It's that wonderful time of year. You know, the snowy time of year! Of course, the snow isn't all grand - oftentimes, the snow and ice can cause dangerous situations on roads, walkways, and stairs. I'm sure at this time of year, we've all seen salt on one of these surfaces at some time or another. Most of us know that this is to melt the snow and keep ice from forming, but most of us do not know the reasoning behind it.
In the synthesis of chemicals (particularly in undergraduate lab courses), a melting point of the product will often be obtained to test the purity of it. This can be done because chemists are aware of the fact that impurities in any material will lower its melting point from that of the pure material. This melting point depression does not rely on what impurities are present in the material, but rather how much of the impurities is present.
From this, we understand why every winter thousands of surfaces are coated in salt - it lowers the point at which water will freeze, and is effective so long as enough salt is present to drive down the freezing temperature below the temperature of the air and the ground. This same method of lowering a freezing point is how we preserve our cars' engines in the cold weather. Anti-freeze chemicals are added to the water in internal combustion engines, which lower the temperature at which the water would freeze.
Sunday, December 21, 2008
Sunday, November 30, 2008
Probably the single most important thought in human history:
A little over 2000 years ago, Euclid gathered up the thoughts of his fellow Greeks on the topic of geometry into a coherent book, which attempted to prove everything about the subject from five little axioms. Euclid's idea was that by choosing a few axioms which were so obvious no one could argue against them, and using them to prove statements about his world, he could be absolutely certain about those statements, and no one could possibly argue against them either. The problem was that one of his axioms wasn't so obvious, or at least some people didn't think it was. There were other issues, regarding minor things which Euclid tacitly assumed, and which therefore had to be stated (and were stated, but only a couple thousand years later) for the proofs to be complete. But those aren't that important right now. What are important are his axioms, which are as follows:
The problem is that if you're looking for something so simple and so obvious that no one could possibly argue with it, something that seems right and that most people agree with, simply isn't good enough. Unfortunately, the fifth postulate was integral to Euclidean geometry, and plenty of people weren't willing to throw the project out completely. So someone had the idea, "What if we could prove the fifth postulate from the others?"
People tried. It was difficult, but people did come up with proofs. But every time someone came up with a proof, someone else realized that they were assuming some other postulate implicitly. Moreover, each of these other postulates was logically equivalent to Euclid's fifth postulate, meaning that, given the first four axioms, you could prove Euclid's fifth postulate if you assumed their implicit postulate, and you could prove their implicit postulate if you assumed Euclid's fifth postulate. Basically, the two are interchangeable, one is just a reformulation of the other. This is certainly interesting, but gets us nowhere in terms of proving Euclid's fifth postulate from the other four.
You can find a list of statements which are equivalent to Euclid's fifth postulate on Wikipedia. The most famous of these is known as Playfair's axiom: "Through a point not on a given straight line, one and only one line can be drawn that never meets the given line." Because of this formulation, which is significantly simpler than Euclid's original axiom, Euclid's fifth postulate is now often known as the parallel postulate. In fact, most modern formulations of Euclidean geometry use this formulation for reasons which should become clear momentarily.
This went on for the next two thousand years. Think about that figure for a second. This was an unsolved problem in mathematics for two thousand years. And it didn't get pigeonholed for a few centuries either. Even when Europe was deep in it's dark ages, Arab mathematicians were still trying to prove it. And still no progress was made until an Italian monk named Giovanni Saccheri decided to pursue an idea (no this isn't the all-important one, we're getting there) originated by Persian polymath Omar Khayyam. His idea was to try to prove Euclid's fifth postulate by contradiction. That is, he wanted to show that if you assumed that the first four axioms were true, and if you assumed that the parallel postulate was false, you would necessarily reach a contradiction. This would in turn show that if the first four axioms are true, the parallel postulate must be true as well, and he'd have a proof.
There are two ways the parallel postulate might be contradicted. The first is to assume that there are no non-intersecting lines. However, from this it can be proven that straight line segments cannot be extended indefinitely, in contradiction with the second postulate. (This is the foundation of elliptic geometry. See below.) The second is to assume that there is more than one line passing through a point not on the given line, which does not intersect the given line. Omar had not been able to find a contradiction given this, but Saccheri thought he might be able to.
Saccheri spent many years exploring the implications of contradicting the parallel postulate, and found that all sorts of wild theorems could be proven, such as the fact that rectangles could not exist, all of which seemed completely absurd to Saccheri, but still he couldn't prove the parallel postulate and he eventually gave up.
That was in the early 18th century. Almost a century later, his work was picked up by a Mathematician named Beltrami. Beltrami saw his work for what it really was - a completely different geometry. Hyperbolic geometry, as it is now known, is a geometry separate from Euclidean geometry, and incompatible with it, in the sense that it assumes the contradiction of one of the axioms of Euclidean geometry, so they cannot both apply to the same world.
But this was the big idea, possibly the most important idea ever: that mathematics was not about studying our world, but about studying possible worlds. Euclid's fallacy, and the fallacy of all of those who tried to prove the fifth postulate, was to assume that these were statements about the world we live in. If you assume that, then you can never prove anything, because you are always working off an assumption which may be wrong. The first four may seem accurate, but there is no way to know absolutely that they are, and so there is no way to be sure that anything you prove from them is true.
But, if we take each of these systems as interesting systems in and of themselves, regardless of whether or not they apply to the world we live in, then we can be certain that anything we prove from these axioms is true within the respective axiomatic system. This idea freed the thoughts of the mathematicians of the 19th century to come up with even more abstract ideas, such as Riemannian geometry, which is neither Euclidean nor hyperbolic nor elliptic (which assumes the contradiction of both the second and fifth axioms). In fact, thanks to Einstein and his professor Minkowski, we know (or are at least fairly certain) that the parallel postulate does not apply in our world, but that the presence of matter or of energy curves spacetime in such a way as to make straight lines (or geodesics, as they are called in Riemannian geometry) curve inward, over time, towards the center of whatever is creating the gravitational field.
Why is this idea so important? Without it, we wouldn't have differential geometry, or general relativity. There might not be any quantum mechanics, or at least not in the way we know it. And of course from quantum mechanics comes modern chemistry, and from that modern biology. How could we ever have discovered the structure of DNA without understanding X-ray diffraction? How could we have gotten as far as we have in quantum mechanics without Hilbert spaces?
This idea, which has revolutionized mathematics, and even, more recently, physics, and has led to so many developments in so many other fields, this, I would say, is probably the most important idea in human history.
- Any two points can be joined by a straight line.
- Any straight line segment can be extended indefinitely in a straight line.
- Given any straight line segment, a circle can be drawn having the segment as radius and one endpoint as center.
- All right angles are congruent.
- If two lines intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines inevitably must intersect each other on that side if extended far enough.
The problem is that if you're looking for something so simple and so obvious that no one could possibly argue with it, something that seems right and that most people agree with, simply isn't good enough. Unfortunately, the fifth postulate was integral to Euclidean geometry, and plenty of people weren't willing to throw the project out completely. So someone had the idea, "What if we could prove the fifth postulate from the others?"
People tried. It was difficult, but people did come up with proofs. But every time someone came up with a proof, someone else realized that they were assuming some other postulate implicitly. Moreover, each of these other postulates was logically equivalent to Euclid's fifth postulate, meaning that, given the first four axioms, you could prove Euclid's fifth postulate if you assumed their implicit postulate, and you could prove their implicit postulate if you assumed Euclid's fifth postulate. Basically, the two are interchangeable, one is just a reformulation of the other. This is certainly interesting, but gets us nowhere in terms of proving Euclid's fifth postulate from the other four.
You can find a list of statements which are equivalent to Euclid's fifth postulate on Wikipedia. The most famous of these is known as Playfair's axiom: "Through a point not on a given straight line, one and only one line can be drawn that never meets the given line." Because of this formulation, which is significantly simpler than Euclid's original axiom, Euclid's fifth postulate is now often known as the parallel postulate. In fact, most modern formulations of Euclidean geometry use this formulation for reasons which should become clear momentarily.
This went on for the next two thousand years. Think about that figure for a second. This was an unsolved problem in mathematics for two thousand years. And it didn't get pigeonholed for a few centuries either. Even when Europe was deep in it's dark ages, Arab mathematicians were still trying to prove it. And still no progress was made until an Italian monk named Giovanni Saccheri decided to pursue an idea (no this isn't the all-important one, we're getting there) originated by Persian polymath Omar Khayyam. His idea was to try to prove Euclid's fifth postulate by contradiction. That is, he wanted to show that if you assumed that the first four axioms were true, and if you assumed that the parallel postulate was false, you would necessarily reach a contradiction. This would in turn show that if the first four axioms are true, the parallel postulate must be true as well, and he'd have a proof.
There are two ways the parallel postulate might be contradicted. The first is to assume that there are no non-intersecting lines. However, from this it can be proven that straight line segments cannot be extended indefinitely, in contradiction with the second postulate. (This is the foundation of elliptic geometry. See below.) The second is to assume that there is more than one line passing through a point not on the given line, which does not intersect the given line. Omar had not been able to find a contradiction given this, but Saccheri thought he might be able to.
Saccheri spent many years exploring the implications of contradicting the parallel postulate, and found that all sorts of wild theorems could be proven, such as the fact that rectangles could not exist, all of which seemed completely absurd to Saccheri, but still he couldn't prove the parallel postulate and he eventually gave up.
That was in the early 18th century. Almost a century later, his work was picked up by a Mathematician named Beltrami. Beltrami saw his work for what it really was - a completely different geometry. Hyperbolic geometry, as it is now known, is a geometry separate from Euclidean geometry, and incompatible with it, in the sense that it assumes the contradiction of one of the axioms of Euclidean geometry, so they cannot both apply to the same world.
But this was the big idea, possibly the most important idea ever: that mathematics was not about studying our world, but about studying possible worlds. Euclid's fallacy, and the fallacy of all of those who tried to prove the fifth postulate, was to assume that these were statements about the world we live in. If you assume that, then you can never prove anything, because you are always working off an assumption which may be wrong. The first four may seem accurate, but there is no way to know absolutely that they are, and so there is no way to be sure that anything you prove from them is true.
But, if we take each of these systems as interesting systems in and of themselves, regardless of whether or not they apply to the world we live in, then we can be certain that anything we prove from these axioms is true within the respective axiomatic system. This idea freed the thoughts of the mathematicians of the 19th century to come up with even more abstract ideas, such as Riemannian geometry, which is neither Euclidean nor hyperbolic nor elliptic (which assumes the contradiction of both the second and fifth axioms). In fact, thanks to Einstein and his professor Minkowski, we know (or are at least fairly certain) that the parallel postulate does not apply in our world, but that the presence of matter or of energy curves spacetime in such a way as to make straight lines (or geodesics, as they are called in Riemannian geometry) curve inward, over time, towards the center of whatever is creating the gravitational field.
Why is this idea so important? Without it, we wouldn't have differential geometry, or general relativity. There might not be any quantum mechanics, or at least not in the way we know it. And of course from quantum mechanics comes modern chemistry, and from that modern biology. How could we ever have discovered the structure of DNA without understanding X-ray diffraction? How could we have gotten as far as we have in quantum mechanics without Hilbert spaces?
This idea, which has revolutionized mathematics, and even, more recently, physics, and has led to so many developments in so many other fields, this, I would say, is probably the most important idea in human history.
Sunday, November 9, 2008
Hello!
Welcome to our little corner of the internet!
My name is Gary and I'm a premed Biomedical Engineering major at the School of Engineering and Applied Science. I do some programming on the side (mostly web related work: PHP, mySQL etc). I'm also in training to become a EMT (Emergency Medical Technician) for CAVA.
Here at Get Quarky, I'll be making posts about interesting stuff happening in biology, talking about random tech stuff, and blogging about some of the projects I occasionally pursue. Whatever catches my eye really.
I promise to include PICTURES in some of my coming posts, and I'll sometimes be including little riddles or puzzles in my posts. There's a really simple one in this post (geared more towards web people *hint*). First to post the solution will get many* hugs.
*number of hugs limited to 3. Cannot be claimed on Mondays, Tuesdays, Saturdays, or on days divisible by 4 or 7.
My name is Gary and I'm a premed Biomedical Engineering major at the School of Engineering and Applied Science. I do some programming on the side (mostly web related work: PHP, mySQL etc). I'm also in training to become a EMT (Emergency Medical Technician) for CAVA.
Here at Get Quarky, I'll be making posts about interesting stuff happening in biology, talking about random tech stuff, and blogging about some of the projects I occasionally pursue. Whatever catches my eye really.
I promise to include PICTURES in some of my coming posts, and I'll sometimes be including little riddles or puzzles in my posts. There's a really simple one in this post (geared more towards web people *hint*). First to post the solution will get many* hugs.
*number of hugs limited to 3. Cannot be claimed on Mondays, Tuesdays, Saturdays, or on days divisible by 4 or 7.
This is Sam's Introduction
Hi, I'm Sam, a Math-Physics double major at Columbia College in Columbia University. I intend to go on to grad school to study theoretical physics, though I'm not sure what my specialty will be yet. I will almost certainly end up being a college professor. I really like explaining things, especially related to math and physics, to pretty much anyone who will listen.
I do quite a bit of programming , but mostly for my own purposes. In my free time, when I'm not doing math and physics problems for fun, I mostly play video games, including most major genres. I'm a huge fan of RPGs, and I'm working on playing through every Final Fantasy game.
I do quite a bit of programming , but mostly for my own purposes. In my free time, when I'm not doing math and physics problems for fun, I mostly play video games, including most major genres. I'm a huge fan of RPGs, and I'm working on playing through every Final Fantasy game.
Can My Air Filter Kill Me?
A product that has been on the market for quite some time is the ionic air filter. Now, one of the chemical byproducts of ionic air filters, beside clear, dust-free air, is ozone. Ozone is typically made when electric discharges pass through a gas that contains oxygen, such as air. Ozone can be made from static electricity, high-powered lightning bolts, and, yes, those ionic air filters. Unlike its cousin regular old oxygen, ozone can actually be smelled: it's that "fresh" smell after a lightning storm, or the scent of fleece-type clothing out of the dryer (not the detergent smell).
Typically, the air that we breathe is made up of inert gases, meaning that they do not readily react with anything we put into them. When an electric charge is put over gasses, however, the energy is sometimes transferred to them and forces them to change. In the case of the ionic air filter, the electric charge is generally low enough that it will not change the gases in the air, but a small amount of ozone does become effected by the charge regardless. While this certainly won't kill you, in order to prevent the effects of minor oxygen deprivation (because the production of ozone does use up some of the oxygen in the immediate area), keep ionic air filters in relatively open areas.
Typically, the air that we breathe is made up of inert gases, meaning that they do not readily react with anything we put into them. When an electric charge is put over gasses, however, the energy is sometimes transferred to them and forces them to change. In the case of the ionic air filter, the electric charge is generally low enough that it will not change the gases in the air, but a small amount of ozone does become effected by the charge regardless. While this certainly won't kill you, in order to prevent the effects of minor oxygen deprivation (because the production of ozone does use up some of the oxygen in the immediate area), keep ionic air filters in relatively open areas.
Spastic Intro!
Hallo everyone! I'm Ivy, a chemistry major who's also planning on studying Germanic languages. Thus, I'll be talking about chemistry on here periodically (not in German, because that would be silly. And I'm not THAT bizarre). I'm currently in an organic chemistry class, so it's likely that I'll be discussing common misconceptions or confusions that occur in the field of organic chemistry. Here at Columbia, along with all my studying and such, I am a cheerleader (psst...so is Mike! He succumbed to my peer pressure last year!). Contrary to popular belief, that doesn't mean I can dance or smile spectacularly, but I do lots of back flips, so it's all good. Ooh, and bagels are AWESOME. Not just because I am a Jew. Just because they are awesome. I love baking, too! But I don't really have time these days to do much of that. But baking is actually chemistry, yes? So maybe periodically I will post recipes! Maybe.
Also, I'd like to think that I'm not pretentious, but it's entirely possible that that is unavoidable since I'm a Columbia student.
Also, I'd like to think that I'm not pretentious, but it's entirely possible that that is unavoidable since I'm a Columbia student.
Hello!
Hey there,
I'm Charline and I'm a creative writing and English double major at Columbia University. For me to say that I am clueless about science is an understatement, but living in a suite with a number of science-oriented people has sparked my interest in the subject more than any high school physics class ever could. When not working on Get Quarky (or doing homework), I am usually off taking photographs, writing in my personal blog, publishing poetry books, or raiding Mike's leftover Halloween candy drawer. I will most likely be writing about how science relates to literature and the humanities, or transcribing some of the nerdy conversations that we all have in our suite.
Cheers!
I'm Charline and I'm a creative writing and English double major at Columbia University. For me to say that I am clueless about science is an understatement, but living in a suite with a number of science-oriented people has sparked my interest in the subject more than any high school physics class ever could. When not working on Get Quarky (or doing homework), I am usually off taking photographs, writing in my personal blog, publishing poetry books, or raiding Mike's leftover Halloween candy drawer. I will most likely be writing about how science relates to literature and the humanities, or transcribing some of the nerdy conversations that we all have in our suite.
Cheers!
Do I have to introduce myself? Can't I just be mysterious?
No, I'm kidding. I don't mind introductions, awkward as they may be. I'm Louise, nicknamed Lissa, I'm a sophomore, and I'm double majoring in Chemistry and Physics. I *wanted* to have a minor in Materials Science and Engineering (MSAE), but The Powers that Be said I can't cross schools. It's complicated and they're mean.
But anyways, even though I'm double majoring, Chemistry and Materials Science are my passions. I've done some research in MSAE at Johns Hopkins University, but none at Columbia (yet!). Like Ivy, I speak German, but she's more serious about it than I am. I also enjoy art, and make art and jewelry in my spare time (not that I have any usually, I mean seriously...) I attempt to sell said art and jewelry online, I think you can see my other blogs/Etsy websites in my profile. Yes, that was a shameless plug, but it won't happen often, promise.
I'm also in the Columbia University Marching Band (CUMB), and I love it. I play bass drum (the big one!) and twirl (pretty much anything I can get my hands on). It's a big part of my life.
To relax when I'm not running around like a chicken with my head cut off I play video games, mostly with Sam and Mike.
Oh, something else that is valuable in this discussion is how we all know each other! Last year, Ivy, Sam, Mike, and I lived in the same place. This year, Sam, Mike, Noah, and Charline live together, and Ivy and I hang around in their suite a lot. A LOT. We're goofy and fun, and generally enjoy hanging out with each other, which is probably good considering we're stuck together a good amount.
I think that's all! Signing off,
Lissa
But anyways, even though I'm double majoring, Chemistry and Materials Science are my passions. I've done some research in MSAE at Johns Hopkins University, but none at Columbia (yet!). Like Ivy, I speak German, but she's more serious about it than I am. I also enjoy art, and make art and jewelry in my spare time (not that I have any usually, I mean seriously...) I attempt to sell said art and jewelry online, I think you can see my other blogs/Etsy websites in my profile. Yes, that was a shameless plug, but it won't happen often, promise.
I'm also in the Columbia University Marching Band (CUMB), and I love it. I play bass drum (the big one!) and twirl (pretty much anything I can get my hands on). It's a big part of my life.
To relax when I'm not running around like a chicken with my head cut off I play video games, mostly with Sam and Mike.
Oh, something else that is valuable in this discussion is how we all know each other! Last year, Ivy, Sam, Mike, and I lived in the same place. This year, Sam, Mike, Noah, and Charline live together, and Ivy and I hang around in their suite a lot. A LOT. We're goofy and fun, and generally enjoy hanging out with each other, which is probably good considering we're stuck together a good amount.
I think that's all! Signing off,
Lissa
Hey
Hey,
My name is Mike. I'm am studying electrical engineering at the School of Engineering and Applied Sciences at Columbia University. I dapple in several areas outside of my major, and I have an avid interest in new and emerging technologies. I enjoy learning how new tools work, or even just learning how to get them to work. Feel free to ask any type of question, and I will do my best to answer it.
My name is Mike. I'm am studying electrical engineering at the School of Engineering and Applied Sciences at Columbia University. I dapple in several areas outside of my major, and I have an avid interest in new and emerging technologies. I enjoy learning how new tools work, or even just learning how to get them to work. Feel free to ask any type of question, and I will do my best to answer it.
Welcome to Get Quarky!
Hello all!
We're a group of Columbia University students dedicated to making science accessible to non-science people. If you've ever been frustrated by dense theoretical physics texts, confounded by common chemical processes, perplexed by the rapid-fire creation of new gadgets, or are simply curious about the intricacies of the world around you, we're here to help.
If you have a specific topic or question you're interested in seeing us address or answer, feel free to e-mail get.quarky@gmail.com.
Enjoy! We'll be introducing ourselves and elaborating on our fields of interest in the next few days, so stay tuned.
Best,
The Get Quarky Crew
We're a group of Columbia University students dedicated to making science accessible to non-science people. If you've ever been frustrated by dense theoretical physics texts, confounded by common chemical processes, perplexed by the rapid-fire creation of new gadgets, or are simply curious about the intricacies of the world around you, we're here to help.
If you have a specific topic or question you're interested in seeing us address or answer, feel free to e-mail get.quarky@gmail.com.
Enjoy! We'll be introducing ourselves and elaborating on our fields of interest in the next few days, so stay tuned.
Best,
The Get Quarky Crew
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