might be easy, so I said "Yes" to helping Graeme. Little did I know what I was getting into.
I thought it might be easy because estimating volumes mathematically goes back thousands of years. The Ancient Greeks were mad keen on geometry, and one of the questions constantly kicking around was how to calculate the volume of regular solids such as the sphere or ball.
I thought that would help because small bubbles are pulled into a nice spherical shape by surface tension. That happens because a sphere or ball is the shape that has the smallest surface area for a given enclosed volume.
Earlier GWR bubbles were apparently measured using this type of approach, and initially I thought I had signed onto an easy project. However, a sphere is a reasonable model for a small bubble, but you can see in the photo that Graeme's bubble isn't all that spherical. Surface tension depends on surface curvature, so for big bubbles the tension is relatively weak, and so they don't get pulled into a nice ball, at least not quickly enough to be useful. I was learning already.
Archimedes (of Syracuse) is credited with the first insight on how to measure volumes of irregular solids, way back before the birth of Christ. It was the original Eureka Moment. He needed to work out the volume of a gold crown but wasn't allowed to melt it down to find out. One day, as he got into his bath, the water overflowed, and he realised that you could measure volume by seeing how much water was displaced. He was so excited by the discovery he leapt out of the bath, and ran into the street naked, shouting "Eureka!" (the Greek word for "I've got it.")Archimedes is famous for many other insights, and is indubitably one of history's greatest mathematicians, but I think this is his best known story. The story ends with Eureka, and so the results of his wet and wild run through Syracuse are not known. Unfortunately for us, his idea doesn't help us with bubbles. Bubbles don't displace, they enclose. If you submerged one, even if the soap film could survive, the bubble would be collapsed by water pressure.
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End View of World Record Bubble
Computer vision has come a long way in recent years. We have automated cars now, that presumably "see" the world around them. Estimating volume should be a snap for robots, compared to the task of driving, surely?
Again, it isn't quite so simple. Computer vision certainly could estimate the volume of a solid. We need photos from multiple points of view, and then we could perform a kind of stereo-on-steroids computation to work out the shape of the surface. It’s how 3D scanners work. But they have a couple of advantages. First, they have time. They need it, because they need pictures from lots of angles (unless you have a lot of cameras -- remember how they did the fight scenes in the Matrix, or am I getting old?)
Second, Computer Vision requires us to be able to find reference points on the surface that can be identified in multiple images, in order to match up these images from different angles. Bubbles provide neither of these two luxuries.
The biggest bubbles don't last long, at least not in Adelaide's dry air. The biggest might last only a few seconds. But more importantly, bubbles don't provide good reference points on their surface. We don't "see" the bubbles surface, so much as we see distorted reflections of various light sources. And what we see are "specular" reflections, which is a fancy way of saying the bubble's surface is a little like a mirror.
The problem with specular reflections is that all the viewpoints see a different set of reflections, so there aren't a lot of points in common between images.
No doubt there is someone clever enough out there to work out a way of inverting everything you see in a mirror to scan it, but a bubble has one more trick. Thin films reflect light from both their outer, and inner surfaces. When they are really, really thin (think a soap film), the two reflections can interact, either constructively reinforcing each other, or destructively interfering. The type of interference depends on the wavelength -- the colour -- of the light, and so when we look at a soap film (or oil slick) we see a rainbow effect. My thought was that these variations in colour would lead to an reflection-inversion problem that was definitely beyond my skills to solve.
So what can we do? We certainly can use images of the bubbles, but we have to understand that some approximation is in order. Mathematicians use approximations all the time, as do Physicists, Statisticians, Engineers and even Doctors (though they don't always know it).
Approximations allow us to take something very complex, and find a simple solution. It might not be perfectly accurate, but it’s simple enough to estimate from the type of photographs you or I could collect.
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Model of Bubble
The model we used here was based on the way Graeme forms his huge bubbles. The technique (used by most Big Bubblers) is to have a "tri-string wand" --- two long wands with a loop of rope suspended between them. The loop is dipped in the soap solution (whose recipe is a fiercely defended secret), and then the wands draw the loop through the air, leaving the bubble behind. It's the reverse of blowing a bubble. The loop is dragged along leaving a bubble with a cross-section shaped roughly like the loop. So the bubble itself is something like a (rather porky) sausage shape.
It would be simple, if the loop formed a perfect circle, to calculate the volume using cylindrical approximations. But the cross section isn't circular. For instance, it has angles arising from the points of connection between the loop and the wands. But the important point is we can see the cross-section. We placed a camera end-on to the bubble, and look down its length, and we can see this shape rather clearly.
The reason we can see a cross-section of a bubble is that when you look through the surface at right angles, the film is very, very thin -- perhaps as thin as ten nanometres. That's at least 1000 times thinner than a human hair. So you can hardly see anything at all. But when we look at the edge, you are looking through a section that is much thicker, and it can refract light away, leading to a change in brightness that is pretty easy to spot. So finding the edge of a cross-section of a big bubble isn't too hard.
We can also measure the length of the bubble, and its height from a side-view camera in the same way. The combination is enough to construct a mathematical model of our sausage-shaped surface. Once we have that, we use a mathematical technique called "integration" to calculate the volume inside the shape. Integration has been around since the days of Newton and his friends (and his enemies), but it’s a lot easier to do today with modern computer packages.
We still need to use Computer Vision techniques to correct perspective distortions introduced by the cameras before we can take accurate measurements. Remember that things further away look smaller -- that's a perspective distortion. We need to be able to reverse this effect to estimate, for instance, the length of the bubble.
To do that, we made sure that we had lots of measurements of the gym where the bubble was created. We measured the lines on the floor, and placed extra measuring scales in several places, so we would have accurate reference points. These allow us to calculate the precise characteristics of the camera (for instance its exact distance from the bubble), and hence estimate the effect of perspective distortions in the images, and then reverse these.
As you can imagine, this is a reasonably complicated series of steps, with the potential for errors at each stage. So one of the tasks I undertook was to make sure these errors were small. For some parts of the process that was easy. We could easily estimate the size of errors in the perspective transformation by taking some of our known measurements, and "re-measuring" them in the photographs. This type of error was pretty small (a few centimeters at worst). The hard part is to measure the errors introduced by the approximations in the model.
Instead, we decided first to eliminate any bubble from contention if it was too irregular. That was a pity -- I think Graeme's biggest bubble was one of these, but if we couldn't measure it accurately, we didn't include it.
Secondly, when we estimated cross-sections and lengths and so on, we did so conservatively. That is, if the bubble had a bulge, or a part that didn't fit the sausage shape (for instance the slightly jagged parts where the bubble finally closed) then we excluded these. So the bubble model we arrive at is actually slightly smaller than the real bubble.
Again, sorry Graeme, because I know my estimate is actually a lower bound -- the real bubble volume is actually a bigger. But I wanted us to be sure this estimate was water-tight (or for bubble's perhaps we should say air-tight).
So how big was big? The final estimate was 19.8 cubic metres. That was more than twice the previous record of 8.5 cubic metres. And from what I can see (and measure) Graeme can blow bubbles about this size all day (or at least until his arms get tired).
Is getting a bigger bubble possible -- it looks like it. The GWR largest bubble (outdoors) is enormous.
It makes it a little easier when you don't have to worry about hitting the ceiling. So Graeme has something to aim for next.
Below is Matt's complete paper on his mathematical work in this project:
References:
https://en.wikipedia.org/wiki/Archimedes
https://en.wikipedia.org/wiki/Eureka_(word)
https://www.scientificamerican.com/article/fact-or-fiction-archimede/