According to the laws of physics laid down by Sir Isaac Newton, all non-accelerating observers witness the same laws of physics. This included an idea of spontaneity, the idea that someone traveling on the highway at 60 mph would witness an event occur at the exact same time as someone who was just sitting on the side of the highway at rest. The transformation from a reference frame in motion to one at rest for Newtonian physics is known as a Galilean transformation, where x is shifted by -vt, or minus the velocity times time. Under such transformations, laws of physics (like Newton's second law, F = ma, remain invariant (don't change).
However, during the 19th century, a man by the name James Clerk Maxwell formulated a handful of equations, known now as Maxwell's equations, that outline a theory known as electromagnetic theory. Of the many new insights this theory gleaned (among these the ability to generate electricity for power which every BZP member uses) one was that light is composed of oscillating electric and magnetic fields; light is an electromagnetic wave. By using his newly invented equations, Maxwell discovered what the speed of light was by formulating a wave equation. When his equations are used to describe electromagnetism, the speed of light is shown to be the same regardless of reference frame; in other words, someone traveling near the speed of light (as long as they weren't accelerating) would see light travel at the same speed as someone who was at rest. According to Newton's laws, this didn't make sense! If you're in your car on the highway and traveling at 60 mph while another car in the lane next to you is traveling at 65 mph, you don't see the other car moving at 65 mph; relative to you, the other car moves at 5 mph. The reason that light is different is because a different theory governs its physics.
This brought about a dilemma: is Maxwell's new electromagnetic theory wrong? Or does Newtonian mechanics need some slight revision? This is where Einstein comes in. He noticed the work of another physicist, Lorentz, who had worked on some new transformations that not only caused space to shift based on reference frames moving relative to each other, but also shifted time. Einstein realized that if light had the same speed in all non-accelerating reference frames, then objects moving faster experienced time differently than those that moved slower. This would come to be known as the Special Theory of Relativity.
How does this make sense? Well, if you have some speed that must remain constant no matter how fast one is traveling, you need time to shift in addition to shifting space to convert between both reference frames, since speed is the change in distance over the amount of time that displacement took place. If you have two reference frames with some relative speed between them, the only way to shift your coordinates from one to another and preserve the speed of light is if both frames experience their positions and times differently. This means that, if something moves fast enough, a journey will take less time in one frame than the other. Special relativity says that moving clocks progress more slowly than clocks at rest, so someone traveling in a rocket at a speed comparable to the speed of light will find that the journey took less time than someone who had been anticipating his arrival at rest. This also means that if someone left Earth in a rocket traveling near the speed of light and came back ten years later would not have aged ten years, but would be younger than someone who was his/her age before his journey took place. Weird, huh?
If you think this is crazy or impossible, there have been experiments done (and are still going) to try to confirm/reject the ideas of special relativity, and they all seem to support it. There's another relativity at play as well known as general relativity, which states that gravitational fields affect spacetime (the combination of space and time into one geometry). General relativity says that the higher up you are in a gravitational field, the faster clocks run (time speeds up). A proof of this theory is GPS; the satellites that help find your position by GPS are all higher up in Earth's gravitational field than we are, and thus their clocks run faster than those on Earth's surface. If general relativity weren't considered in the calculations to figure out where you are on Earth, your GPS would be off by miles.
Most of us are used to the real numbers. Real numbers consist of the whole numbers (0, 1, 2, 3, 4, ...), the negative numbers (-1, -2, -3, ...), the rational numbers (1/2, 2/3, 3/4, 22/7, ...), and the irrational numbers (numbers that cannot be represented by fractions of integers, such as the golden ratio, the square root of 2, or π). All of these can be written in decimal format, even though they may have infinite decimal places. But, when we use this number system, there are some numbers we can't write. For instance, what is the square root of -1? In math class, you may have been told that you can't take the square root of a negative number. That's only half true, as you can't take the square root of a negative number and write it as a real number. This is because the square root is not part of the set of real numbers.
This is where the complex numbers come in. Suppose I define a new number, let's call it i, where i2 = -1. We've now "invented" a value for the square root of -1. Now, what are its properties? If I take i3, I get -i, since i3 = i*i2. If I take i4, then I get i2*i2 = +1. If I multiply this by i again, I get i. So the powers of i are cyclic through i, -1, -i, and 1.
This is interesting, but what is the magnitude of i, i.e. how far is i from zero? Well, the way we take the absolute value in the real number system is by squaring the number and taking the positive square root. This won't work for i, though, because we just get back i. Let's redefine the absolute value by taking what's called the complex conjugate of i and multiplying the two together, then taking the positive square root. The complex conjugate of i is obtained by taking the imaginary part of i and throwing a negative sign in front. Since i is purely imaginary (there are no real numbers that make up i), the complex conjugate is -i. Multiply them together, and you get that -i*i = -1*i2 = 1, and the positive square root of 1 is simply 1. Therefore, the number i has a magnitude of 1. It is for this reason that i is known as the imaginary unit!
Now that we have defined this new unit, i, we can now create a new set of numbers called the complex numbers, which take the form z = a + bi, where a and b are real numbers. We can now take the square root of any real number, e.g. the square root of -4 can be written as ±2i, and we can make complex numbers with real and imaginary parts, like 3 + 4i.
How do we plot complex numbers? Well, complex numbers have a real part and an imaginary part, so the best way to do this is to create a graph where the abscissa (x-value) is the real part of the number and the ordinate (y-axis) is the imaginary part. This is known as the complex plane. For instance, 3 + 4i would have its coordinate be (3,4) in this coordinate system.
What is the magnitude of this complex number? Well, it would be the square root of itself multiplied by its complex conjugate, or the square root of (3 + 4i)(3 - 4i) = 9 +
We can think of points on the complex plane being represented by a vector which points from the origin to the point in question. The magnitude of this vector is given by the absolute value of the point, which we can denote as r. The x-value of this vector is given by the magnitude multiplied by the cosine of the angle made by the vector with the positive part of the real axis. This angle we can denote as ϕ. The y-value of the vector is going to be the imaginary unit, i, multiplied by the magnitude of the vector times the sine of the angle ϕ. So, we get that our complex number, z, can be written as z = r*(cosϕ + isinϕ). The Swiss mathematician Leonhard Euler discovered a special identity relating to this equation, known now as Euler's Formula, that reads as follows:
eiϕ = cosϕ + isinϕ
Where e is the base of the natural logarithm. So, we can then write our complex number as z = reiϕ. What is the significance of this? Well, for one, you can derive one of the most beautiful equations in mathematics, known as Euler's Identity:
eiπ + 1 = 0
This equation contains the most important constants in mathematics: e, Euler's number, the base of the natural logarithm; i, the imaginary unit which I've spent this whole time blabbing about; π, the irrational ratio of a circle's circumference to its diameter which appears all over the place in trigonometry; 1, the real unit and multiplicative identity; and 0, the additive identity.
So, what bearing does this have in real life? A lot. Imaginary and complex numbers are used in solving many differential equations that model real physical situations, such as waves propagating through a medium, wave functions in quantum mechanics, and fractals, which in and of themselves have a wide range of real life application, along with others that I haven't thought of.
Long and short of it: math is awesome.
Seriously though, my apologies for the lack of updating of my comics. I have not had as much time this semester to do things I enjoy as I would like (we kinda hit the ground running). I do hope to inspired soon and start working on a comic.
Another thing sort of slowing me down is that I'm getting used to using GIMP, which is quite different from Photoshop, which my new computer does not have. Thus, I have to make do.
In other news, this semester I'm taking quantum mechanics version 2.0 and classical mechanics. In both classes (ironically) we're working with Lagrangian mechanics, as the classical Lagrangian (the difference of kinetic and potential energy of a system) is useful in deriving equations to describe systems in both the classical and quantum mechanical regimes. In fact, when one uses the Lagrangian as a way to formulate wavefunctions of quantum mechanics, Hamilton-Jacobi equation (a classical physics equation) pops out of the Schrödinger equation! It's as though physics is self-consistent or something...
What happens is that the sun, whilst supplying our lovely planet with energy for us to live (yay, sun!)
also gives off these lovely rays in the energy domain of ultraviolet light, which most of you probably know as the reason we get sunburns when we are outside too long in the summer (boo, sun. ). Another detrimental effect of this UV radiation is the fading of LEGO pieces. Why/how does it fade them? Well, Earth is extra special in that its atmosphere has wonderfully healthy amounts of oxygen (O2) gas which we need to breathe and live (yay, oxygen!). Not only do we like oxygen, but so does ABS plastic, from which LEGO is made. The plastic has a compound in it that possesses bromine which, for those of you who do not know your periodic tables, is a halogen in the second to last column of the periodic table and is, thus, highly reactive when on its own. Fortunately, it is nestled in the ABS compound, but this doesn't quite satiate its need for buddies to bond with it (because it's greedy that way), so it decides to find more buddies to bond with in our own air – the very oxygen we breathe!
What does UV radiation have to do with this? Well, it turns out that bonding takes energy, and the bromine within the ABS does not have the energy by its lonesome to absorb a buddy oxygen from the air (since oxygen gas is fairly stable and thus requires more energy to separate). So, the UV radiation of the sun is just the kick it needs to bond with oxygen, thus producing this:
Ugly, huh? But, someone discovered that our friend hydrogen peroxide (with a catalyst found in oxi-clean detergents) is able to reverse this process with the help of – guess what – UV radiation. That's right, the same thing that triggers the fading is also what allows it to reverse! Weird, huh? I decided to try this process on some of my faded white and gray pieces from some of my older sets (circa 1998-2000, mostly Adventurers theme) and this was the lovely result:
The difference is like night and day. For those of you wondering if it affected the printing on the skulls of the skeletons or the minifig bodies, the answer is no, it did not. Truly remarkable and a relief that my old pieces can shine as the pearly whites they were meant to be.
Night Lord's Castle: Awesome. It's pretty much everything a creepy castle run by a vampire and witch should have: bats adorning the entrance and tower, a prison cell in the tall tower for the good guys, and the eerie crystal ball predicting the doom of anyone who dares oppose them make the atmosphere wonderfully appropriate. Basil's throne is a nice touch in the main room of the castle, and the large oak doors on the side give it a sinister majesty (note, I have it in the configuration of the front of the instructions/box, for those curious). Also, the fact that it's swarming with guards add to the "do not mess with us because we're evil and magical" vibe.
Alien Avenger: The epitome of a UFO. It's a giant (for LEGO) flying saucer manned (?) by three aliens and an android guy with two extraterrestrial buggies to explore various worlds. Oh, and Alpha Dragonis' ship detaches from the main vessel. Pretty darn cool. The rotating laser canons on the front and the magnetic buggy-lifting hose are great touches and makes me want to reenact a scene of aliens abducting cattle (I need to obtain some LEGO moos!).
Sorry for the lack of pictures. I promise when I get a decent picture-taking apparatus (the iPhone 3G's camera is surprisingly lacking compared to my old phone) I will post pics.
EDIT: Picture of the Alien Avenger:
I should also review some of my other 90s sets some other time while I'm at it. I miss those days.
My roommate dressed as Sherlock Holmes, and I dressed as his colleague, Dr. John H. Watson. I had a fun time. 8D
Oak Log Bans
Akano Toa of Electricity
Stone Champion Nuva
+2 for Premier Membership
+1 from Pohuaki for reporting various things in Artwork
Real Name: Forever Shrouded in Mystery
Likes: Science, Math, LEGO, Bionicle, Ponies, Comics, Yellow, Voice Acting
Notable Facts: One of the few Comic Veterans still around
Has been a LEGO fan since ~1996
Bionicle fan from the beginning
Misses the 90's. A lot.
Hieroglyphs And The Like
Awesome CartoonsNigel Thornberry - Feb 17 2015 10:43 AM
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Leave your oak logs at homeAkano Toa of Electricity - Feb 13 2015 12:58 PM