You may have encountered the popular claim that \( 0.999... = 1 \), where the three dots signify that the decimal continues forever. This is a somewhat weird claim, since it would mean that mathematics is broken. There should be no way for two different numbers to have the same value. What makes it weirder is that this is quite popular claim. I've even seen mathematicians say that it's true! But is it though? One popular proof is to first denote \( S = 0.999...\) and then multiply by \(10\) to get \( 10S = 9.999...\) and subtract \( S \) from it, to get \( 10S - S = 9.000...\) and finally dividing by \(9\) yields \( S = 1.000... = 1 \) and we see that \(0.999... = 1\)! However, there's a problem. This short derivation is not strictly speaking correct. It is veeeery close to being correct, and to see why let's look at finite decimals first. Let's say that \(S = 0.999\) (note that this is not the same as \(S = 0.999...\) ). Let's do the same trick as ...
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The series of tubes that we call the Internet
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Maybe some people still remember the infamous statement made by the late Republican Senator Ted Stevens during the net neutrality debates in 2006, that the Internet is "a series of tubes." He used this statement to criticize the bill on net neutrality, because according to him, the Internet is not a "big truck" and tubes can get clogged. After the statement he was met with ridicule, because of a very limited understanding of the Internet, even though he was in charge of its regulation.
To me, the extreme criticism that Stevens got for the tube statement is a little weird, because let's face it, the Internet is, in all actuality, a series of tubes. And no, I don't mean in a metaphorical sense. Although, with the word tube, I don't mean plumbing pipes or whatever. I have no idea what Stevens meant with tubes, and I don't think that he made a good argument against net neutrality. The argument was just ridiculed for all the wrong reasons. I mean come on, he said that "an Internet was sent by my staff at 10 o'clock in the morning on Friday. I got it yesterday [Tuesday]." Well, if he was sent an Internet, I would expect that to take some time! And as for the part where he says "the Internet is not something that you just dump something on. It's not a big truck," he is comparing information that is literally moving at the speed of light, to trucks.
But anyway, back to the point. The Internet we have today is formed by a vast network of optical fibers. You can think of an optical fiber as a kind of "tube," because it functions exactly like one. Both fibers and tubes are just paths, along which things can flow. In optical fibers, the things that flow are pulses of light, and in plumbing pipes... Well, its something else. And as I mentioned in an earlier post, the modern internet would not be possible without optics. But why is this?
You might think that if we didn't have optical fibers, we could just connect everything with electrical cables, and there, problem solved. Since we have a cable for every job, such as the USB, HDMI, DP, ethernet, coaxial, and so on, then surely we have an electrical cable that could replace the optical fiber, right? The cable that has the highest potential to do this is the coaxial, but it has two large limitations: 1. the coaxial cable is not capable of ultra high bandwidth transmission on par with the optical fiber and 2. it is made of copper, which makes it expensive. In other words, if the Internet was based on coaxial cables, it would be really slow and reeeally expensive. The two things that people don't want to hear in the same sentence with Internet, oh dang it.
What makes optical fibers so special, is that they are cheap to produce, have very low losses, and information can be sent in the form of optical pulses, which travel at (unsurprisingly) the speed of light. Those pulses are being held in check by a phenomenon known as total internal reflection. This effect causes light to get confined inside a material that is optically denser than its surroundings, given that the angle of incidence is larger than the critical angle. For example, glass and water are more optically dense (high refractive index) than air (low index). Total internal reflection is quite easily achieved, and one of the coolest demonstrations of this effect is light that is guided by a stream of water, as shown here by James Dann.
In order to connect the world, these fibers have been laid all over, even across the Atlantic as undersea cables. The fibers themselves are really small, the core that guides the pulses can be just 9 micrometers in diameter (a human hair is about 30 to 100 micrometers) and there are usually about six of those in one cable. But all of the protective layers on top make the whole thing quite hefty, about the size that fits snugly in a sharks mouth. Oh right, sharks like to chew on them for some reason, which is a major problem.
Sharks are not the only issue, believe it or not. In my Master's studies we had a course on optical telecommunications, and back then, the professor would start the course by saying "the optical fiber features nearly unlimited bandwidth!" Now, just a few years later, I talked with the same professor about this topic again and he told me that he no longer says that in the course. We have almost reached the bandwidth limit of the optical fiber, even though we have used almost every trick in the book. We need to find some new bits if we want to increase the bandwidth further!
As a side note on net neutrality (US citizens, pay attention!), the whole point of that particular law is to keep corporations from making their own rules; a classical user vs. provider situation. That is, net neutrality is one government regulation that staves off nearly an unlimited amount of corporate regulations, and now the US seems to have decided that they prefer the latter.
You may have encountered the popular claim that \( 0.999... = 1 \), where the three dots signify that the decimal continues forever. This is a somewhat weird claim, since it would mean that mathematics is broken. There should be no way for two different numbers to have the same value. What makes it weirder is that this is quite popular claim. I've even seen mathematicians say that it's true! But is it though? One popular proof is to first denote \( S = 0.999...\) and then multiply by \(10\) to get \( 10S = 9.999...\) and subtract \( S \) from it, to get \( 10S - S = 9.000...\) and finally dividing by \(9\) yields \( S = 1.000... = 1 \) and we see that \(0.999... = 1\)! However, there's a problem. This short derivation is not strictly speaking correct. It is veeeery close to being correct, and to see why let's look at finite decimals first. Let's say that \(S = 0.999\) (note that this is not the same as \(S = 0.999...\) ). Let's do the same trick as ...
It's that time of the year again, the time to look back on what was accomplished. I couldn't find the time or the energy to write this sooner, and I kind of left this to the last minute. But well, here it is now, the photonics news that really caught my eye this year: OSA Photo Contest winner Here is the photo that won this years annual OSA Photo Contest! Shot by Tobias Tieß from Leibniz-IPHT, Germany, the photo features a glass cup filled with fluoresciing liquid and a UV-laser is coupled into the handle of the cup. Quite nice effect, I must say. The rest of the contestants can be found here . Nobel prize The Nobel prize in physics was awarded "for groundbreaking inventions in the field of laser physics," To Donna Strickland, Gerad Mourou and Arthur Ashkin. I made a whole post on this, which you can find here for more info. SI system redefinition The SI system underwent a major overhaul! Or to be more precise, it was decided that it will be o...
Another year, another review. 2019 was quite a wild year for me, and whew, what a decade it has been! It's hard to imagine that at the beginning of the decade, I was still in high school. In chronological order, starting from 2010: I graduated from high school, served in the army, moved across the country, got my BSc, met five Nobel laureates, started a scicomm platform, got my MSc, moved abroad for work, moved back, got my PhD, got married, moved again across the country, and got a postdoc position. I don't think there will be another decade quite like this one for me! What was your year/decade like? Share in the comments! The winner of this years OSA photography contest, by Dr. Pascale Müller and Prof. Dr. Dan Curticapean: the cross section of a bean, imaged with a scanning electron microscope and digitally colored. Anyway, let's get to the sciencey stuff; here are the news that caught my eye from the past year: Nobel prize in physics This years Nobel prize...
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