Seriously, though, Spirit of Justice ruled. It definitely made up for Dual Destinies' lack of awesome. Also, the puns were taken up to eleven. No complaints here.
Although Case 4 was VERY out of place. Basically, the only thing worthwhile there was Blackquill's appearance. I didn't really like how they portrayed Athena in that one.
Yesterday I stumbled across this image (which I recreated and cleaned up a bit). It's a beautiful image. Arranged around the edge is the circle of fifths, which in music is a geometric representation of the twelve tones of the Western scale arranged so the next note is seven semitones up (going clockwise in this figure). The notes are all connected in six different ways to the other notes in the "circle," known as intervals, which are color-coded at the bottom. I thought, "Wow, this is a really cool way to represent this geometrically. How neat!" However, I found the original website that the image came from, and it's a pseudoscience site that talks about the fractal holographic nature of the universe. While fractals do show up in Nature a lot, and there are legitimate theories proposing that the Universe may indeed be a hologram, what their site is proposing is, to put it lightly, utter nonsense. But instead of tearing their website apart (which would be rather cathartic), I instead want to point out the cool math going on here, because that sounds more fun!
Looking at the bottom of the graphic, you'll notice six figures. The first (in red) is a regular dodecagon, a polygon with twelve equal sides and angles. This shape is what forms the circle of fifths. The rest of the shapes in the sequence are dodecagrams, or twelve-pointed stars. The first three are stars made up of simpler regular polygons; the orange star is made up of two hexagons, the yellow is made up of three squares, and the green one is made up of four triangles. The final dodecagram (in purple) can be thought of as made up of six straight-sided digons, or line segments. These shapes point to the fact that twelve is divisible by five unique factors (not including itself): one set of twelve, two sets of six, three sets of four, four sets of three, and six sets of two! You could say that the vertices of the dodecagon finalize the set as twelve sets of one, but they're not illustrated in this image. So really, this image has less to do with musical intervals and more to do with the number 12, which is a rather special number. It is a superior highly composite number, which makes it a good choice as a number base (a reason why feet are divided into twelve inches, for instance, or why our clocks have twelve hours on their faces).
The final dodecagram in cyan is not made up of any simpler regular polygons because the number 12 is not divisible by five. If you pick a note in the circle of fifths to start on, you'll notice that the two cyan lines that emanate from it connect to notes that are five places away on the "circle," hence the connection to the number 5. In fact, it would be far more appropriate to redraw this figure with a clock face.
This new image should shed some more light on what's really going on. The dodecagrams each indicate a different map from one number to another, modulo 12. The only reason this is connected to music at all is due to the fact that a Western scale has twelve tones in it! If we used a different scale, such as a pentatonic scale (with five tones, as the name would suggest), we'd get a pentagon enclosing a pentagram. Really, this diagram can be used to connect any two elements in a set of twelve. The total number of connecting lines in this diagram, then, are
where the notation in parentheses is "n choose 2," and Tn is a triangular number. This figure is known in math as K12, the complete graph with twelve nodes. And it's gorgeous.
So while this doesn't really have anything to do with music or some pseudoscientific argument for some fancy-sounding, but ultimately meaningless, view on the universe, it does exemplify the beauty of the number 12, and has a cool application to the circle of fifths.
where ħ is the reduced Planck constant, μ is the reduced mass of the electron-nucleus system, Z is the number of positive charges in the nucleus that the electron is orbiting, e is the charge of a proton, τ is the circle constant, ε0 is the vacuum permittivity, and ψ is the wavefunction. Solving this equation (which is nontrivial and is usually done after a semester of Advanced Quantum Mechanics) yields a surprisingly simple formula for the energies of the atom,
where h is Planck's constant, c is the speed of light, me is the rest mass of the electron, and n is any integer larger than or equal to 1. The constant R∞ is known as the Rydberg constant, named after Swedish physicist Johannes Rydberg, the scientist who discovered a formula to predict the specific colors of light hydrogen (or any hydrogen-like atom) would absorb or emit. Indeed, the formula I gave, En/hc, is equivalent to the inverse wavelength, or spatial frequency, of light that it takes for the atom in its nth energy state to free the electron of its atomic bond. Indeed, this was a puzzle in the early 20th century. Why was it that hydrogen (and other atoms) only absorbed and emitted specific colors of light? White light, as Isaac Newton showed, is comprised of all visible colors of light, and when you split up that light using a prism or similar device, you get a continuous rainbow. This was not the case for light emitted or absorbed by atoms.
The equation above was first derived by Niels Bohr, who approached solving this problem not from using the Schrödinger equation, but from looking at the electron's angular momentum. If electrons could be considered wavelike, as quantum mechanics treats them, then he figured that the orbits of the electron must be such that an integer number of electron wavelengths fit along the orbit.
Left: Allowed orbit. Right: Disallowed orbit. Image: Wikimedia commons
This condition requires that
The wavelength of the electron is inversely related to its momentum, p = mv, via Planck's constant, λ = h/p. The other relation we need is from the physics of circular motion, which says that the centripetal force on an object moving in a circular path of radius r is mv2/r. Equating this to the Coulomb force holding the proton and electron together, we get
Plugging this into the quantization condition, along with some algebra, yields the energy equation.
What's incredible is that hydrogen's energy spectrum has a closed-form solution, since most problems in physics can't be solved to produce such solutions, and while this equation only works exactly for one-electron atoms, it can be modified to work for so-called Rydberg atoms and molecules, where a single electron is highly excited (large n) and orbits a positive core, which need not be a nucleus, but a non-pointlike structure. In my lab, we consider two types of Rydberg molecules.
The example on the left is an electronic Rydberg molecule, while the one on the right is called an ion-pair Rydberg state, where a negative ion acts as a "heavy electron" co-orbiting a positive ion. To model the energies of these kinds of states, we use a modified energy equation.
where I.P. represents the ionization energy of the electron, and the new quantity δ is known as the quantum defect. It's a number that, for electronic Rydberg states, has a magnitude that's usually less than 1, while for ion-pair states can be quite large (around –60 or so in some cases); it in some sense contains information of how the core ion, e.g. H2+, is oriented, how the electron is spread over space, how its polarized, and so on. It's a vessel into which we funnel our ignorance in using the approximation that the molecule is behaving in a hydrogen-like manner, and it is surprisingly useful in predicting experiments. Currently my research involves studying electronic Rydberg states of molecular nitrogen, N2, and looking at heavy Rydberg states of the hydrogen molecule, H2 to gain a better understanding of the physics of certain states that have been experimentally observed in both systems.
Indeed, Horned Serpent is my Ilvermorny house by a decent margin, with my second place house being Pukwudgie, then Wampus, and Thunderbird being my least compatible house. (lolololololol)
My Hogwarts house, however, is not Ravenclaw, despite my scholarly ways. I'm a Gryffindor. Ravenclaw was indeed my next most compatible, followed by Hufflepuff and finally, by a large margin, Slytherin.
These results make much more sense to me. Remember, kids, larger sample sizes are better.
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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.