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Introduction
Introduction
This blog is best suited for undergraduates getting started in science, but I do my best to make it accessible to everyone. That being said, the topics included will probably focus on math, physics, showing what is capable with Mathematica, and anything else that comes to mind.
Benderama
Benderama
I love Futurama. It's just one of those shows that I can rewatch time and time again, even though I already know the plots. It combines my love of science and being able to sit down and just watch some TV and laugh for a while.
That being said, Futurama is littered with science in-jokes; and for those who may not have a math degree, these might go unnoticed. To that end I chose to do my final project for my recent science communication class, hosted by the awesome Dr. Regina Barber DeGraaff (who also hosts the super great Spark Science podcast and blog (which just got a pretty cool, new blogger) outlet.), on explaining some of these math in-jokes in the episode Benderama.
Final.wmvThe Beauty of Interactive Mathematics
The Beauty of Interactive Mathematics
Static Math = Boring Math
Static Math = Boring Math
Usually when most people think of math, they think that it's just numbers drawn on a board with black ink or chalk, or in the case of geometry, some circles with some notes about how one side is equal to another. This is boring, or at least not too engaging, you know it and I know it. But there are also some, in my own opinion, gorgeous depictions of results from mathematics. Even then, most presentations of these are static. We're going to see, with the help of the mathematical computation software Mathematica, that just by introducing interactivity in the CDF, a whole new world of mathematical allure.
Nodes of Revolving Arcs
Nodes of Revolving Arcs
When objects move in circles, they can move at constant angular speed (i.e. they go around the circlc at constant rate), but the actual distance they travel depends on the distance from the center of the circle (the radius).
We can see here by circular arcs that subtend the same angle (or rather, take the same mount of their respective circles), and how they move as time goes on, that there are sort of ''resonances'' that occur, or rather every so often there is a smooth line that forms and bends formed by the lack of arc. These resonances are best seen with a low density of arcs, while with a higher density, a more contiguous spiral forms (Which is also pretty).
Orbiting Points on Arbitrary Planes
Orbiting Points on Arbitrary Planes
Here we see two points connected by an emphasizing line that are stuck on the surface of spheres, but are orbiting in arbitrary planes. Basically, the points follow a path of constant distance from the center of their respective spheres, but the flate surface on which they travel can be arbitrarily oriented with respect to the conventional "xy" plane.
Now, I look at this and see the union of something that is random (the orientation of the planes), with something that is constant (the distance from each respective sphere), and that, again to me, is quite beautiful.
Another example, or rather a different application of this same idea, is to Bohr model of the atom.
Here we can see, if we employ a strong suspension of disbelief, that the two electrons (green) of a Helium atom orbit the nucleus (red). The same idea can be applied to the Hydrogen atom of course, but that's not very interesting, and heavier elements as well. That being said, when going to heavier elements, the number of electrons also increases (Cause that's how atoms work (i.e. not ions)), and this makes the visualization a bit more messy.
Orbits
Orbits
Another inspiration is from astronomy. Here we see the orbits of two planets, or just two points revolving on fixed circles, and how the distance between them changes over time. The interesting part here is that the distance, while certainly oscillatory, forms certain "anti-resonances" where there are significant places where the distance is minimized every revolution.
While the initial inspiration for this project was to model the motion of planets, it quickly evolved into seeing what patterns I could create with just specifying the different speeds of the points. In the end, I ended up making these models and patterns to which I am very satisfied. One of my favorites is the animation just above on the right, where the emblem traced out in the end looks like the rebel symbol from Star Wars.
Something should definitely be said about how particular choices of speeds yields particular designs. For instance, the figure in the middle above is mad from the speeds varying just slightly (this is also one of my favorites as well), whereas some others are chosen such that the speeds are rational multiples of pi different. Can you guess which ones?
All of these animations are available for you to interact with in the CDF linked at the end of the page, so you too can see what patterns you can create!
Triginometric Projections
Triginometric Projections
Another area of that some people may struggle with is trigonometry. Here we see exactly how the trigonometric functions relate to the circle. We do this by exploring the relationship of the projection of the circle onto each side of the bounding box. This really is just what trigonometric functions do. This is emphasized by the identity sin2(t)+cos2(t)==1, which is the same as the Pythagorean theorem for a unit circle. If we think about the Sin and Cosine as the fundamental building blocks, just like the legs of a right triangle, this relation ship is even more evident.
Since there are multiple system in which to think about trigonometry and circle geometry, we first investigate two dimensions.
Here we can see the actual relationship between the evolution of the trace of a circle and the evolving Sine and Cosine functions. The Sine function is just the projection of a circle that is being traced out onto the vertical axes, while the Cosine function is the projection onto the horizontal axis.
While the above demonstration shows the relationship of the standard trig functions to the circle in a kind of superimposed way, below we can see exactly how the graphs of the function over the interval [0,2π] relate to the position of a point traveling along the perimeter of a circle.
I find this demonstration to be a little more insightful because you can actually see the full plots of the Sine and Cosine functions, and a point traveling along both curves and the circle as well at the same time. Go figure, the evolution of the Sine and Cosine points exactly matches (at least in phase) with the point tracing out the circle.
Visual Decomposition of Fourier Series
Visual Decomposition of Fourier Series
In 1807, Jean-Baptiste Joseph Fourier came up with the rigorous idea to represent functions as an "infinite sum" (I put this in quotes because it's not technically correct to say that a series is an infinite sum, but rather an infinite sequence of partial sums), or series, of trigonometric functions. The name of this series is aptly named the Fourier Series (Wikipedia, Mathworld). The original idea was applied to the theory of how heat dispersed in a metal, but the idea soon became an entire area of both pure and applied math.
Here we explore how exactly the Fourier series of a function ''looks'' like. Really what we're looking at is the function itself on the left-most side of the picture, along with the trigonometric functions that, when added together, form the function you want to represent.
In this example, we see the Fourier expansion (out to the first few terms) of the exponential function. I picked this specific function to represent because it certainly is not periodic on this interval [-pi,pi], and it gave a good representation of the constituent functions that go into modeling this function on this interval.
Partitioning the Circle
Partitioning the Circle
Finally, the newest piece of the collection, we explore the patterns that arise when we look at points that are equally spaced on a circle. It turns out, as demonstrated below, that if a number of circles with equally spaced points are concentrically (one around the other) placed, the set of second dots form a horizontal parabola. I think the most interesting point about this is that, at least for me, I would never guess this would happen. So when I first discovered this phenomenon, I just thought to myself "Huh...That's a parabola. That's weird.".
Moving Math = Fun Math < Interactive Math
Moving Math = Fun Math < Interactive Math
We've explored a lot of different topics today, and hopefully you played with most of the pieces because that's what they're there for. The whole point of this demonstration is that you can tweak and play with different parameters to see how things will change, and most importantly to discover the patterns that arise for yourself. While looking at a graph of a function, or geometry on board, can be illuminating, I argue it can only be so to an extent. The most of math in this day and age can come from inquiry based learning, i.e. discovering it for yourself. Though it should not be lost on us to remember why we were interested in this particular article in the first place.
Step 0: To view the interactive content in the CDF, you must first download the free Wolfram CDF (Computable Document Format) player here.
For even more fun and adventure to play with, and even tweak the source code for these animations, check out the actual interactive version here.
All the images here were created by me, Nathan Chapman with the help of Mathematica 8.0 at Western Washington University.
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