Andy Barr's Blog

So, in spite all of the incredibly fun and apparently non-productive times I have had in Brazil, I also managed to do quite a bit of reading. Those many days I kept in the library or studying, alongside those long hours in the bus allowed me to tackle quite a bit of material.

One of the things that have been in my attention the last few months are complex numbers. I remember helping out my little sister with some complex number multiplication near the end of last school year and that kind of got me on the subject a little. I had done some exploration of the e^ipi = -1 formula a while ago and had learned to work a little with complex logs but that was just really algorithmic and without any understanding so at the end of spring quarter I decided to look into that topic. Starting with “An imaginary tale: The Story of “i” (by Nahin). If you haven’t read that book, I encourage you to go here and buy it right now. For less than 10 dollars you get hours of entertainment and a pretty good understanding of complex numbers in addition to great explanations of some pretty amazing mathematical magic. After tearing through that book and getting a good grab on most of the equations, I found “Hypercomplex Numbers” at my school (University of Washington) library’s bookshelf. After reading about complex numbers, I also became intrigued about the idea or possibility of “supercomplex” or 3d numbers and made some lame attempts to discover them by myself. hahaha Within a few days, I gave up and cracked open the “Hypercomplex Numbers” book and that was like the discovery of a whole new world to me. Like, for Seattle area people, I would put it into these terms. Redmond would be like just the normal 1-d number line, you know, that is what I grew up with. It is very familiar, but unfortunately, not all that exciting. I would put the complex numbers like me getting my license and getting to know Bellevue and the surrounding areas. Much cooler than just Redmond and I was free to do a lot more things. Then the quaternion system was like learning about Seattle.
Anyways enough of the talk, let me attempt to explain this concept for a non-math reader to get.
For those not familiar with the complex number systems, lets start with the natural numbers.

We can think of them as progressing in a line, like below:

The natural number line

It is a line. One dimensional, fairly-straightforward.

Now with the complex numbers, those weird things that you probably had to learn how to multiply some time in high school for some strange reason that was probably never revealed to you, are pretty straightforward too. The only real thing we have to add/define is the property that i = ( -1 )^2. Without really thinking how something like a negative square root would work, you still can figure out most of the properties of complex numbers, using only that one identity.

For those wanting a quick refresher, here is a short one below. If you feel comfortable with this, feel free to skip it.
i is just another number that you can multiply or add, just with a few added rules to the processes. For example, 4i + 9 is not something you can simplify any further. Those two terms, 4i and 9, are the two parts to only one complex number. Lets call that complex number z. z has both a complex part and a real part. Like everyone learns in 7th grade algebra, you can only add and subtract like terms. That is exactly the same in the complex system.
Multiplication is also pretty easy. For example, 7 * i just becomes 7i. 7i * i becomes 7*(i^2) and since i^2 = (-1), the product 7i * i ends up equaling (-7). Multiplying two complex numbers, lets call them y and z, is just done with the same FOIL (first-outside-inside-last) algorithm that you use with expressions like (4+x)*(5x-11). The only difference is you have to substitute any (i)^2 terms for (-1). So, if you haven’t worked with these for a while, a good exercise is to figure out what the product of complex numbers (a+bi) and (c+di) is.

So here is where the two dimensional extension of the number line comes in. Lets add another axis to our number line and call it the imaginary axis. The only difference with the numbering is that instead of using the real numbers we are used to, we use multiples of i. What is cool about that is now we have a visual way of representing, real, imaginary, and complex numbers on the same graph.

Three numbers plotted on the complex plane

Now that we have already gone over multiplication of two complex numbers, lets think of them as vectors. Since a complex number has two parts (a + bi), we can plot it on the plane. The difference with a vector is that it is not just a point, but rather a line that has a direction to it (represented by an arrow at one end). So lets multiply two numbers and plot them on the complex-plane. We shall do (2+i)*(1+i). The answer is (1+3i).

The multiplication of two complex numbers

Rather than just tell all the rules and properties, I encourage you to figure them out by yourself. Like see what happens if you multiply a complex number by itself. Is there something geometrical you notice that could make the process simpler than going through the FOILing? One thing to note is that “i” can also be referred to as the rotation operator. Why? Because multiplying by “i” rotates a point 90 degrees in the complex plane, enabling you to get back to where you started, just by successive multiplication. So it could be worth it to think about some of the implications of this for square roots, cube roots, and raising something to a power.

The most important part about the addition of “i” though, in my opinion, is the extension of the number system into two dimensions. You can get some really interesting functions with this new 2d system of numbers. It is worth trying to plot something simple (like z+2-i) at every point to see what it does. Is there anything you foresee being difficult about doing this. What dimension graphs do you need to represent a function of the complex numbers?

As you experiment and get a feel for the complex number system, you will see that it makes many two dimensional operations fairly easy. Things like rotations and description of transformations are easy to figure out. Since we live in a 3d world, there was an immediate desire to come up with a 3d system of numbers following the introduction to the complex plane. There is a famous story about the discovery of these and I encourage any reader of this to check out the Wikipedia entries on the Hamiltonian and the Quaternions. See, in order to work in two dimensions, mathematicians only needed to add the imaginary axis to have a complete set of complex numbers. However, things get a little more difficult in 3 dimensions and surprisingly, you end up needing to work with ordered groups of 4 to get an analogous expansion of the number system into 3 dimensions. These are called Quaternions. Well, technically quaternions are made from a four-dimensional basis so they are all 4d (a+bi+cj+dk) but they work really good for 3d problems too.

I am going to talk about Quaternions more tomorrow along with their relative, the octonions! For now, I am going to have a little sleeping break.

Well, I had some ideas for some music related scripts I wanted to make for programming practice and music practice purposes.

My first was something where you would really easily be able to hear and create complex polyrhythms. The idea was that you could define a time interval and then create different tracks, each with a noise, each playing a tone at a certain division of time. So, say your time interval is 2 seconds (rather long for most musical purposes but stick with me), and you wanted to hear a piano playing an A 4 times during that interval and a clarinet playing a C 3 times during that interval.

Well, with and MIDI sequencer, working with 4 over 3 (anything really related to 2,3,4, and 6) is fairly easy. The example I was talking about would look like this on a piece of sheet music.

Actually, as complex and potentially difficult as that can seem, that polyrhythm is found fairly commonly in music.

One famous example among piano players is in Debussy’s first Arabesque. The main theme features triplets over duplets and is the subject of much suffering for many piano students. However, it is certainly not impossible, and with practice, the rhythm becomes quite natural!

That is far from being the only case (Beethoven frequently changes from duple time to triple time within sonatas, Lyapunov’s etude 6 “The Storm”, Scriabin’s first prelude from Op. 16 has a particularly tricky case due to the accentuation in the left hand, 1812 overture with the measure of supertriplets in the climax, many cases!)

Actually a good exercise that I frequently do on the piano is to start major scales in both hands, with one doing the scale in 2 (or 4) and the other in 3. So one hand will end up doing 3 octaves and the other doing 4, coinciding perfectly at least at the start and end.

The reason I wanted to make the program was because I wanted to extend this thinking into other fractions. As someone will quickly find out upon sitting at the piano, this can be kind of tricky!

Not that this is at all a new idea. Chopin did this stuff all the time. Actually, he did this to quite a wild scale. Check out this excerpt:

11 over 6 and 22 over 12! You can find many examples of this.

At the end of op. 27, no. 2

My point is that these things are very cool! When played precisely, they give a really cool almost clockwork sound (depending on the piece of course) that seems superhuman or at least some form of transcendent.

When I first heard Khaikosru Sorabji’s music, I realized that a lot of his magic comes from his creative use of rhythms. Those unique and exotic textures are created just as much from his rhythmic experimentation as his tonal experimentation.

Check out this excerpt from his first sonata.

So my whole point is, that as a person who likes to play music, I wanted to make this easier for myself.

Luckily, a bit of googling showed me that someone else with a little more programming prowess had already came up with a solution. Actually there were quite a few. If you were willing to pay, there is Bounce Metronome (link). I liked that program quite a bit and am still on my free 30 day test drive
I also have done some Ruby work in the past so I was able to use the results if a Ruby Quiz. (link) The cool thing was that they gave you all the code so it is very easy to vary to suite your own needs.
The rest of what I found out there were just people who manually programmed their MIDI in certain polyrhythms. Not much control out allowed with those and when exported to something like Noteworthy they end up looking kind of yucky (as in, approximating with 16th and 32nd notes! -depending on your settings of course).

So my newest goal now is to be able to play smooth scales in 3 and 5 at the same time! There is something about 5 time that I just really like. It could be the inherent instability to it that resolves nicely to any other time (though best with 6). For example, there are quite a few examples of songs where the verses are in 5 time but the chorus is in 6. Yes! Even heavy rock too! Songs by Thrice and Dir en Grey do this to great effect. Anyways, I really want to get the ins-and-outs of five time and would certainly love to hear about any pieces in 5 that you enjoy.

Another thing I want to do is really be able to do more rhythms than just straight polyrhythmic beats. Like this:

I don’t know the technical way to describe it but I hope it makes sense. Ha when I try to do this stuff, I normally just write down all the fractions, obtain numbers relating to when I have to play things, and then play it a bunch of times until it actually is what is written.

Do any of you have tips or things that you do for playing these kinds of interesting rhythms?

Another interesting place to hear examples of polyrhythms is in the music of Meshuggah. If you check out their song “Bleed” (here) you can hear the vocalist singing in a meter independent of the guitars and around 1:40, you can hear a really cool interplay between the bass drum and the instruments.

Actually, a new program I want to make would be to aid in the creation of Math Rock, but that is a whole different post for a different day!

So my mom sometimes gets kind of upset when I talk smack about Perú (her country) and I think is not so happy that I can now speak Portuguese better than Spanish, but, if you look at the standings right now for the South American World Cup Qualifiers (Eliminatorias-just 1 easy word in Spanish), THAT TALK IS TOTALLY JUSTIFIED. I mean, seriously, how in the world can Perú be at the bottom of the list?! They have a population of 30 million people and resources to do a whole lot better. Look at Paraguay. How is a largely agrarian country with 6 million people able to be higher on the ranks than Argentina and Colombia?? For goodness sakes, the largest city, Asunción, is smaller than Seattle. Massive props to them for being able to do what they have done. Don’t get me wrong, I love Paraguay and hope to return soon (I just went to Ciudad del Este for a few hours) but it just is incredible that they can be doing so well, (and Perú be doing so poorly). Thank goodness though, they beat Uruguay (1-0). And of course the Colombians had to rub it in by telling the news that they paid Perú 120,000 USD for the victory (http://depor.pe/noticia/339182/colombia-otorgaria-120-mil-dolares-peru-su-victoria-sobre-uruguay). I actually think that it is plausible, though you would hope that the desire to be on top of Bolivia in the rankings would be incentive enough (though very difficult with the serious goal differential).
With the Peruvian team in Venezuela now, it looks like they are getting quite the treatment. hahaha they are in a place where the temperature is hovering around 95-100 degrees and the hotel’s air conditioning and electricity is suddenly not working… You can imagine how happy the team is now (or read about it here). Do I think they are talking now to make it a little more acceptable when they lose yet again? kind of, but then again, you got to sympathize with the guys for being in that heat all day and having a game to play at night.

Interesting video of what appears to be liquid explosives blowing up a plane (link).

Everyone has been inconvenienced by the “no liquids on planes” rule many times and who doesn’t remember the rebuttals by scientists (1 & 2)

hmmm, the guy in the BBC video was an expert and it appears he was just pouring stuff from like a gatorade bottle.

I definitely don’t know enough to make a call about this one.

Ok, so I was not planning on making my first real post about the dolphins/whales, but after reading a few separate stories I felt compelled to put them up here.

Details of Japanese dolphin slaughter here.

Europeans who are also guilty of this here

Good link to a video trailer of the recent documentary, “The Cove”

You can’t help but wonder if humans are in some way responsible for something so bizarre and tragic as this.

So now that I have kind of got this blog running for over a month from my trip, I am going to start talking about some other things that I have been wanting to blog about.

I read a lot of math books and want to blog about what I read there. Not because I am an expert but because:
-Explaining something helps me understand things better
-If I find something interesting, there is a good chance that other people will too

This blog will be to talk about interests like math, physics, astro, world traveling, and fashion. Hopefully some others will find what I write interesting as well

Ok, so I left to meet Rodrigo and little did I know how much of a crazy weekend it would be. So before going to meet him (I only knew him from couchsurfing) I gave him a call and he told me he was listening to music and drinking wine with some friends. Ok, sounds interesting. So I decided to get something for the man who would be hosting me in his cool apartment so I stopped by a supermarket on the way there, grabbed the cheapest bottle of wine I could find, and then made my way to Copam.

Oh man, that building was really something else. From the bottom, there are 5 entrances since it is divided into different blocks. The street level floor is somewhat of a mall, with shops and internet cafes. Well, I get to his place, on the 32nd floor, and my goodness, he has the coolest view of the city! It was a really sweet place. He had a nice couch and a good music set-up. I very quickly felt at home. After staying there for a while and talking, we decided to hit up some Mexican food with his friend Renata as well. It was really good Mexican food. Surprisingly, it was not very different than what I am used to for Mexican food. I had some great nachos and some steak tacos. It was a little expensive (like 35-50 reais) but well worth it. After that, the two of them went to a super market to get some alcohol for a party we were going to later that night. I stayed in the car and waited while the two went in and got the drinks. We picked up a friend, Gabriella, and then were off to the party.

I should mention, Rodrigo didn’t want to be early to the party so we get there at like 12:30 (0:30) and we still were fairly early! The party was at this film director/producers house and it was really awesome. For people that don’t drink alcohol like me, there was just water, but they also had hotdogs so I wasn’t completely without options. I got to chat with a bunch of very cool people. One guy I talked to for some time was trying to convince me of the merits of indigenous science and how the ancients sometimes knew more about certain phenomena than even modern science. As soon as he learned I was a physics/astronomy major I got asked all the usual questions, but then at a point he started to argue with me about something, and it was really frustrating since he did not know what he was talking about and eventually I tried leaving him to talk with someone else. I met a very cool girl named Jaquê and as soon as I learned that she made her clothes I wanted to talk with her for the rest of the night. Luckily, it all worked out well and I got to spend the next day with her. I also talked to this bearded guy wearing a leather jacket (built just like the guy from Crystal Castles!) but I forgot his name and didn’t get his orkut. Anyways, the party went on and on for a while. I got to show some people some stars and planets in the sky. I went back inside and at one point took a 20 minute nap on the couch. After that, I got up and we were getting ready to leave. About an hour later, we left the house, and when I stepped out the door, I was shocked to see that it was already light outside! It was 6:30 am!! So we walked around, looking for a taxi, and actually had a little trouble finding some. All of them we saw pass were full. Luckily we eventually got one and made it back to Rodrigo’s. I slept for a short time and got up again in about two hours, ready for my last day of Brazil.

So I got up around 8:30 and without even showering, called Jaquê up and we decided to go around to some markets. We went to Liberdade (Japantown) and this time the place was packed for the Sunday market! The streets at some places were full like a concert. It was really cool though. There were all sorts of good smells and nice foods being cooked outside. We found a cool self-service and ate there for lunch. I also got some last minute shopping done from the little vendors.

Another highlight. I couldn’t leave Brazil without trying the infamous acaraje so I got one after lunch. Acaraje is made from a ball of black-eyed peas that are mashed and fried in dendê (a red palm oil). A very strong orange sauce and a couple of shrimp are put on top. Everyone told me I would have violent stomach problems as a result of the dendê. Fortunately, I didn’t have any troubles later in the day and the acaraje was really tasty.

Afterwards, we made our way to Bexiga, the bohemian district, for the antique fair. That was really cool, but I wasn’t really that much of a shopper there. We passed by the Italian street fair but didn’t eat anything since I had been like eating all day and was really full. Oh, we also picked up an antique 3 piece luggage set for 5 reais ($2.75)! That was very useful since my one bag was absolutely packed! After that, we got picked up to Renata and it was time for one last party. The party was a fashion line launch at someone’s very cool house. I looked at the cool clothes, tried on some, but unfortunately didn’t buy any. I spent most of the time talking to these two wonderful woman who I gave my orkut info to, but they didn’t add me!
Anyways, I soon had to say bye to Renata and Jaquê as Rodrigo and his family came to pick me up. We swung by their house to grab the rest of my stuff and then rushed and made it just in time to the airport for me to say good bye and thank the wonderful Palma family and make my flight!

So today, I woke up at a more normal hour, got ready, had a small
sandwich of ham, salami, cheese, and requeijão (the plasticky looking
cheese stuff that tastes so darn good). Then it was out for shopping!
I am glad that Marily and I share a passion for shopping since I think
many people would not have been able to handle the amount that we did.
We went to the Camelodromo and had a blast at the many cool stores.
Marily and I got stuff for all of my family that I am sure everyone
will love! I also got some really cool clothes for myself and I think
many of you will be seeing me wearing my “Rio de Janeiro” shirts for a
while. The prices were very reasonable too, everything being about a
half to two-thirds of US prices. Anyways, after doing the shopping, it
was revealed that I had accomplish one of my goals in coming to
Brazil; the emptying of my checking account. The result of which has
been an unforgettable time, many wonderful experiences, and some great
friendships. Luckily I do have credit so I am good for the last 5 days
of being here. Anyways, after stopping in the nearby Shopping Iguatemi
where Marily bought my mom a sassy, brazilian style bikini, it was
back home for lunch of rice, beans, and meat.
After that, we didn´t waste a second before heading to Corcovado to
see the famous Cristo Redentor. It was a nice car ride. We started by
passing the favela of Borél, then continued to the forest of Tijuca
that I had seen, and then went out the steep mountain road, deep into
the forest until we made it almost all the way to the top. From there,
you buy a van ticket (13 reais), that takes you to the statue. The
view from there was simply incredible. We were worried it would be
cloudy, since it had been raining in the morning and all last night,
but thankfully the sun came out right before we got there and the
clouds were gone fast. The statue itself was very impressive, down to
the face of Jesus. It is incredible to think how the thing got moved
up there. We had a lot of fun taking pictures there. That was also
another place where I saw some tourists from England and Germany. I
also met a very nice looking Mineira (girl from Minas) who is also a
biologist, and with the help of Marily, got her Orkut. So now I have
yet another buddy to help practice Portuguese with. Anyways, after
that, we came down the mountain, and then went to visit some relatives
of Marily and Roberto. Somewhere along the way, Marily got the idea of
taking me into a favela, and as it was rapidly getting dark, I was a
little apprehensive about the idea. After visiting Roberto´s mom and
two of Marily´s sisters, we went to a place down the street to grab a
soup. I tried the wonderful Angu a Baiana, which is a red, possibly
tomato based soup,that had pieces of liean brisket in it. It was also
mixed in with a good amount of polenta. It was one of my favorite
things I have tried while in Brasil. I talked to the chef afterwards
and she was very happy I liked it but unfortunately didnt give me the
recipe! (I didnt really ask, just kind of hinted). I will certainly
try to make it (or force Bianca to) when I get back. After that, I had
the chance to meat the lovely Ingrid, who was oringinally going to be
my tour-guide for the day, but had some other arrangements so she
couldnt come. We had a chance to talk for a little bit before she was
off to see a show of O Rappa, a very popular, dread-locked, musician.
When she left, Marily´s friend who lived in the favela came and
announced it was time to go in. I think Marily noticed my apprehension
and got a little nervous but then his buddy with his motorbike came
and as soon as I saw I was going in on motorcycle, it was a done deal.
His buddy with the motorbike was a delivery boy who knew the favela
like the back of his hand so I was much less worried. As soon as I
entered much of my fear left me. I saw it was just people living their
lives in the best way they could. It was cool to see the patchwork
ways that the electricity and cables ran but things looked like they
were working. Also, unlike another favela I passed by, this one
thankfully had a system of plumbing, since the other one literally
smelled like feces, even from blocks away. So, the first thing I
noticed was that the favela had a much higher than average rate of Lan
Houses, arcades, and snooker (pool) places than the rest of Brazil
that I had seen. I also saw a few casinos that were just a few slot
machines and card tables. The place was decently busy with activity
and the street was sometimes a little hard to navigate. It would have
been impossible to go in car so the motorbike was a good idea. The
next thing I noticed was that there were a large number of very
attractive girls walking around. The downside and sad part was that
many of them were quite obviously teenagers which were quite obviously
pregnant. Literally, I think I saw a 13 or 14 year old with a belly
that was almost half the size of her still developing body. The favela
had many things you would expect to see in a normal city, including
super markets, dry cleaning, clothes stores, open air fruit markets.
There were also a little more than average of the people that were
laying down on the street, high on probably shoe polish.
Unfortunately, a large percentage of those were also children.
Anyways, I went really really deep in, with my driver taking me
through all the becos (alleys), some of which were barely wide enough
for a bike and a person to fit through. Of course, since it was a
favela, there were a number of arms or people walking with arms that I
saw. The vast majority of people I saw with guns were the store
owners, some of which sat at the counter with a machine gun at their
side. Others looked kind of like they were standing guard, though not
looking particularly agressive or threatening, just kind of chatting
with their pistols, shotguns, and machine-guns at their sides. I also
saw some tables, around which some armed men stood, where I think the
drugs were traded. The most stunning thing were some of the bazookas
that I saw literally leaned up against a wall at the front of the
store. I think those were used for when a police helicopter would come
over the city. I certainly would not want to be a soldier that had to
enter the favela. It was definitely a hive of smalls streets, some of
which were connected only by small alleys. I definitely would not have
entered not on the back of a motorcyle driven by someone that knew the
place and the people (many of the people waved at him and said hello
to him). Unfortunately I did not taken any pictures from the inside.
One way of keeping the police away is that they have a strict no
pictures rule (as I was told) so they could not have an idea of how
the exact geography to make plans for raids. I also do not think I
would ever take pictures of someone armed with a machine gun who was
not a cop or soldier. Anyways, I survived my trip, and was realy glad
that I got to see that side of Brazilian life. Many many people of Rio
de Janeiro live in favelas. Everyone from the cleaning lady of the
house, people from the feiras, and a few relatives and friends of
Marily live there. They are not a hive of people who are evil or
intent on breaking the law. That is what the drug dealers do who live
at the very top of them. Rocinha, the very biggest of Rio´s favelas
(which I got to see from the bottom), is home to 2 million people!
That is more than the amount of people living in Seattle, all packed
on a hill. After that, we stayed to chat for a little, made our way
home, I revised my schedule to stay an extra day here in Rio, and now
I am here writing.

So, my last few days in São Paulo were a real adventure! I got back
and knew the Palma family had visitors so I managed to find a place to
stay with a connection of Marily’s. It was really cool. The family was
composed of Helio (the father), Heloisa (the daughter), and the mother
named Maisa. They were incredibly nice and I had a wonderful time with
them. I spent nearly an entire day at the Centro Cultural in São
Paulo. That was like my dream come true of a place. It was a really
big library that also had public showings of films and music events. I
really loved it and had a great time reading and watching some
interesting music performances. I spent another day going to Santo
André to have a piano lesson with the famous Marisa Lacortes. She is
the best piano teacher in all of Brazil and I was very lucky to get a
lesson with her. So I took the train to Santo André, a neighboring
city, and got to her place with all my money drawn from my account to
pay for what surely would be an expensive lesson. I had an incredible
2 hour lesson with her and in the end, she charged me NOTHING for the
whole time spent with her. I already have put into practice many of
the things she has told me and it has already helped. I would move to
São Paulo just to have lessons with her. Anyways, after that, I had a
little lunch with the great Amarita, my friend who had arranged the
lesson for me, and said my goodbyes to her. Then I was off to Helio’s
place where I met up with him and went out to see his friend perform
music. It was a lot of fun. He sang lots of songs that were familiar
to even me, including “What a Wonderful World”. I also got to meet
Heloisa’s very cool boyfriend. Oh, I forgot to mention that she spoke
nearly perfect english so that was a lot of fun. She is such an
amazing, talented person. Her boyfriend is a lucky guy. So we spent
the night having many espetinho’s and enjoying our time together. The
next day, we all went out to Ibirapuera Park, one of my favorite
places in all of São Paulo. I played frisbee with Heloisa and her
boyfriend, had some suco de cupuaçu, enjoyed the sun and agua de coco,
and altogether just had a good time. After that, I got all my stuff
together, and took the subway, during rush hour, to Copam to meet my
Couchsurfing friend Rodrigo.