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Akano's Blog


Equation of the Day #10: Triangular Numbers

Posted by Akano , in Math/Physics Apr 04 2014 · 415 views
Triangular number and 1 more...
I like triangles. I like numbers. So what could possibly be better than having BOTH AT THE SAME TIME?! The answer is nothing! 8D

The triangular numbers are the numbers of objects one can use to form an equilateral triangle.

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Anyone up for billiards? Or bowling? (Image: Wikimedia Commons)

Pretty straightforward, right? To get the number, we just add up the total number of things, which is equal to adding up the number of objects in each row. For a triangle with n rows, this is equivalent to

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This means that the triangular numbers are just sums from 1 to some number n. This gives us a good definition, but is rather impractical for a quick calculation. How do we get a nice, shorthand formula? Well, let's first add sequential triangular numbers together. If we add the first two triangular numbers together, we get 1 + 3 = 4. The next two triangular numbers are 3 + 6 = 9. The next pair is 6 + 10 = 16. Do you see the pattern? These sums are all square numbers. We can see this visually using our triangles of objects.

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(Image: Wikimedia Commons)

You can do this for any two sequential triangular numbers. This gives us the formula

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We also know that two sequential triangular numbers differ by a new row, or n. Using this information, we get that

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Now we finally have an equation to quickly calculate any triangular number. The far right of the final line is known as a binomial coefficient, read "n plus one choose two." It is defined as the number of ways to pick two objects out of a group of n + 1 objects.

For example, what is the 100th triangular number? Well, we just plug in
n = 100.

T100 = (100)(101)/2 = 10100/2 = 5050

We just summed up all the numbers from 1 to 100 without breaking a sweat. You may be thinking, "Well, that's cool and all, but are there any applications of this?" Well, yes, there are. The triangular numbers give us a way of figuring out how many elements are in each row of the periodic table. Each row is determined by what is called the principal quantum number, which is called n. This number can be any integer from 1 to infinity. The energy corresponding to n has n angular momentum values which the electron can possess, and each of these angular momentum quanta have 2n - 1 orbitals for an electron to inhabit, and two electrons can inhabit a given orbital. Summing up all the places an electron can be in for a given n involves summing up all these possible orbitals, which takes on the form of a triangular number.

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The end result of this calculation is that there are n2 orbitals for a given n, and two electrons can occupy each orbital; this leads to each row of the periodic table having 2⌈(n+1)/2⌉2elements in the nth row, where ⌈x⌉ is the ceiling function. They also crop up in quantum mechanics again in the quantization of angular momentum for a spherically symmetric potential (a potential that is determined only by the distance between two objects). The total angular momentum for such a particle is given by

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What I find fascinating is that this connection is almost never mentioned in physics courses on quantum mechanics, and I find that kind of sad. The mathematical significance of the triangular numbers in quantum mechanics is, at the very least, cute, and I wish it would just be mentioned in passing for those of us who enjoy these little hidden mathematical gems.

There are more cool properties of triangular numbers, which I encourage you to read about, and other so-called "figurate numbers," like hexagonal numbers, tetrahedral numbers, pyramidal numbers, and so on, which have really cool properties as well.

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I lied

Posted by Akano , Apr 01 2014 · 143 views

This is why it's the best April Fool's Day ever.

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Best April Fool's Day ever

Posted by Akano , Apr 01 2014 · 173 views
BZPower, April Fools
Why, you ask? Because it gave me Unikitty as my spinny!

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Oh, hey, it's that topic I haven't updated in over a year.

I made myself sad.

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Together, We Ride

Posted by Akano , in Video Games Mar 21 2014 · 193 views
Fire Emblem
So, my roommate finally coaxed me into playing the first Fire Emblem (actually the seventh, but the first one released in the US) for GBA, and I'm loving it. After finishing it, I'm definitely looking forward to playing Fire Emblem: Awakening, because I've heard nothing but good things about it.

The music is great in that awesome nostalgic way, the characters are fun and memorable, the magic wielders are freakin' awesome, and the gameplay is fantastic – in order to keep all your troops alive, obtain everything worth getting in each level, recruiting all new troops, and beating the levels is a fun challenge.

Other things I've done in the past week or so include:
  • Finishing MetalBeard's Sea Cow (Awesome!)
  • Learning how to derive the formula for the volume of an n-dimensional sphere (really clever trick!)
  • Drinking lots of tea.
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Happy Half-Tau Day!

Posted by Akano , in Math/Physics Mar 14 2014 · 209 views

Read here to find out why π is wrong! 8D

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My Weekend

Posted by Akano , in Life Mar 11 2014 · 171 views
Boston, playin all the bars and 1 more...
So this weekend two of my friends and I journeyed to the New England city known as Boston. We didn't actually do anything downtown, but we did get to see some history in Concord, such as the Old North Bridge (site of the fabled "shot heard 'round the world") and walked past Sleepy Hollow Cemetery (we didn't have time to walk through it, unfortunately), and went to a birthday party for a friend of a friend. We also saw one of my grad school professors who left my school due to his wife and him getting a job somewhere that didn't involve one of them taking a nearly three hour commute to work, and there was Indian food.

All in all, a good weekend with good company. Now I'm just relaxing and recovering from the drive, and will soon get to go home and build a pirate ship.

My body is ready.

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So, I just ordered Metal Beard's Sea Cow...

Posted by Akano , in LEGO Mar 06 2014 · 313 views
There goes my, life savings
Worth it!

My wallet may be in a bit of pain, though...

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Equation of the Day #9: Time Dilation and Length Contraction

Posted by Akano , in Math/Physics Mar 04 2014 · 239 views
Special Relativity, Einstein
Today I want to talk about something awesome: Special Relativity. It's a theory that was developed by this guy you may have heard of, Albert Einstein, and it's from this theory that arguably the most famous equation in physics, E = mc2, comes from. I'm not going to talk about E = mc2 today (in fact, I've already talked about it, but it's not the whole story!), but I wanted to talk about two other cool consequences of Special Relativity (SR), time dilation and length contraction.

First and foremost, the main fact from which the rest of SR falls out is the fact that the speed of light is the same for all observers moving with constant velocity, regardless of what those velocities may be. Running at 5 m/s? You see light traveling at the same speed as someone traveling 99% the speed of light.

Wait, how can that be? This idea originally came from Maxwell's equations, which govern electromagnetism. When you solve these equations, you can put them into a form that results in a wave equation, and the speed of those waves is equal to that of light. This finding brought on the realization that light is an electromagnetic wave! But here's the interesting thing: Maxwell's equations do not assume any particular frame of reference, so the speed of the waves governed by Maxwell's equations have the same speed in all reference frames. Thus, it makes sense from an electromagnetic point of view that the speed of light shouldn't depend on how fast someone is traveling!

Now, we're still in a bit of a pickle; if all observers see light traveling at the same speed, how do things other than light move? Think about it. If you're driving down the highway at 60 mph and the car next to you is driving 65 mph, they appear to be moving 5 mph faster than you, don't they? So why doesn't this work with light? If I'm traveling 5 mph, shouldn't I see light moving 5 mph slower than normal? No; the problem here isn't that the speed of light is the same for all observers, but the fact that we think relative velocities add up normally. In fact, this relative velocity addition is simply a very good approximation for objects that are much, much slower than light, but it is not complete.

The answer to this conundrum is that length and time are different for observers traveling at different velocities. These two principles are governed by the equations

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The first equation determines time dilation, and the second equation determines length contraction, when shifting from a frame moving at speed v to a frame moving at speed v' (β and γ are both physical parameters that depend on the velocity of the frame in question and the speed of light, c). From the first equation, we can see that the faster someone is moving in frame S (moving at speed v), the slower their clock ticks away the seconds in frame S' (moving at speed v') and the more squished they look (in the direction that they're traveling). These ideas are the basis for the famous "barn and pole" paradox. Suppose someone is holding a pole of length L and is running into a barn, which from door-to-door has a length slightly longer than L. If the person runs fast enough, an outside observer will see that the person running with the pole will completely disappear into the barn before emerging from the other side. But from the runner's frame of reference, the barn is what is moving really fast, and so the barn appears shorter than it did to the outside observer. This means that, in the runner's frame, a part of the pole is always outside of the barn, and thus he is always exposed.

What if the observer outside the barn had the exit door closed and the entrance door open and rigs it such that when the runner is completely inside the barn, the entrance door closes and the exit door opens? Well, in the outside observer's frame, this is what happens; the entrance door closing and the exit door opening are simultaneous events. But in the runner's frame, there is no way for him to fit inside the barn, so does the door close on the pole? No, because the physics of what happens has to be the same in both frames; either the door shuts on the pole or it doesn't. So, in the runner's frame, the entrance door closing and the exit door opening are not simultaneous events! In fact, the exit door opens before the entrance door closes in the runner's frame. This is due to the time dilation effect of special relativity: simultaneous events in one reference frame need not be simultaneous in other frames!

Special relativity is a very rich topic that I hope to delve into more in the future, but for now I'll leave you with this awesome bit of cool physics. :)

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Posted by Akano , in D&D Feb 25 2014 · 286 views
Fad, Wizard, Lightning Bolt!
Not sure what started this fad, but I'm okay with this.

I Am A:

True Neutral Human Wizard (3rd Level)

Ability Scores:
Strength- 9
Dexterity- 12
Constitution- 12
Intelligence- 16
Wisdom- 14
Charisma- 12

True Neutral- A true neutral character does what seems to be a good idea. He doesn't feel strongly one way or the other when it comes to good vs. evil or law vs. chaos. Most true neutral characters exhibit a lack of conviction or bias rather than a commitment to neutrality. Such a character thinks of good as better than evil after all, he would rather have good neighbors and rulers than evil ones. Still, he's not personally committed to upholding good in any abstract or universal way. Some true neutral characters, on the other hand, commit themselves philosophically to neutrality. They see good, evil, law, and chaos as prejudices and dangerous extremes. They advocate the middle way of neutrality as the best, most balanced road in the long run. True neutral is the best alignment you can be because it means you act naturally, without prejudice or compulsion. However, true neutral can be a dangerous alignment when it represents apathy, indifference, and a lack of conviction.

Humans are the most adaptable of the common races. Short generations and a penchant for migration and conquest have made them physically diverse as well. Humans are often unorthodox in their dress, sporting unusual hairstyles, fanciful clothes, tattoos, and the like.

Wizards- Wizards are arcane spellcasters who depend on intensive study to create their magic. To wizards, magic is not a talent but a difficult, rewarding art. When they are prepared for battle, wizards can use their spells to devastating effect. When caught by surprise, they are vulnerable. The wizard's strength is her spells, everything else is secondary. She learns new spells as she experiments and grows in experience, and she can also learn them from other wizards. In addition, over time a wizard learns to manipulate her spells so they go farther, work better, or are improved in some other way. A wizard can call a familiar- a small, magical, animal companion that serves her. With a high Intelligence, wizards are capable of casting very high levels of spells.

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Physics Conference

Posted by Akano , in Math/Physics Feb 24 2014 · 208 views
As the fates would have it, the day after my birthday I hopped on a plane and went to a physics conference. I'm now sitting here in my lovely hotel room waiting for today's poster session at which I am presenting a poster on my research thus far. It involves stuff from my first published paper and some current "in the works" calculations that I'm doing to help our analysis along.

The talks up to this point have completely left me in the dust, so I'm hoping there will be discussion during the poster session that's more to my level of understanding on the various topics I've been exposed to.

Also the food is quite good.

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Oak Log Bans

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About Me

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

Name: Akano
Real Name: Forever Shrouded in Mystery :P
Age: 25
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

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My Lovely Topics

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Hieroglyphs And The Like

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