Jump to content

  • Log in with Facebook Log in with Twitter Log In with Google      Sign In   
  • Create Account

Welcome to BZPower!

Hi there, while we hope you enjoy browsing through the site, there's a lot more you can do if you register. Some perks of joining include:
  • Create your own topics, participate in existing discussions, and vote in polls
  • Show off your creations, stories, art, music, and movies
  • Enter contests to win free LEGO sets and other prizes
  • Participate in raffles to win LEGO prizes
  • Organize with other members to attend or send your MOCs to LEGO fan events all over the world
  • Much, much more!
Enjoy your visit!

Akano's Blog



Photo

Equation of the Day #18: 12

Posted by Akano , in Math/Physics Sep 07 2016 · 91 views
star polygon, dodecagon, music and 2 more...

Posted Image


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.

Posted Image


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

Posted Image


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. :)

Posted Image


Photo

Just beat the original Legend of Zelda last night

Posted by Akano , in Video Games Aug 09 2016 · 109 views
Zelda, Gannon banned, Link and 2 more...
Am I a true fan now?

Posted Image


Photo

Equation of the Day #17: The Rydberg Formula

Posted by Akano , in Math/Physics Aug 04 2016 · 109 views
Hydrogen, Atoms, Molecules and 2 more...
Hydrogen is the simplest and most common neutral atom in the universe. It consists of two particles – a positively charged proton and a negatively charged electron. The equation that describes the hydrogen atom (or any one-electron atom) in the nonrelativistic regime is the Schrödinger equation, specifically

Posted Image


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,

Posted Image ,


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.

Posted Image

Left: Allowed orbit. Right: Disallowed orbit. Image: Wikimedia commons


This condition requires that

Posted Image


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

Posted Image


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.


Posted Image


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.

Posted Image


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.

Posted Image


Photo

Gen 2

Posted by Akano , in LEGO Jul 29 2016 · 41 views
Johto, Ethan, Lyra, Kris and 1 more...
Guys, Gold and Silver were released almost sixteen years ago, why are you all—

OH! You mean BIONICLE...

...

There was a gen 2?!


Posted Image


Photo

Sorting

Posted by Akano , Jul 06 2016 · 24 views
Harry Potter, Pottermore and 2 more...
So, I decided to take the Pottermore Sorting quiz with all questions available on some third party quiz site, because while I love the Thunderbird, the house's traits do not really fit me (I can be adventurous, but let's face it: I'm a schooler). So, I took the full Pottermore quiz, and my results were somewhat predictable.

Posted Image

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)

Posted Image

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.

Posted Image



Photo

New Packermen

Posted by Akano , in Pokémon Jun 30 2016 · 44 views
Tapu Koko, New Electric types!
Posted Image

Well, I know whom I'm raising. (Dat color scheme, tho!) Kuwaganon looks pretty sweet, too.

Posted Image


Photo

Happy Tau Day!

Posted by Akano , in Math/Physics Jun 28 2016 · 108 views
C/r, circles
Eat twice the pi(e)! Light some fireworks! Marvel at the beauty of circles!

I love math. ^_^

Posted Image


Photo

My past three weeks

Posted by Akano , in Life Jun 10 2016 · 84 views

I had a whirlwind of a time these last few weeks. I'll try to break it down simply.

Week of May 23: Attended the DAMOP conference in Providence, RI to present my research in poster form. It was all right, but probably my least favorite DAMOP thus far. (Last year it was in Columbus, OH, which meant I got to see friends during the week, and the year before it was in Madison, WI, which was an absolute joy because Madison is a rarity in that it's a city I actually kinda like.) My roommate and I left the conference on Thursday to attend a wedding, which happened Thursday evening. The ceremony was one of my favorites I've attended thus far.

The following day I left on a plane to go to Columbus to attend the bachelor party, rehearsal, and subsequent wedding of a friend I've known and kept in touch with since 6th grade. It was wonderful. :)

Week of May 30: Went back to my childhood home and had a Memorial Day dinner with family and friends, played the role of babysitter with Tekulo (we earned major brownie points, both with the kids and their mothers). The next day I chilled at home and spent time with my family, with the evening punctuated by an awesome bonfire that was probably against local fire codes. Saw my next door neighbor unexpectedly and got a chance to catch up. Wednesday featured more catching up with some college friends who live about 40 minutes away from my parents' place. We played some Harry Potter Trivial Pursuit and watched YouTube videos.

Thursday was the beginning of a roadtrip from Ohio to Wisconsin, with the ultimate goal to meet up with KK and hang out for a few days. Among the people we saw along the way were two of my fellow grad students who were working in Chicago, my roommate's (Friend: Toa of Friendship) cousin, KK, and friends of Friend: Toa of Friendship's girlfriend (did you keep track of that? :P). It was awesome, but rather short, and we'll definitely have to do something again.

I got back to my apartment Wednesday night and am now enjoying time off relaxing before I go back to the lab on Monday to start up summer research mode! Huzzah!

Posted Image



Photo

In Honor of Astronomy Day

Posted by Akano , May 14 2016 · 103 views



In the highest exalted way.

Posted Image


Photo

Thoughts on Fates: Conquest

Posted by Akano , in Video Games Mar 21 2016 · 175 views

So much more enjoyable. I'm on Chapter 10 of Conquest (decided to switch it up), and already I feel much more positive. The characters have fun quirks and interactions, the plot is interesting, and I'm invested in finding out what happens next.

Glad that this version is my physical copy.

Posted Image







Oak Log Bans

Posted Image

About Me

Akano Toa of Electricity
Posted Image
Posted Image
Premier Members
Stone Champion Nuva
Posted Image
1,500+ posts
Posted Image
Proto
+2 for Premier Membership
+1 from Pohuaki for reporting various things in Artwork


Name: Akano
Real Name: Forever Shrouded in Mystery :P
Age: 27
Gender: Male
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.
Twitter: @akanotoe

Posted Image

My Lovely Topics

Posted Image
Posted Image
Posted Image

Hieroglyphs And The Like

IPB Image
IPB Image
IPB Image
IPB Image

Recent Comments

Approvals

Posted Image

Posted Image