Posts Tagged ‘Mathematics’
This book was a captivating read… but not exactly what I was looking for when I read it. Though flavorful — and I can appreciate that this is book is written for a specific audience that I might not be a part of — I felt that Aczel could have dared to present a little more mathematics in a few places. There were about two or three pages devoted to silhouetting Cantor’s diagonal proofs for the countability of the integers and reals, but besides occasionally inserting a statement of the continuum hypothesis he shied away from presenting anything much deeper than a layman’s explanation of some very important mathematics.
I’m glad I read this book, and I still would have if I’d known more about the content ahead of time. It was well-composed and gave me lots of interesting trivia and historical context. Just know that if you’re looking for something that tells you much more about infinity in the mathematical sense than the first paragraph of the Wikipedia page, you’ll want to find a different book.
Last year, I learned about the layman’s essence of quantum mechanics, and I wrote a post about it on this blog. This semester, my big topic in physics class has been the Lagrangian and Lagrangian Mechanics. So, like last time, I’m going to write a nice lengthy post about it because:
- It might help someone else who wants a basic introduction, and
- It will definitely help me sort it out in my head, seeing as how my exam is this Tuesday.
Recall: Kinetic and Potential Energy
You might remember in previous physics classes a discussion concerning kinetic and potential energy. Recall that for any conservative force, E = T + U, where E is the total energy of the system, T is the kinetic energy, and U is the potential energy (I’ll use these conventions for the rest of this post).
You also might remember that using conservation of energy made some problems much easier to solve compared to using Newtonian methods. Consider, for example, the point-particle baseball thrown directly up in an air-resistance free world; we can find the maximum height by recongizing that at its maximum height the ball has zero velocity relative to the ground. Thus it has zero kinetic energy (because T=(1/2)mv2) and so its potential energy U=mghmax is equal to the total energy of the system. As a consequence, we can find the ball’s maximum height by setting E = E, or, equivalently, Umax = Tmax.
mghmax = (1/2)mv02
Where v0 is the initial velocity of the thrown ball. Note that we could also find the initial velocity by knowing the final height….
Principle of Least Action
To derive the basics of Lagrangian mechanics, we need to understand the calculus of variations, which is a topic beyond the scope of a blog post (and certainly beyond the scope of barebones HTML formatting).
En anglais, the principle of least action says this: a body moving from point A to point B will take the path that minimizes required action. Calculus of variations teaches us how to minimize an action integral in the general case. Now we can apply that to physics.
The Lagrangian L (usually written as a script L) is defined as:
L = T – U
Aside. I think it’s also valid to define L = U – T since we’ll be setting derivatives of the same function equal to each other so that signs will be irrelevant.
Then we can use the Euler-Lagrange formula (a result of calculus of variations) to say that for each generalized coordinate (xi) in our configuration space:
dL/dxi = d/dt [ dL/dxi' ] (*)
If you have trouble reading that, just look up the Wikipedia article on Euler-Lagrange; I don’t feel like going through the trouble of LaTeX’ing on this not-yet-configured machine just to point out that the partial derivative of the Lagrangian with respect to the generalized coordinate xi equals (under the condition of minimizing the action integral) the time derivative of the partial derivative of the Lagrangian with respect to xi‘ (or xi dot, the derivative of xi with respect to time).
These mysterious generalized coordinates can be whatever you want them to be as long as they can fully describe the system you’re concerned with. With a simple pendulum, you might just have the angular coordinate phi, which can alone describe any state of the pendulum. With a pendulum on a spring, you might have phi and x, the length of the spring. With an Atwood machine, you might just have one coordinate again that describes the length of the rope on one side (which describes the entire system given an ideal rope).
The set of your generalized coordinates forms the basis of a configuration space in which every possible state of the system is in the set spanned by those coordinates… I think. We didn’t really cover that in class too much.
Equations of Motion
So anyway, now we have these equalities as defined by the equation (*), and each equality for some coordinate y should include y” (because we’ve taken a time derivative of y’). Now we can re-arrange these as second order differential equations! Hurray!
I’ve been trying to come up with an idea for an application to write for the math4 program. I’ve mentioned this program before, and you can find more information at their wiki page: http://wiki.sugarlabs.org/go/Math4Team.
My idea is inspired by a leaning game we used to play in elementary school in 4th and 5th grade. We had these devices reminiscent of balance scales and we had to put different weights on one end to balance the weights on the other. The trick, of course, is that while the left side of the balance may have had a 10 gram weight, we could only balance it with a discrete number of non-10 gram weights on the other end. Mathematically, you might think of it like this:
10 = X1 + X2 + … + Xi
Where each X was some weight. The first idea of my Sugar activity was just to emulate this behavior with some basic drag-and-drop. I think that’s a good start, but then I wondered: can this be expanded to include variables?
I think the answer is that it can. Consider the polynomial (6X + 13). If we need to balance this on the right side of our scale, then we need two types of weights: weights of the form nX and weights of the form n. There’s really no way around this. This can be explained formally with linear algebra and vector spaces, but that would be beyond the scope of our 4th grade curriculum. On the other hand, grasping that concept in a more a priori way is something that I feel can be important to grasping the bigger idea of variables and equations.
That’s not that hard then. We can simply have a new class of weights which take the form nX. In fact, it could be easily expanded to include weights of the form nY, nZ, et cetera; once the kid gets the concept, they should be able to balance a polynomial of any order.
From there, we can probably do even more, but I don’t have any solid ideas yet. I just think that the framework of using a seesaw format like this has lots of applications for understand equality.
As I wrote that last sentence, I realized that we could have a dynamically changing equality sign that demonstrates how the equation is currently balanced (a >, <, or =). With that in mind, I’m off to start coding!
Disclaimer #1: This is not a political blog post. If you try to interpret it as one, then you’re interpreting it improperly.
Disclaimer #2: I’m not about to try and bash business majors or investment bankers. I’m just making a generic point.
Many Americans probably saw President Obama on Jay Leno last night. It was fun enough to watch, and I think was actually a good move by the president to bolster public support by dropping back down to the friendly, laughing American citizen status he had on the campaign trail instead of the all-powerful presidential aura he’s had to take on lately.
I agreed and disagreed with some things that he said, but one thing that I agree enough with to blog about is his take on the role of education and, perhaps more importantly, the tone given to the importance of certain career choices following from education.
I’m going to quote a large part of what he said last night, just so there’s some context. You can find a full transcript at the Huffington Post website (http://www.huffingtonpost.com/2009/03/20/obama-on-tonight-show-wit_n_177206.html).
Well, and part of what happened over the last 15, 20 years is that so much money was made in finance that about 40 percent, I think, of our overall growth, our overall economic growth was in the financial sector. Well, now what we’re finding out is a lot of that growth wasn’t real. It was paper money, paper profits on the books, but it could be easily wiped out.
And what we need is steady growth; we need young people, instead of — a smart kid coming out of school, instead of wanting to be an investment banker, we need them to decide they want to be an engineer, they want to be a scientist, they want to be a doctor or a teacher. And if we’re rewarding those kinds of things that actually contribute to making things and making people’s lives better, that’s going to put our economy on solid footing. We won’t have this kind of bubble-and-bust economy that we’ve gotten so caught up in for the last several years.
I can’t ignore that, as a physics major, I probably have some unavoidable bias here, but I think his argument also makes good sense. We need to reward jobs that contribute to society and make everyone better off rather than jobs which are focused on monetary and personal gain. This doesn’t mean that we don’t need bankers and business men; they’re obviously quite important to the business model that’s been in place for a long time. But we do need people to want to contribute to humanity in more real ways. At the same time, I think we would have a more enlightened culture as a whole if we focus on these types of fields. But that’s quite possibly my bias sneaking out.
And since this blog is syndicated on Fedora Planet, where everyone’s an open source contributer, I’m curious to see what everyone else thinks. Could this sort of mentality work for the country the same way our open source communities do? (As a side note, I’ve read some stuff recently about the government looking into adopting open source software, but that’s a blog for another day.)
By the way, this line of thought reminded me about the XO developer program that David Nalley talked about way-back-when. You can find the project page at http://wiki.sugarlabs.org/go/Math4Team.
I’ve just started reading Beyond Geometry, a collection of “classic papers from Riemann to Einstein”, which outlines the progression of the interpretation of geometry from Riemann’s time in the 1800s through the early 20th century. The introduction opens with a quote from Albert Einstein:
Only the genius of Riemann, solitary and uncomprehended, by the middle of the last century already broke through to a new conception of space, in which space was deprived of its rigidity and in which its power to take part in physical events was recognized as possible.
The beautiful thing about this book is the metaphysical nature of the papers. There are often technical details included, but on the whole the reader is presented with the philosophical divergence of some of the greatest thinkers since the Enlightenment. I have only had the chance to read the introduction and two of the eleven or so papers in the compilation, but some of what I’ve read so far is fascinating. Overly dramatic as I may sound, I think there’s something even romantic about debates over the nature of space.
The truths of geometry were synthetic a priori, meaning that their validity did not stem from experience, but from a synthesis conditioned by the nature of the mind itself.
Kant argued that we naturally intuit space to be Euclidean because it is “hard-wired” into our brain to so based on the information we receive from our eyes. This powerful argument, which is undeniably both reasonable and sensible, would undergo more than a century of critique (many geometers still debate the nature of space today). Science and mathematics would force the immediately logical Kantian view to be replaced with the bizarre (but interesting) ideas of modern physics and geometry. These arguments would start with Riemann, presenting his lecture “On the Hypotheses That Lie at the Foundations of Geometry” to Gauss and other German thinkers of his time, followed by Helmholtz, Klein, and eventually Einstein’s realizations of General Relativity with experimental proof to show that space is, indeed, not Euclidean.
This compendium describes what I find to be an astonishing timeline, outlining the triumph of human intuition over instinct, and a demonstration of how the groundwork of physics and mathematics is often moved by words and philosophy more than by equations and numbers. Definitely a suggested read for the interested academic.
We finally got to the beginnings of quantum mechanics today in our physics class. Today, for me, classical physics died; but I’ve also discovered how incredibly exciting modern physics is going to be. See, I knew about most of this stuff, like the uncertainty principle, and wave-particle duality and whatnot, but it just doesn’t mean as much to you until a professor – after telling you the truth all year and answering all your questions – simply can’t give you anymore answers.
For those of you who aren’t physicists – and for my own benefit of writing out the notes in my head – I’ll explain this problem in the simplest terms that I can here. You probably won’t get the full effect of this problem unless you’ve actually taken a class on it, though. And if you are a physicist, and I get something wrong, well… sorry. I’m just an undergrad.
The Double Slit Experiment
Light – not just visible light, but also radio, microwave, infrared, ultraviolet, X-ray, and gamma ray – is a wave. You may already know this. It’s definitely a special kind of wave, but it is a wave. So just like dropping two pebbles a pond, we can shoot a laser through two very small slits in a board and the light wave coming from each slit will interfere with light from the other slit.
You can see this roughly in the first image; and it happens (you can work it out if you like) that this interference pattern defines straight lines where the waves interfere constructively (two maxima on top of each other) and destructively (two minima on top of each other). When this light hits the wall, then, it doesn’t illuminate the wall evenly; it creates a diffraction pattern, which looks like the patterns in the second image.
So you might say – “Okay, fair enough. Light’s a wave and it interacts with other light waves. I haven’t taken much physics and I understand what’s going on; what’s the big deal here?”. Well, here’s big deal number one. It turns out that light is also a particle, called a photon. Photons have no mass and move at the speed of light, and they’re critical to how physics works. ”Are you sure?”, you might ask, “Because this wave thing works well enough, and I don’t see how particles could interfere like waves.”
Well, nobody thought they could before the 20th century. But here’s some proof that light really is a particle; you can have a light with a very low intensity that only emits a few photons per second. Take a look at the image to the right. This is a time lapse image of a light detector screen (like photo film) where the bright spots are photon hits on the screen.
Initially, the pattern looks pretty random. But as we get more and more light, the pattern we saw in the other images reveals itself. So it seems that even though the photons come one at a time, the wave interference pattern still occurs. This is the first step into the concept of wave-particle duality.
You might suggest that we put photon detectors inside our double slits so we know which slit the photon is going through and where it should end up. It turns out, however, that nature is a fickle trickster, so to speak, because the diffraction pattern disappears.
Let me say that again; as soon as we try to measure what’s happening with the photons between the laser light and the projection screen, the entire refraction pattern disappears, and it’s as if we were just shining a light at the wall the way we classically think of it happening.
This is because measuring the light interferes with it, messing up the entire wave pattern and scattering it, so it looks like light from a flashlight instead of the perfectly coherent light that comes from a laser.
Keep suggesting ways to avoid this, and I’ll keep telling you how you’re messing with the light. It just doesn’t work. We can’t know what’s happening between the laser and the screen, and we don’t know where the light will hit the screen until it does. In a way of speaking, the light exists everywhere at once from the slits to the screen until it gets to the screen… at which point it’s forced to isolate into a single position that we observe as the photon collision with the screen.
Big deal number two – mass refracts exactly like light. It just so happens that mass is also a wave.
“Whoa!”, you might say (that’s what I said), “hold on a second. I’m made of matter, but I don’t diffract when I walk through a doorway; isn’t that basically the same thing?”. Well, yes, and it just so happens that you do diffract when you walk through a doorway. But to avoid mathematics on what was going to be a fairly short blog post, I’ll just say that the diffraction is absurdly negligible. But much smaller things like electrons and even atoms diffract noticeably (with the right detection equipment). If you shoot electrons through a diffraction grating like a crystal, the atomic lattice works just like the double slit board and you will observe a diffraction pattern. Strange, but true.
You would think that since we’re using matter and not light, we can actually watch the electrons going through the air… right?
Wrong. Watching the electrons going through the air means that we need to shine light on them (to see them), which would interfere with the photons (because light actually has momentum). So we can do this… but the pattern collapses. You can try to think of any other way to observe the electrons, but the fact of the matter is that it won’t work. For us to get the information, something has to interfere with our sensory organs which was affected by – and, thus, reciprocally affected – the electrons, so the pattern collapses. All we can know is what happens at the end. This is the basic version of the uncertainty principle, although formally it involves the relation between observing mass and momentum.
If you’re still reading this, you might have some philosophical questions lined up. The uncertainty principle was most certainly (no pun intended?) something that had to be dealt with philosophically. The idea that light affected matter and wasn’t simply a vehicle for sensory observation certainly, to my knowledge, impacts several philosophers’ takes on perception and space. The Kantian view is certainly out, as I see it; space, in this case, is not simply a medium for sensory transaction, because it definitely cannot transmit a sensory experience of this mysterious waveform between the slits and the screen. If we cannot sense it, does it exist? For now, this problem is probably better answered over coffee than over a lab table.
Perhaps this is best approached from the views of eastern mysticism. In a way, this whole mystery seems like a zen koan – contradictory (wave or particle?), seemingly illogical – but it makes perfect sense when enlightened to the secret of the koan. We just need to be enlightened. Either way, the mystery of wave-particle duality has caught me today, and I’m looking forward to my future as a physicist.
I’ve always been a big fan of the entanglement between nature and mathematics, and one of the most famous associations between the two lies in the simple irrational number Φ (Phi). To define it in rational terms, we can simply call it Φ=(1+sqrt(5))/2. But there are plenty of less rigorous ways that we can discover the number in nature. Phi can be found in the logarithmic spiral of the nautilus shell, for example.
Less known to most people is that it can also be found, along with the Fibonacci sequence, in plant growth. As leaves spiral up around the stem of many plants, for example, you may find that the angle between two leaves as you travel up the stem of a plant is (a/b) * 360 degrees. The fact that the leaves spiral is simply a mechanism of plant growth so that higher leaves don’t block resources from lower ones, but, surprisingly, a and b are Fibonacci numbers for a very large subset of plants. The the Fibonacci numbers, of course, are related to Phi by a simple function which is, nonetheless, too complex to write here without LaTeX.
So anyway, I think Phi is interesting, so I picked up this book my Mario Livio called The Golden Ratio. Truth be told, I’m not done with it yet, but it’s pretty interesting so far, and it’s a very easy read. Altogether, this means cool I’m getting cool trivia knowledge for a low effort overhead, and that’s what I like. You’re welcome to look through the first few pages with Amazon’s book reader. I may check out some other books by this guy in the future.
If you have a chance to look into Math/Nature relationships, you should definitely do it. Fractals, irrational ratios, nonlinear dynamics… it’s all pretty neat. It starts to be less and less of a surprise that this number that governs so much of nature, including the human body (seen in this diagram), is the answer to so many unique mathematical problems. Φ^2 = Φ + 1, and Φ = 1/ Φ + 1, and… you get it. Beyond just the macroscopic proportions of the body, Phi also is found in the finger, hands, and face (among other places, to be sure).