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# Akano's Blog

## Equation of the Day #16: The Pentagram of Venus

Posted by Akano , in Math/Physics Feb 16 2016 · 852 views
Resonance, Pentagram, Venus and 1 more...

The above image is known as the Pentagram of Venus; it is the shape of Venus' orbit as viewed from a geocentric perspective. This animation shows the orbit unfold, while this one shows the same process from a heliocentric perspective. There are five places in Venus' orbit where it comes closest to the Earth (known as perigee), and this is due to the coincidence that

When two orbital periods can be expressed as a ratio of integers it is known as an orbital resonance (similar to how a string has resonances equal to integer multiples of its fundamental frequency). The reason that there are five lobes in Venus' geocentric orbit is that 13–8=5. Coincidentally, these numbers are all part of the Fibonacci sequence, and as a result many people associate the Earth-Venus resonance with the golden ratio. (Indeed, pentagrams themselves harbor the golden ratio in spades.) However, Venus and Earth do not exhibit a true resonance, as the ratio of their orbital periods is about 0.032% off of the nice fraction 8/13. This causes the above pattern to precess, or drift in alignment. Using the slightly more accurate fraction of orbital periods, 243/395, we can see this precession.

This is the precession after five cycles (40 Earth years). As you can see, the pattern slowly slides around without the curve closing itself, but the original 13:8 resonance pattern is still visible. If we assume that 243/395 is indeed the perfect relationship between Venus and Earth's orbital periods (it's not; it precesses 0.8° per cycle), the resulting pattern after one full cycle (1944 years) is

Click for hi-res image.

Which is beautiful. The parametric formulas I used to plot these beauties are

Where t is time in years, r is the ratio of orbital periods (less than one), and τ = 2π is the circle constant.

## Gravitational Waves at LIGO

Posted by Akano , in Math/Physics Feb 11 2016 · 370 views
Gravitational waves, LIGO and 2 more...
Direct detection of gravitational waves has been confirmed! Another milestone for physics!

Also, Einstein was right again, for those of you keeping track at home.

Dangit, Einstein.

## Lemuria

Posted by Akano , in Video Games Feb 04 2016 · 390 views

When I first played Golden Sun, I had no idea that this was actually a mythical place that some supposed existed in the real world. I also had no idea that lemurs were named with a common etymology.

## Golden Sun

Posted by Akano , in Video Games Jan 25 2016 · 577 views
Golden Sun, Isaac, Mia and 2 more...
Just finished replaying the first game in the series, and it's still awesome. The world, the music, the characters, the story – all are so wonderful and charming. The soundtrack has to be some of the best music in video game history.

After the long, gruelling process that is inputting a Gold level password from my Clear Data (I don't have a GBA link cable with me), I'm now starting The Lost Age. I will miss Isaac and Co. for now.

## Portal 2

Posted by Akano , in Video Games Jan 19 2016 · 414 views
GLaDOS, ATLAS, P-body and 2 more...
Just started playing this with my roommate (Friend: Toa of Friendship). I love it so much. The single-player campaign was awesome, and now we're doing the extra "future" co-op levels.

I love ATLAS' design. I wish he and P-body had opposite eye color.

Also, the free DLC is called Peer Review. This speaks to me as a scientist.

Cave Johnson is the best. I could listen to him talk forever about ignoring human rights in the name of advancing science.

## January update

Posted by Akano , in Life Jan 15 2016 · 475 views

Haldo, BZPorples,

I hope everyone had some awesome holiday funtimes! Mine were packed with traveling, visiting friends, gift giving and receiving, and all the food. All of it.

I also saw The Force Awakens twice while I was home. It was pretty fantastic. The part where we find out that Chewbacca is Rey's father was quite the twist![/trololololol]

Now I am back at school. Though classes don't start until next week, I'm in my lab typing this and sorta doing work. (I've been at a loss to find a certain physical quantity for the past week and have been trying to cope with this by watching various videos. Right now I'm watching Cosmos: A Spacetime Odyssey. When Knowledge Conquered Fear = WIN!)

Otherwise, things are going the way they've always been going.

## Equation of the Day #15: The Earth-Moon system

Posted by Akano , in Math/Physics Dec 16 2015 · 555 views
Equation of the Day, planet, moon and 1 more...
The definition of a planet has been under scrutiny several times, and with New Horizon's recent visit to Pluto, the discussion of Pluto's demotion was on everyone's minds (at least, back in July). But I'm not going to talk about Pluto's demotion (though I think it was totally appropriate from a scientific perspective). Instead, I'm going to talk about the Moon.

Should the Earth-Moon system be considered a binary planet? This sounds outlandish at first, since the Moon is a moon, obviously. It orbits the Earth as a natural satellite, just as the Galilean moons (Ganymede, Callisto, Io, and Europa) orbit Jupiter, Titan orbits Saturn, Triton orbits Neptune, and so on, right?

The definition of a moon is vague, and thus there are multiple ways of determining whether or not a planet-moon system is really a binary planet. One way of drawing the line between the two descriptions is by finding the barycenter (or center-of-mass) of the system. The center of mass of a collection of N masses is given by

where M is the total mass of the system, and mi and ri are the mass and position of the ith object, respectively. If the center of mass of a two-body system lies outside the larger object in that system, call it a binary planet. This makes sense, right? This means that the smaller body doesn't orbit the larger body, but instead they both orbit some point in space. For instance, the barycenter of the Pluto-Charon system lies outside Pluto (0.83 Pluto radii above Pluto's surface), the larger of the two bodies, while the Earth-Moon barycenter lies within the Earth (just under 3/4 of an Earth radius from the planet's center). By this definition, the Pluto-Charon system is a binary (dwarf) planet system, while the Earth-Moon system is is a planet-moon system. (Although, we are slowly losing our moon due to tidal acceleration. In a few billion years, the Moon will have drifted far enough away that the barycenter of the Earth-Moon system will leave the interior of our planet.) However, when you plug in values for the Sun-Jupiter system, you find that the center of mass lies outside the Sun! Indeed, Jupiter is the only natural satellite of the Sun for which this is true. (Does this mean Jupiter should have a different classification from the rest of the planets? Not really; the Sun is around 1000 times more massive than Jupiter, so the reason for this is that Jupiter is very distant from the Sun.)

Maybe a different definition is needed to distinguish planet-moons from binary planets, then, since the Sun-Jupiter system is not a binary star (Jupiter is slightly too small to generate nuclear fusion). Another proposition is to look at the so-called tug-of-war value of a body. The tug-of-war value of a moon determines which Solar System object has a stronger gravitational hold, the Sun or the moon's "primary" (the Earth is the Moon's primary). Using Newton's law of gravitation

we can take a ratio of the Sun's pull on a satellite to the primary's pull. The result is the tug-of-war value, proposed by Isaac Asimov.

Here the subscripts s and p refer to the Sun and the primary, respectively; m is the mass of the body referred to by the subscript; and d is the distance between the moon and the body referred to by the subscript. If the tug-of-war value is larger than 1, then the primary has a larger hold on the moon than the Sun, whereas if it's less than 1, the Sun's gravity dominates. For the Earth-Moon system, it turns out this number is 0.46, which means that the Sun pulls on the Moon with more than twice the force of Earth's pull. This is an oddity among moons, but is not unique. It does mean, though, that the Moon, when viewed from the Sun, never undergoes retrograde motion; it moves across the solar sky without changing direction. Another way to put this is that the Moon is always falling toward the Sun (like the planets), and never in its orbit does it fall away from the Sun (unlike most moons). If you look at the orbits of the Earth and Moon from the point of view of the Sun, they dance around each other in careful step, which is unlike most other moons in the Solar System. For Asimov, this was reason enough to consider the Earth and Moon as a binary planet system.

This tug-of-war value does not, however, classify Pluto and Charon as a binary dwarf planet system (they're too far from the Sun for their tug-of-war value to be less than 1). Perhaps the definition of a binary planet is a difficult one to pin down.

Should the Moon be promoted to planet, just as Pluto was renamed as a dwarf planet? I don't know, but it gives us something to think about as we look up at the starry night, watching the dance of all the chunks of rock and gas hurtling through space in our sky, to music written by nature and heard through science.

## Equation of the Day #14: Chaos

Posted by Akano , in Math/Physics Nov 08 2015 · 771 views
Chaos, deterministic, PRNG and 1 more...
I'm taking a second pass at this one. Instead, I'm going to talk about chaos.

Chaos is complexity that arises from simplicity. Put in a clearer way, it's when a deterministic process leads to complex results that seem unpredictable. The difference between chaos and randomness is that chaos is determined by a set of rules/equations, while randomness is not deterministic. Everyday applications of chaos include weather, the stock market, and cryptography. Chaos is why everyone (including identical twins who having the same DNA) have different fingerprints. And it's beautiful.

How does simplicity lead to complexity? Let's take, for instance, the physical situation of a pendulum. The equation that describes the motion of a pendulum is

where θ is the angle the pendulum makes with the imaginary line perpendicular to the ground, l is the length of the pendulum, and g is the acceleration due to gravity. This leads to an oscillatory motion; for small angles, the solution of this equation can be approximated as

where A is the amplitude of the swing (in radians). Very predictable. But what happens when we make a double pendulum, where we attach a pendulum to the bottom of the first pendulum?

Can you predict whether the bottom pendulum will flip over the top? (Credit: Wikimedia Commons)

It's very hard to predict when the outer pendulum flips over the inner pendulum mass, however the process is entirely determined by a set of equations governed by the laws of physics. And, depending on the initial angles of the two pendula, the motion will look completely different. This is how complexity derives from simplicity.

Another example of beautiful chaos is fractals. Fractals are structures that exhibit self-similarity, are determined by a simple set of rules, and have infinite complexity. An example of a fractal is the Sierpinski triangle.

Triforce-ception! (Image: Wikipedia)

The rule is simple: start with a triangle, then divide that triangle into four equal triangles. Remove the middle one. Repeat with the new solid triangles you produced. The true fractal is the limit when the number of iterations reaches infinity. Self-similarity happens as you zoom into any corner of the triangle; each corner is a smaller version of the whole (since the iterations continue infinitely). Fractals crop up everywhere, from the shapes of coastlines to plants to frost crystal formation. Basically, they're everywhere, and they're often very cool and beautiful.

Chaos is also used in practical applications, such as encryption. Since chaos is hard to predict unless you know the exact initial conditions of the chaotic process, a chaotic encryption scheme can be told to everyone. One example of a chaotic map to disguise data is the cat map. Each iteration is a simple matrix transformation of the pixels of an image. It's completely deterministic, but it jumbles the image to make it look like garbage. In practice, this map is periodic, so as long as you apply the map repeatedly, you will eventually get the original image back. Another application of chaos is psuedorandom number generators (PRNGs), where a hard-to-predict initial value is manipulated chaotically to generate a "random" number. If you can manipulate the initial input values, you can predict the outcome of the PRNG. In the case of the Pokémon games, the PRNGs have been examined so thoroughly that, using a couple programs, you can capture or breed shininess/perfect stats.

Dat shiny Rayquaza in a Luxury ball, tho.

So that's the beauty of chaos. Next time you look at a bare tree toward the end of autumn or lightning in a thunderstorm, just remember that the seemingly unpredictable branches and forks are created by simple rules of nature, and bask in its complex beauty.

## Halloween approacheth

Posted by Akano , in Video Games, Life Oct 28 2015 · 507 views
Halloween, Autumn, Fall, Mario and 1 more...
'Sup, BZPeoples?

I've been away a while attending friends' weddings, visiting various peeps, rereading Harry Potter (still as magical as always), and working on research. If all goes well, this Friday will see me submitting a paper to the Journal of Chemical Physics. All in all, it's been a good few months. But now is the season of autumnal haunts, and I have grown very happy with the drop in temperature and the swap of palette.

Over my fall break, I visited KK in the land of cheese, Wisconsin, and he introduced me to the joys of Punch Out!! for Wii and Super Mario Maker. I now own both, and have been having loads of fun playing others' levels and designing levels of my own (mostly goofing off). When I've finalized a level I'll post it for you all to play if you like. In particular I've been working on a Ghost House.

It's been super fun. Also, RubberRoss' levels are utterly evil.

EDIT: Untimely Haunt v2: CD1D-0000-00C8-8D19

Happy Halloween!

## A Brief History of Everything

Posted by Akano , in Math/Physics Jul 20 2015 · 372 views

feat. Neil deGrasse Tyson.

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Akano

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Name: Akano
Real Name: Forever Shrouded in Mystery
Age: 29
Gender: Male
Likes: Science, Math, LEGO, Bionicle, Comics, Yellow, Voice Acting, Pixel Art, Video Games
Notable Facts: One of the few Comic Veterans still around
Has been a LEGO fan since ~1996
Bionicle fan from the beginning