More Brillouin Zones with origami

These are the first and second brillouin zones of the bcc crystal structure.  This is a geometrical shape that is sometimes used in condensed matter physics, which is my field.

These models are much smaller and simpler than the model I created for the second brillouin zone of the fcc crystal structure:

That's great. Simpler is better.  Anyway, now I can cross this off my bucket list:

1. Be a physicist.
2. Get in the top 25 of the US Puzzle Championship.
3. Write a novel.
4. Learn to play "Pyramid Song".
5. Create the second brillouin zones of the fcc and bcc structures using origami.

I realize that I haven't written about physics in a few months (depending on what counts).  Writing about physics takes a lot more work than writing about other things.  Also, it is hard to be in grad school and not be all cynical and jaded.  But I wanted to write something about "coffee cups and donuts" topology in condensed matter, because that's fun.

Brillouin Zones with origami

(Click for a bigger version.  Guitar pick included for scale.)

As I said I would, I created the second Brillouin Zone of the fcc structure using modular origami.  It's a shape that's related to some crystal structures we study in condensed matter physics, though truth be told it is not a very useful shape.  It's pretty though.

It took 96 squares of paper, and four different kinds of modules.  It holds together, but I wouldn't want to toss it around like some of the more stable shapes.

It turns out that designing your own modular origami is nontrivial.  I started out by making the shapes exactly as they were in my origami book, and then I started making small modifications.  But the first few things I tried (with simpler shapes) were not very successful.  It's difficult to make them sufficiently stable.

Among the things I tried was the second Brillouin Zone of the bcc structure.  I have a small one pictured below, but it doesn't count because I cheated.  It completely fell apart and I had to cover it in tape.  I could probably just create 48 triangle modules, but I'm trying to think of a more elegant way.

Science and Religion illustrated

There are maybe three or four major views on the relationship between science and religion.  I'm in an artistic mood, so I decided to illustrate these views.  It's also an exercise in understanding one's opponents, though it is inevitable that I misrepresent them in some way.

1. Science and religion are non-overlapping magisteria.

NOMA is simple enough, and the easiest to represent.  I don't have a scanner, can you tell?

2. Science and religion should interact for mutual benefit

People with this view ask why there must be so much conflict between science and religion.  Clearly, we can only answer our most profound questions if they team up.

3.  Science and religion are in conflict, and religion is in the right.

This representation is the one I'm least sure about, since who knows what creationists really think?  I guess they think that science is only honest if it confirms religious precepts.  I'm not sure how to draw that.

4. Science and religion are in conflict, and science is in the right.

Since this is the camp I'm in, I can vouch for the accuracy of this representation.  However, I know it does not represent all views in this category.  Some might say religion should merely have a reduced role, or science should have an expanded role.

Platonic origami

Check this out:

I now own copies of Beginner's Book of Modular Origami Polyhedra: The Platonic Solids by Rona Gurkewitz and Bennett Arnstein, and Exquisite Modular Origami by Meenakshi Mukerji.

 

At this rate, I will soon be able to complete another one of my life goals.

miller's life goals:

1. Be a physicist.
2. Get in the top 25 of the US Puzzle Championship.
3. Write a novel.
4. Learn to play "Pyramid Song".
5. Create the second brillouin zones of the fcc and bcc structures using origami.

Pascal's swindle

Blaise wanted to borrow some dough
For an investment he said would grow.
"I'll pay back tenfold,
Before fall takes hold!"
But I frowned, and let my doubt show.

To every concern he'd reply,
"How can you ignore stakes so high?"
Then he'd up the reward
Infinite! Absurd!
All I could say was, "Nice try."

Suddenly I think we should turn all arguments for and against God into limericks.  Give them the level of seriousness they deserve.  But I don't like poetry, so someone else would have to do it.

The thief's assistant

Two college students on vacation were walking in through a small town, absorbed in an incomprehensible debate.

"...doubt that the soul operator would commute with the Hamiltonian."
"Why should--hey what's that over there?"

There had been a single figure walking up ahead of them, but now there was a second figure which looked much more threatening. A little old lady, the kind that always gets her purse stolen in comic books, was getting her purse stolen.

"HEY! What are you doing!" The thief looked up, and then ran away with the purse. One student stayed to make sure the lady was okay, while the other one chased after the thief into an alley.

But he ran straight into a big guy who had stepped in his way. "Whoa there. What's the rush?" The college student looked him up and down. He looked quite friendly, not threatening at all despite his size.

"I'm chasing a thief, he stole a woman's purse back there! Come on, help me!" The student tried to go around the big man, but it was a narrow alley, and the guy didn't seem to want him to go past. The student suddenly realized how odd it was that there could be such a narrow alley in such a small town.

"A thief? Here?" The big man looked the college student up and down. He looked like the kind of young adult who was always trying to be a hero, always imagining up dragons to fight. Idealistic. Arrogant. "There are no thieves in Townsville. Townsville is a friendly community."

"But he's right there! Hurry, before we lose him!"

The man turned his head towards the escaping thief. He doesn't react. "There are no thieves in Townsville. Don't you dare insult us."

The student sighed. There was no way he'd catch up anymore, not even to get a good look at him. He started to get a little angry at the big man blocking him. "Well then what do you call that guy who ran away?"

"If that were really a thief, he couldn't be a citizen of Townsville, much less a good citizen. He must be a traveler, much like yourself."
"But that's beside the point! You should at least have let me chase after him, if you weren't willing to help!"
"Why are you getting so angry at me? I am not a thief! How dare you accuse me."
"I'm not accusing you. It's not your fault if there is a thief in Towns--"
"I am a citizen! Citizens of Townsville never steal because they always ask politely first."
"I think you're misunder--wait, what was that?"
"I said that no proper Townsville citizen is a thief. I think you are misunderstanding me."

The student was quite exasperated and disturbed at this point. He was about to give up. "And yet there are still thieves."

"No one here is a--"

The student felt he had heard enough from this guy. "Okay, okay, no one in Townsville is a thief. Could you just direct me to the police already, so I can report this?" The man blocking him, satisfied that he had defended the dignity of his town, directed him to a dusty old building in the furthest corner of town.

Fractal Results

These are the results of the requests I got for Newton's fractals. Each function generates a fractal that, in principle, covers the entire plane, but I only show a small window of it. When I talk about the "range" of the fractal, I am referring to the location and dimensions of the window I chose.

Susan asked for the hyperbolic trig functions. Actually, they look more or less the same as the regular trig functions, but that's okay because the trig functions turn out well.

This is the function cosh(z) in the range -.5 to .5 on the real (horizontal) axis, and -.5 to .5 on the imaginary (vertical) axis. Note that only the first six roots get unique colors--the rest are all black.


And this is the function tanh(z) in the range -3 to 3 in the real axis and -2 to 2 in the imaginary axis.

An anonymous commenter asked for the function e^-(ixcosx)+e^(xsinx). This is a complicated one, graphed from 0 to 2 in the real and imaginary axes. I suspect those black comb-shaped things are actually artifacts of my program, but it took such a long time to generate the fractal that I wasn't going to try to figure out how to get rid of them. Besides, they look cool.

Request a Fractal!

In a previous post, I explained how to make fractals using Newton's method. These fractals can be generated by a java program I wrote for a high school project. The main input is a mathematical function.

So you give me a mathematical function, and I will make a fractal out of it!

You do not need to understand how Newton's Method works to make a request.

The rules:

• Be creative! A simple function (like f(x) = x²) might not produce anything interesting. More complicated functions may not produce anything interesting either, but I can adjust them to make them interesting.
  
• You may use any of the following in your function:
• Any arithmetic: addition, subtraction, multiplication, division, in absolutely any combination.
  
• Imaginary numbers (represented by "i")
• Trigonometric functions: sin, cos, tan, etc.
• Inverse trigonometric functions: arcsin, arccos, arctan, etc.
• Exponentials: e^x, x^(-1/2), x^x, x^i, and so on.
• Logarithms
• Any, absolutely any combination of the above. Even something like x*(cos(tan(x))^(i/x))-log(i+arctan(x)) is possible. But please don't suggest anything that crazy.

More complicated functions tend to generate more complicated fractals; simpler functions tend to generate simpler (more elegant?) fractals. For examples refer to the bottom of my previous post.

Oh, and here's one more example to throw out there. This Mandelbrot look-alike was generated with the function (x+1)*sqrt(x).

I will post a bunch of these at a later date.

A time traveler's anecdote

5:00 PM, the dining hall

Me: "You know what? I just remembered..."

Suddenly, in the past...

9:30 AM, my room

Roommate: "Just so you know, Sarah is coming by later today. She's going to invite us to the group dinner."

Me: "Okay."

Meanwhile, in the future

5:00 PM, the dining hall

Me: "We forgot about Sarah. She was going to come over."

Roommate: "What about Sarah?"

Me: "Remember? You're the one who told me!"

Roommate: "I did?"

Me: "Yeah, we were going to eat dinner, but we forgot about it. And now, here we are, eating dinner."

Roommate: "Dude, it's lunch."

Time skips backwards

11:30 AM, the dining hall

Me: "Oh."

Me: "I have a poor sense of time."

Loosely based on a true story

Scratch paper art

Oh, so I never really said how the math camp went. It was really cool. Some of the lessons were really interesting. The kids were a lot of fun, most of the time anyways.

And now, I have a large stack of used scratch paper.

I think scratch paper is an art. It is very chaotic, yet there are hidden patterns everywhere. It is at once simple yet complex. It is a glimpse of the human mind, frozen in time.

Let me show you some scraps.

If you look closely, you might see the puppies and kittens. Also, base Fibonacci.

Exactly what it looks like: a bunch of random circles.

I am not actually sure that having an ellipsis full of sigmas is proper notation.

If you try to guess what this is, you are doomed to fail. I think I was trying to draw a polyhedron?

This one is actually very straightforward. It has to do with quadratic residues.

Base Fibonacci returns! Now with arithmetic.

This is an abstract drawing of something more concrete--namely, a cyclic string of binary digits.

And that's it for now. Hope you've enjoyed it.

No molecule of love

It's true, I'm a reductionist. Everything can ultimately be explained by its constituent parts. We're all just a bunch of quarks and leptons, as they say.

But I'm told this leaves no room for love. I'm told that there is no molecule of love. I'm told that they sure got me there.

It also leaves no room for milk. There's no molecule of milk, you know. Sure, there's lactose, but that by itself isn't milk. How can a reductionist like me live without milk? They sure got me there. I better fix my philosophy up before I break my bones.

I'm also told about God. God is like a puppy that hasn't learned its name yet. Here I am, talking about reductionism and love, when God suddenly pokes his nose in, as if to say, "You called?" And then suddenly everyone acts like we were talking about God all along. Oh, obviously, he came because he heard the word "love". So cute! And isn't Love a good name for our puppy? Oh, but you don't believe in Love. Do you kick puppies too?

Only if I can't see them.

Incidentally, there is no puppy molecule. The particle zoo has no room for puppies. No room for molecules either for that matter. Molecules are made up of many quarks and leptons, interacting in an intricate quantum dance. Molecules aren't elementary particles at all. That's why molecules can't exist.

A paradox! I see reductionism collapsing around me, as if it were an electron spiraling into a nucleus. How could it ever have been stable in the first place? How did I ever believe the contradicting ideas of milk and reductionism? That's as incoherent as believing in both the brain and consciousness, or both pages and books.

A solution suggests itself. Or rather, someone suggests it to me, as if they had been reading my thoughts. There are no pages or books. There can only be libraries. Libraries, love, and God. These are all eternal cosmic forces, who only occasionally manifest themselves into the illusion of molecules. Pages are not constituent parts, but consequences of libraries. People and their feelings aren't constituent parts of love, but consequences. Spiritu-

But the suggestion dissipates, becoming as ethereal as a puppy. Realizing what just happened, I back away slowly. Carefully. But I still end up kicking it.

Secrets of a blogger

I'm anonymous, of course. But I'm not really. I don't really hide my blog from anyone I know. I just don't feel the need to disclose my boring identity to the internet. I don't really blog about the personal details of my life much anyway. Utterly unrecognizable allusions are more my style.

Though I don't hide my views from the people I know, I don't usually make them clear either. Why should I? If I were a fervent liberal blogger, I wouldn't go around talking about that either. All this doesn't really matter that much, except on the internet, where talk is cheap. What matters in real life is how well you can reference pop culture.

It makes me wonder. To what degree are we anonymous denizens of the internet like the people who send postcards to PostSecret? Living double-lives... With that thought, I present my own not-a-secret. Why yes, I do enjoy making you work to find it.

Googleoetry

This little meme has been making the rounds in the blogosphere. It's Googleoetry! I basically look at all the Google search terms that have landed people on my blog, and then organize them into a poem. Here's one example on Greta Christina's blog (slightly NSFW language). Of course, Science after Sunclipse has been doing this since way back, but now I get to jump on the bandwagon!

Now, my blog is relatively small, so I have less search terms to choose from (none of which are dirty). Also, I don't know the first thing about poetry, nor do I even like poetry. Mostly, it's just fun to look through search terms to see what odd things people are searching for. So I guess this will be more fun for me than for anyone else. Well, here goes my random little poem.

Math is made up

"math is made up
a play called the leap of faith
what happen when you see the sunclipse blind
can you blink when your blind
changing the universe by simply observing it

where can i find skeptical for glasses
skeptic hell
hell skeptic
bad skeptics
critizing negativity

what is mankind without god
"pharynguloid minions"
we are alone in a uncaring universe
homeopathic precession
meaning of without care
picture of summer from tilt of earth

10% of the brain other 90% penguins
i am a cool guy
brick t. miller
"just because" "no reason" "no purpose"

-------------------------------------------

Maybe next time, I'll use my own search terms, which are pretty crazy too, and much more varied.

The mountain theologians

For the scientist who has lived by faith in the power of reason, the story ends like a bad dream. He has scaled the mountains of ignorance; he is about to conquer the highest peak; as he pulls himself over the final rock, he is greeted by a band of theologians who have been sitting there for centuries.

-Robert Jastrow, God and the Astronomers

After staying a while to share some stories with the theologians, the scientist begins to explore the surrounding area. Soon he realizes that the mountain goes much higher, but the path is poorly marked and obscured in fog. He points it out to the theologians, but they cannot see the markings.

"How did you get this far?"

"God guided us here."

"Can God guide you further?"

They cannot agree amongst themselves. Some declare they are already at the peak. Others speculate that there is no peak, and thus no reason to continue. Still others say, "Yes, God will guide us," and begin to wander in the direction pointed out by the scientist, to become forever obscured in the mist.

The scientist prepares to leave, bringing only a few theologians with him. He slowly continues to scale the mountain, meticulously checking every rock, and occasionally backtracking for days at a time.

Those would not be the last theologians he would pass by.

[Incidentally, Robert Jastrow died shortly before I wrote this, but I didn't hear about it until afterwards.]

Fractal Christmas Tree

Merry Christmas!
This is a fractal generated by a program written by yours truly. I like to show off. It would take a while to explain how I created this, but suffice it to say that it involves using Newton's method of approximating roots, as applied to the function f(x) = log(x²) over the complex plane.

Actually, since this tree was created using a mathematical process invented by Isaac Newton, perhaps it would more appropriately be called a Newtonmas tree. See, Sir Isaac Newton, unlike Jesus, was actually born on December 25th under the Julian Calendar. As a result, some people with a bad case of hero worship like to talk about "Newtonmas". Of course, the Newtonmas tree is usually an apple tree.

I see contemporary art

Quite a while ago, I visited a local art museum. I have a better appreciation for these things after I took an art history class in the summer. However, I only studied modern art, which refers to a period from around 1770 to 1940. The most interesting part of the museum for me was the contemporary (post-modern?) art exhibit. I don't understand what is up with art these days (after WWII). What is up with it?

There was some pretty cool stuff, like a creepy car sculpture. And there were also the stereotypical minimalist stuff like a blank canvas, an empty room, a glass cube. And then there's everything in-between. For example, there was a black canvas with a white rectangle in the middle, but upon closer inspection the rectangle was a detailed drawing of a house.

I think the minimalist stuff is what gives modern art a bad name in the public's mind. I'm inclined to agree with the critics. I mean, maybe the first time, it was novel and all. But there are only so many blank canvases you can see before they all look the same--just one. Unless you think it has some value outside of novelty, it's more or less worthless.

Oh! But here I am talking about it, which means that it must be good art! In my mind, this argument only really discredits its own premise. "Good" art does not mean "promoting discussion". Besides, if you think about it, this is just a very brief discussion of an entire category of artworks.

One thing that's good about minimalist art is that after thinking, "This is art?", my next thought is, "I should be an artist!" Behold! I call it The Net.
Forget rectangles! Hypercube net silhouettes are the new deal, because I, an artist, said so! And it's not "ms paint", it's my personal style! This piece is a perfect blend of minimalism, technology, and four-dimensional geometry. Did I mention that you have to see 261 of these "paintings" before you've seen them all?

Bonus points if you can identify the hypercube net in question. In fact, I'd be impressed if anyone had any clue what I'm talking about.

Blind coin-sorting

Difficulty: 4 of 10

I pull out a stack of pennies. I use pennies because I don't trust you. Last time, you stole a quarter. I know you did! I counted. You wonder where I got a stack of pennies, and what kind of person still carries around such worthless things. But before you can ask, I produce an ugly green and blue handkerchief and say, "Blindfold yourself." While you blink incredulously at me, I explain: "It's in the name of Reason!" I make sure to capitalize "Reason" as I speak. "Or Fun. Or Whatever Else it is that you like."

After some more incredulous blinking, you finally tie the handkerchief around your head. It smells faintly of skepticism. You hear the clatter of a stack of pennies being scattered on the table. I say, reassuringly, "Preliminary tests suggest that this is a fun 'party trick,' as they call it." I make quote motions with my fingers as I say "party trick," but you can't see them, which is why I need to type out what I'm doing. You wonder why it is that you are performing the party trick and not I.

I boldly declare, "There are five tails showing among the pennies on the table! Your task is to sort all of the pennies into two groups, each group showing an equal number of tails!"

I ignore your incredulous blinking, almost as if I could not see through your blindfold. After a bit of hesitation, you pick up a penny, and try to feel whether it is heads or tails. "No!" I shout, surprisingly close to your ears. "There is an easier way, without tricks! Think!" You start to mumble that you feel uncomfortable, but I cut you off with another, "Think!" I go on to reassure you, "If you can't figure it out, don't worry. You'll only be doomed to blindly flip and sort coins until I post the solution in a few days."

<blatant genre switch>
Are you unable to solve the puzzle? Turn to page 2.
Do you solve the puzzle? Turn to the comments section and choose your own adventure.
</blatant genre switch>