So there are four fundamental forces of nature, supposedly. Gravity, electromagnetism (which pretty much is responsible for every force we experience except gravity), the strong force, and the weak force. For each type of force, particles can have a field around them of that force (e.g. gravitational field, magnetic field, electric field, etc), which affect other particles in the field. But what I've recently discovered is that PEOPLE have a field of force around them that ISN'T ANY OF THE ABOVE. Yes, we have gravitational fields around us, and electric and magnetic fields (weak though they may be)... but there's another force of nature surrounding each human being. It seems to always cause a repelling force most of the time (like the electromagnetic force with like-charged particles, or like dark energy, not so much like gravity); and like gravity and electromagnetism, this force is inversely proportional to some power of distance (most likely 2, but I have yet to experimentally prove this... a possible experiment is discussed below)
Evidence? I've got plenty. Everything I'm going to say, you already know to be true - you've just never realized the implications of it.
FACT 1: People in a small enclosed space (such as an elevator) spread out so as to evenly fill the space.
EXPLANATION: This is exactly what a repelling force, inversely proportional to distance, would be expected to do. A system of electrons constrained to a given space, will naturally develop over time into a configuration of least potential energy - essentially, when the electrons are as far apart as possible. Once it reaches this position of equilibrium, any change in the system will result in two charges being brought closer together, giving a higher potential energy - which can only happen if some external force is applied. Of course, the electrons may not actually find the "best" arrangement; the most one can say is that they will achieve a local minimum of potential energy.
Likewise, in an elevator, people naturally arrange themselves to be in a position of furthest distance from their neighbors - maybe not the best possible configuration, but a configuration in which any change would only bring two people closer together. If my theory is correct, this can easily be interpreted, as above, as finding a local minimum in the potential energy of the system. And when new particles (humans) are introduced to the system, or when particles (humans) are removed, the system rearranges to find a new local minimum in potential energy.
FACT 2: Bringing people together causes complex phenomena.
EXPLANATION: Considering the above result, one may then ask "if the force field surrounding humans is a repelling force, then what brings people together?" The simple answer is that there are many other variables, many other forces at work besides just the predictable, natural "human force" I'm postulating. For example, a force of friendship may easily overcome the natural human force field; and the force of gravity will clearly overcome this force field if one human is positioned directly above another with no solid object in between. The human force is a comparatively weak force - its effects only come out in situations in which no other force can override it (such as in a closed elevator, in which the only other significant forces are vertical - gravity and the normal force - so the horizontal effects of the human force field may show).
But a more complex and enlightening answer is a direct result of the repelling nature of the force - In any system in which a repelling force is the dominant force in question, then smaller distances will result in higher energies, and thus more complex behavior. Consider, for example, how two colliding molecules don't just bounce off each other if a certain energy level (the activation energy) is reached; they may undergo a chemical reaction. And in high-energy particle collisions, higher energies will result in more complex results, which is why further research can only happen when higher-energy particle accelerators are built.
So what does this mean for humans? Well, as mentioned in the explanation of fact 1, without the application of some external force, humans will naturally stay apart from each other. But if some force is applied, two particles can actually be brought together; and if the inverse relationship between force and distance holds, this will result in a higher potential energy. Potential energy will equal the ability for a system to do work - so when people are brought together, and the energy of the system increases, more work can be done by the system.
This, essentially, is how civilization was possible. This is the physical basis of sayings such as "two heads are better than one" and "the whole is greater than the sum of its parts." Bringing people together results in, quite literally, a large potential. Potential energy can be converted into many different forms - which is why people in groups can achieve so much. Rather than seeing the natural repelling force as a reason for humans to stay apart, it should be seen as a reason to come together - because for forces inversely proportional to distance, energy is stored in the force field, and the energy is larger if the distance between particles is smaller.
FACT 3: Animals tend to "bounce away from" humans, even at a distance.
EXPLANATION: This was actually the inspiration behind the discovery of this new kind of force. Animals too, seem to be affected by this human force field (perhaps all conscious life forms actually have this field around them?). There are actually many different reactions of animals towards humans - some attack, some run up with wagging tail, some instantly dart away, etc. - as mentioned above there are countless other forces at work in determining how a given animal responds to a given human, and most will override this realtively simple human force. But we can find a "control" case in a particular species which is neither domesticated nor wild - neither friendly to nor completely fearful of humans - the pigeon. This species likely has the simplest, most repeatable (and therefore testable) reaction to human beings out of any vertebrate; barring the imposition of additional variables (such as bread crumbs), the behavior of pigeons can give us the most raw data about the way this force surrounding human beings truly works.
Imagine approaching a stationary pigeon at a relatively low velocity. There will be a certain distance, an "event horizon" if you will, at which the pigeon will begin to move away, but then return to being stationary. The "event horizon," in this case, is the distance at which the human force is equal to the other forces acting on the pigeon (such as the unwillingness-to-move force), resulting in a net force of 0 on the pigeon; once the distance of separation is smaller than this distance, the human force becomes stronger than the other forces acting on the pigeon, causing a net acceleration away from the human, until it is far enough away that the other forces once again overcome the human force.
An analogy to magnets would be helpful. Imagine slowly bringing one magnet A closer to another magnet B, with like poles facing each other. Up until a certain point, magnet B will remain stationary. Only once the distance of seperation is small enough so that the magnetic force can overcome the force of static friction acting on magnet B will magnet B move away from magnet A, just to, once again, remain stationary.
Now, if you move magnet A close to magnet B really quickly, magnet B will respond with a higher acceleration away from A, because the quicker you are, the closer A can get to B before B can accelerate away, resulting in a stronger force on B and more pent-up energy, which will give B a stronger push. This behavior can also be seen in pigeons - the faster you approach them, the faster they move away. In fact, up to a certain critical distance, the behavior of pigeons near humans is very much like the behavior of a magnet in another magnet's magnetic field. They literally "bounce away" from the human's force field, and bounce off faster if the human moves towards them faster.
At a certain critical distance, something unexpected happens - the pigeons change modes of transportation and take flight. But this is exactly what I mentioned in Fact 2: once a certain energy threshold is reached, behavior may drastically change. In fact, in a way that is very analogous to the pigeon, string theory predicts that at high enough energies, a collision between high-speed particles may actually send gravitons out of our regular three dimensions, so as to be unobservable. Analogous to the pigeon, I say, because a high-energy encounter with pigeon and human will send the pigeon out of the regular two dimensions of interaction (confined to staying on the ground) to a third dimension, resulting in completely different behavior.
The above fact leads to a possible experiment. Roughly stated (I'll have to work out the precise details on my own), we would approach a given pigeon at different constant velocities, and observe the velocity of the pigeon at various points in time throughout the interaction. By calculating the acceleration from the velocity data, we will be able to calculate the force acting on the pigeon at each point in time. Comparing this force as a function of time to the distance of separation between human and pigeon as a function of time, we may be able to determine some form of relationship between force and distance.
This would have to be tested on many different pigeons, because each pigeon will have a different reaction to the human force - likely due to an as yet unknown property, similar to mass and charge, that can be calculated into the force formula. However, when we do the above analysis on a given pigeon's motion, this property (assuming it is constant) should cancel out when various data points are compared with each other, yielding a pure ratio between forces and distances - the relationship between forces and distances can be analyzed to see what the precise relationship between force and distance is. (I hypothesize that it is, like gravity and electromagnetism, an inverse square relationship).
I still have a long way to go before this theory is completely developed - I hope to perform the experiment sometime soon (though I lack sufficient tools to perform it with much accuracy), and this will give me the mathematical foundation to this otherwise very qualitative theory. But I'm convinced that either the data will support this theory, or lay the groundwork for an even more profound, effective theory.
Call me crazy if you will. But mark my words: by the time I'm done... you'd be right.
Friday, September 23, 2011
Squirrels
I saw a squirrel army-crawling through the grass today. It found a giant dead leaf (bigger than the squirrel), and pounced on it, rolled over on its back like a cat playing with a ball of yarn and started munching on it. I almost laughed out loud.
Friday, September 16, 2011
Math homework
"ugh" is probably the most common reaction to the title of this post (including synonyms). Most people just don't like math homework. In fact, you may think "only people who love math would like math homework." Well... even some people who love math don't like math homework. Good example: me in high school (I'll get to me in college later on).
Yes, that's right, I didn't like math homework. Somehow or other, I managed to put aside my dislike of the problems and focus on the math itself, which I found much more beautiful, quirky, and open to exploration and discovery. I did far more math than math homework in my high school years (as evidenced by literally hundreds of pieces of paper covered with tiny mathematical scribbles, none of which had anything to do with what I was learning in school - the following is one from back in 8th grade):
So people may ask - how? How could I put aside my dislike of the boring set of problems that had to be solved every day? How could I look through the giant obstacle of daily assignments that blocks so many other people from ever appreciating the cool stuff behind it?
And what I've recently realized is... I kind of cheated a bit.
A bit of background first. I've always liked math. From before I was even in elementary school, math was something I found interesting. Sometimes, it was just because I felt I did it well, so it made me feel good about myself. Sometimes it was because I felt it connected things well, or had interesting results. There were various reasons for my enjoyment of it... but it wasn't until 8th grade that I REALLY started getting into math. Up until 7th grade, I was doing the stuff everyone else was doing. Perhaps I did it better than average, or got the concepts quicker, but I was learning 7th grade stuff - figuring out how to work with variables, dealing with different types of numbers (negatives, irrationals), plotting points on the coordinate plane, and so on. By the end of my 7th grade year, I was really good at 7th grade math - no more than that. But within half a year, I had taught myself calculus. That's... a big jump.
Almost all the credit for this rapid change goes to my 8th grade math teacher, Mr. Finlay. I didn't realize it at the time, but what he did for me, essentially, was pulverize any ideas I had that math was to be learned in a strongly structured way. I started the school year at a new school in Canada (I was in Japan previously) and was given a 9th grade math textbook - no testing out of 8th grade math, no asking me if I thought I was ready, nothing - I was just put into a class of 8th graders learning 9th grade math (though some were working on 10th or even 11th grade math at the time). I accepted the challenge, and started working on 9th grade math. About a month into the year, Mr. Finlay pulls me aside and says "You don't seem to be having any trouble with this stuff. Why don't you take the 9th grade test and we'll get you started on 10th grade math?" And so, a week later, I got the 10th grade textbook. I officially stayed at that level through the year - but that sudden jump, from 7th grade to 10th grade math in just a couple months, gave me a lot of momentum, and I just kept moving. Kept exploring, kept discovering - until, as I mentioned, I learned calculus over Christmas vacation. Not well enough to take the AP test, definitely - I couldn't integrate anything more complicated than a polynomial, for example. But for the first time, I was hungry for math, not just interested in it. I basically ignored the textbook that whole year - and Mr. Finlay didn't mind at all. He knew I was learning more than the textbook could teach me.
Fast forward to the beginning of 9th grade - return to Japan, to a smaller school with fewer resources, and therefore a more structured learning environment (it takes a lot of resources - time, people, books, training - to provide a program that can be personalized to fit every student's learning style). After some hard work, I managed to get into 10th grade math (again). A bit of a step back in some ways, but it was definitely a new experience to be learning with people a year older than me, and I learned a lot that I never would have otherwise. I had a math teacher named Mr. Mhlanga, and he, like Mr. Finlay, understood that I didn't need the textbook problems in order to learn, that I was getting somewhere on my own. So he let me get away with not doing much work. Same in my 10th grade year, when I took AP Calculus AB - again, Mr. Mhlanga was the teacher, and he emphasized taking charge of your own education, meaning "do what you know you need to do." He graded very little - so I did very little, and did fine. 11th grade, I took AP Calculus BC... which wasn't actually a class at my school, so it turned into a study hall in which I basically didn't do calculus. And then in 12th grade, I had no math class.
So what do I mean by "I kind of cheated a bit?" Basically... I haven't done math homework regularly since 7th grade. :p
And it was after 7th grade that I started really getting into math.
Coincidence? Maybe. But that's not my point. What I meant to talk about is where I am now...
Do I like regularly assigned math homework now that I'm out of high school? I actually don't know. After not having done it regularly for five years, maybe my thoughts on it have changed. Maybe all the time I spent away from it, getting a solid foundation in other ways of doing math, will make me appreciate the math behind the assigned problems more. Maybe the fact that college problems are more interesting than high school ones will give me more motivation. But it could be the other way too... five years without regular math homework means five years of losing my touch. I'm not used to doing regular homework problems any more. Getting back on track could be very hard, very painful, and just make me hate regularly assigned problems even more. Which will it be? So far, I've done one assignment for each of my two math classes (both a week early!), and it's actually been pretty fun so far. But the real test will come once I have to start doing them over and over and over and over again. Will the novelty wear off? Will I get bored or frustrated? Or will I continue to find the problems invigorating? We have yet to see. I'll let you know in a couple of months.
Tuesday, September 13, 2011
Obviously.
So this past week has been packed with stuff. I arrived at my dorm, had an incredibly busy six-day orientation (each day started at 9 AM and only one ended before midnight), met up with a friend from Japan for a day, and started attending courses. And now I have a bit of time to write... but, as you probably know, the more there is to write about, the harder it is to choose a topic.
So I'll just talk about an interesting thing I noticed in one of my classes - Algebra I it's called (and no, this is not the same as the Algebra I you learn in middle or high school). At one point, the professor was defining a union of sets - basically, if you have two collections of objects, then the union is what you get from putting all those objects together into one collection. More formally, "the union of X and Y contains all elements z such that z is an element of X or z is an element of Y." Now, he had a lot to cover, so he just zipped through the definition, and throughout the talk, he kept referring back to it, but I noticed something interesting - he never clarified what he meant by "or."
Now, I've seen unions defined before, but the first time I learned it, it was carefully explained that "X or Y" in math meant "at least one of them." This is very different from how or is usually used in English - when someone asks "coffee or tea," "both" is usually not considered as an answer. You may talk about "the answer is B or C" - that means one or the other, not both. If you ask "should I give money to Steven AND Stephanie," the answer is yes or no, but if you ask "should I give money to Steven OR Stephanie," it's not a yes-or-no answer - you have to choose one or the other. In regular English, OR means "pick only one," which is very different from the "or" mathematicians use.
I really wasn't sure what to do - I'm sure most people in the class knew what the professor meant. But if anyone in the class thought that the "or" meant "only one, not both," they would be stuck with that for a long time - small misconceptions such as this, I know for a fact, can cause huge frsutration and confusion about math. So I decided to bring this up - I asked the professor "Just to clarify, the 'or' you're talking about is an inclusive or, right?" hoping he would use the opportunity to explain the language more clearly.
And he did - but in an odd way. He explained to the class that yes, in fact, "or" can have two meanings - and that in logic, sometimes people use the exclusive or, "xor," which means to pick one or the other but not both. "But," he said, "for this, I'm using the usual version of 'or,' meaning in one, the other, or both."
[oomph moment here]*
The usual version? To professional mathematicians maybe... but what about to students who've never gone beyond basic high school math, and have never seen Boolean operators or logical statements? Yes, if these people are going into math, they'll have to be taught what "or" means in a math context - but that's something that needs to be taught, not assumed.
Basically, the professor forgot what it's like to be "normal" - when you spend your life around a specific group of people, and you learn to think like them, you forget about what the "normal" way to think is (and I won't go into a huge discussion here about "what is normal" - just bear with me on that)
Nothing against the professor - in many ways, he teaches really well. The main reason I bring this anecdote up is because it's a great illustration of what happens more generally - there are many people that get so involved in a specific group that they forget what it's like to be outside the group, even if they were outside it once. They assume that what's obvious for them would be obvious to anyone. I'm sure I do the same thing - the scary part is that, by definition, I wouldn't know if I'm doing it. If I think something's obvious, I wouldn't give it a second thought. So basically, this served as a reminder to question the obvious - not necessarily to question its truth, but to question its obviousness. Maybe it'll turn out not to be true - then good for me, I've learned something. But if it is true, then questioning its obviousness will make me realize that this truth is not something to take for granted - that it's much deeper, more amazing and profound than I ever would have thought.
*I'm sure there's a better term for it, but I'm going to use "oomph moment" to mean a point at which I say something that's really mind-blowing, bizarre, incredible, strange, surprising, ridiculous, or otherwise thought-provoking - it means "stop and think about what I just said - if you understand the significance, it should be a big deal, maybe even a bit of a shock. If you don't understand, don't worry, I'll explain it right away"
So I'll just talk about an interesting thing I noticed in one of my classes - Algebra I it's called (and no, this is not the same as the Algebra I you learn in middle or high school). At one point, the professor was defining a union of sets - basically, if you have two collections of objects, then the union is what you get from putting all those objects together into one collection. More formally, "the union of X and Y contains all elements z such that z is an element of X or z is an element of Y." Now, he had a lot to cover, so he just zipped through the definition, and throughout the talk, he kept referring back to it, but I noticed something interesting - he never clarified what he meant by "or."
Now, I've seen unions defined before, but the first time I learned it, it was carefully explained that "X or Y" in math meant "at least one of them." This is very different from how or is usually used in English - when someone asks "coffee or tea," "both" is usually not considered as an answer. You may talk about "the answer is B or C" - that means one or the other, not both. If you ask "should I give money to Steven AND Stephanie," the answer is yes or no, but if you ask "should I give money to Steven OR Stephanie," it's not a yes-or-no answer - you have to choose one or the other. In regular English, OR means "pick only one," which is very different from the "or" mathematicians use.
I really wasn't sure what to do - I'm sure most people in the class knew what the professor meant. But if anyone in the class thought that the "or" meant "only one, not both," they would be stuck with that for a long time - small misconceptions such as this, I know for a fact, can cause huge frsutration and confusion about math. So I decided to bring this up - I asked the professor "Just to clarify, the 'or' you're talking about is an inclusive or, right?" hoping he would use the opportunity to explain the language more clearly.
And he did - but in an odd way. He explained to the class that yes, in fact, "or" can have two meanings - and that in logic, sometimes people use the exclusive or, "xor," which means to pick one or the other but not both. "But," he said, "for this, I'm using the usual version of 'or,' meaning in one, the other, or both."
[oomph moment here]*
The usual version? To professional mathematicians maybe... but what about to students who've never gone beyond basic high school math, and have never seen Boolean operators or logical statements? Yes, if these people are going into math, they'll have to be taught what "or" means in a math context - but that's something that needs to be taught, not assumed.
Basically, the professor forgot what it's like to be "normal" - when you spend your life around a specific group of people, and you learn to think like them, you forget about what the "normal" way to think is (and I won't go into a huge discussion here about "what is normal" - just bear with me on that)
Nothing against the professor - in many ways, he teaches really well. The main reason I bring this anecdote up is because it's a great illustration of what happens more generally - there are many people that get so involved in a specific group that they forget what it's like to be outside the group, even if they were outside it once. They assume that what's obvious for them would be obvious to anyone. I'm sure I do the same thing - the scary part is that, by definition, I wouldn't know if I'm doing it. If I think something's obvious, I wouldn't give it a second thought. So basically, this served as a reminder to question the obvious - not necessarily to question its truth, but to question its obviousness. Maybe it'll turn out not to be true - then good for me, I've learned something. But if it is true, then questioning its obviousness will make me realize that this truth is not something to take for granted - that it's much deeper, more amazing and profound than I ever would have thought.
*I'm sure there's a better term for it, but I'm going to use "oomph moment" to mean a point at which I say something that's really mind-blowing, bizarre, incredible, strange, surprising, ridiculous, or otherwise thought-provoking - it means "stop and think about what I just said - if you understand the significance, it should be a big deal, maybe even a bit of a shock. If you don't understand, don't worry, I'll explain it right away"
Sunday, September 4, 2011
Thoughts on an Airplane
First, reasons I like window seats.
- the view. I loved it as a little kid, and I still love it now. This past trip (flying from Calgary to Toronto) was probably one of the best I've had in a long time- as we took off, there was a perfect view of almost the entire city of Calgary, with the snow-capped Rockies in the distance (truly awe-inspiring). Flying over the prairies is an interesting experience... you know when you look at satellite images on Google Maps or Google Earth and the land looks like a haphazard patchwork of rectangles? Well, the prairies actually look like that. And completely flat in all directions. Kind of dizzying. The approach to Toronto was cloudy, but very high clouds - once we got in low enough, there was a perfect view of the huge city. We actually had to fly past the airport, turn around, and come back, so from my window, I got to see the whole city, and many parts of it twice - resorts, amusement parks, vast neighborhoods of houses, complex road junctions, huge train yards, factories, etc. And off in the distance, by a body of water that stretches off to the horizon, a black silhouette shrouded in cloud, the lone spike of the CN Tower, stood proudly yet ominously in the distance. So exciting.
- you can have a larger carry-on. I always pack my backpack too full, so it doesn't fit under the seat. When I have an aisle seat, the flight attendants always get concerned that it's a hazard - but by the window, no one cares if the backpack doesn't fit.
- you never have to be bothered by people who want to get out to go to the washroom or by flight attendants reaching over you to give other passengers refreshments.
- related to the two above, but you can make your space "yours" - keep your tray down, take your shoes off, do whatever without worrying that it'll get in the way of other people.
- while there's not quite as much leg room as in an aisle seat, the wall of the airplane curves outward a bit, so there's room to nestle an extended leg between the seat in front of you and the wall.
- you can take your time getting off the plane, without worrying about whether the people further in need to get somewhere quickly.
In the end, the only main downside to having a window seat I can think of is guilt - whenever you need to go to the washroom or get something from a flight attendant, you feel bad about bothering the people beside you.
Now, my particular window seat wasn't as great this ride because it was the only seat in the plane where the in-flight entertainment system wasn't working. But every annoyance is a blessing in disguise - I used the chance to sleep. Or to try.
School buses look like yellow pastels from thousands of feet in the air. Especially when there's a bunch parked beside each other.
Lesson learned - if there's not room for your carry-on in the overhead compartments directly above you, find an empty spot closer to the front of the plane, not further in the back. I learned this the hard way, and had to wait until every other passenger was off the plane before I could go back and get my stuff. First time being the last passenger on a plane.
Sorry, I tried to think of some deep philosophical connection between an airplane and life, but nothing profound came to mind. I hope you enjoy these simple, down-to-earth (pun intended) thoughts instead.
- the view. I loved it as a little kid, and I still love it now. This past trip (flying from Calgary to Toronto) was probably one of the best I've had in a long time- as we took off, there was a perfect view of almost the entire city of Calgary, with the snow-capped Rockies in the distance (truly awe-inspiring). Flying over the prairies is an interesting experience... you know when you look at satellite images on Google Maps or Google Earth and the land looks like a haphazard patchwork of rectangles? Well, the prairies actually look like that. And completely flat in all directions. Kind of dizzying. The approach to Toronto was cloudy, but very high clouds - once we got in low enough, there was a perfect view of the huge city. We actually had to fly past the airport, turn around, and come back, so from my window, I got to see the whole city, and many parts of it twice - resorts, amusement parks, vast neighborhoods of houses, complex road junctions, huge train yards, factories, etc. And off in the distance, by a body of water that stretches off to the horizon, a black silhouette shrouded in cloud, the lone spike of the CN Tower, stood proudly yet ominously in the distance. So exciting.
- you can have a larger carry-on. I always pack my backpack too full, so it doesn't fit under the seat. When I have an aisle seat, the flight attendants always get concerned that it's a hazard - but by the window, no one cares if the backpack doesn't fit.
- you never have to be bothered by people who want to get out to go to the washroom or by flight attendants reaching over you to give other passengers refreshments.
- related to the two above, but you can make your space "yours" - keep your tray down, take your shoes off, do whatever without worrying that it'll get in the way of other people.
- while there's not quite as much leg room as in an aisle seat, the wall of the airplane curves outward a bit, so there's room to nestle an extended leg between the seat in front of you and the wall.
- you can take your time getting off the plane, without worrying about whether the people further in need to get somewhere quickly.
In the end, the only main downside to having a window seat I can think of is guilt - whenever you need to go to the washroom or get something from a flight attendant, you feel bad about bothering the people beside you.
Now, my particular window seat wasn't as great this ride because it was the only seat in the plane where the in-flight entertainment system wasn't working. But every annoyance is a blessing in disguise - I used the chance to sleep. Or to try.
School buses look like yellow pastels from thousands of feet in the air. Especially when there's a bunch parked beside each other.
Lesson learned - if there's not room for your carry-on in the overhead compartments directly above you, find an empty spot closer to the front of the plane, not further in the back. I learned this the hard way, and had to wait until every other passenger was off the plane before I could go back and get my stuff. First time being the last passenger on a plane.
Sorry, I tried to think of some deep philosophical connection between an airplane and life, but nothing profound came to mind. I hope you enjoy these simple, down-to-earth (pun intended) thoughts instead.
Saturday, September 3, 2011
Number Sense Tester
As mentioned in 1, 2, 3... not the best place to start? (Part 2A), I decided to make a basic number sense tester. It's inspired by the one found here - but with a few things I thought should be changed.
So without further ado, here it is. Note that pressing Space Bar (when you're not in the middle of a test) lets you fiddle around with some options if you want to.
A few basic modifications I made...
(1) more dots. The other number sense tester could only go so far; this one can show up to 500 of each colour, allowing extremely small ratios to be tested. If you want to limit yourself to 20 anyways, press space bar and edit the "max. number of dots of each colour" field. For most purposes I've found you don't really need more than 100.
(2) different background colour. The original uses #666666, or 40% gray (from 0% black to 100% white). But with this, the yellow stands out too much - some very basic testing showed that of all my incorrect answers, about 80% would be because I saw too much yellow. Conversely, if the background were white, the blue would stand out too much - 75% of my incorrect answers were because of seeing too much blue. The colour I ended up choosing - #AAAAAA (67% gray) was the best I could find to show blue and yellow fairly equally.
(3) ratios. Simple - just divide blue dots by yellow dots (or the other way around, depending on which has more)
(4) calculating skill level. This was the hardest part - to find a way to take all the results and calculate about where your skill level is. Much thanks to the Mathcamp alumni mailing list for helping me figure out how to go about calculating that.
(5) providing tests based on skill level. Whatever your skill level is, you'll be given a couple questions just above it and a couple just below it, so you're always at a place you're comfortable with.
(6) options. Press space bar to change things... maybe you'll discover something by tweaking the options a bit. Can you get a better score if you have fewer (or more) dots? Does the relative size of the dots make a difference? Try it out for yourself.
This was mostly just for fun, so chances are I won't come out with an updated version, even if I do find ways to make it better... but if you have any comments or suggestions, I'd love to hear them!
So without further ado, here it is. Note that pressing Space Bar (when you're not in the middle of a test) lets you fiddle around with some options if you want to.
A few basic modifications I made...
(1) more dots. The other number sense tester could only go so far; this one can show up to 500 of each colour, allowing extremely small ratios to be tested. If you want to limit yourself to 20 anyways, press space bar and edit the "max. number of dots of each colour" field. For most purposes I've found you don't really need more than 100.
(2) different background colour. The original uses #666666, or 40% gray (from 0% black to 100% white). But with this, the yellow stands out too much - some very basic testing showed that of all my incorrect answers, about 80% would be because I saw too much yellow. Conversely, if the background were white, the blue would stand out too much - 75% of my incorrect answers were because of seeing too much blue. The colour I ended up choosing - #AAAAAA (67% gray) was the best I could find to show blue and yellow fairly equally.
(3) ratios. Simple - just divide blue dots by yellow dots (or the other way around, depending on which has more)
(4) calculating skill level. This was the hardest part - to find a way to take all the results and calculate about where your skill level is. Much thanks to the Mathcamp alumni mailing list for helping me figure out how to go about calculating that.
(5) providing tests based on skill level. Whatever your skill level is, you'll be given a couple questions just above it and a couple just below it, so you're always at a place you're comfortable with.
(6) options. Press space bar to change things... maybe you'll discover something by tweaking the options a bit. Can you get a better score if you have fewer (or more) dots? Does the relative size of the dots make a difference? Try it out for yourself.
This was mostly just for fun, so chances are I won't come out with an updated version, even if I do find ways to make it better... but if you have any comments or suggestions, I'd love to hear them!
Friday, September 2, 2011
Question Time! Tonedeaf Overtones
"just wondered do you think its mathematically/physically possible to make a single note thats in disharmony with itself? like the close overtones in it are not in harmony?" - Philip Chuah
Brief intro to overtones: Everything you hear, you hear because something is moving, causing the air around it to move, which pushes a wave of air into your ear where it is heard. Musical notes come from a specific type of movement - vibrating at a constant speed. When you pluck a string, blow into a tube, hit a metal bar, or anything like that, something vibrates at a regular speed, producing a note. The faster it vibrates, the higher the note you get.
Here's a weird thing though - Only the simplest waves - sine waves (like in http://www.youtube.com/watch?v=R7D1f6U6TpU) - come from just a smooth vibration back and forth. Every real-world instrument actually vibrates in a more complicated way. But the cool thing is that these complicated vibrations can actually be split up into a bunch of simple vibrations - called overtones - that add up to the original!
You can see overtones at work quite easily if you have a stringed instrument like a guitar, or violin, or piano. Touch a string lightly at exactly the halfway point and play that string - you will see that the string still vibrates, but at a higher pitch. This is one of the overtones of the string. The same thing happens if you hold your finger one third of the way down the string, or one fourth, or one fifth. In fact, all the overtones of all instruments that vibrate along a straight line (strings, pipes, etc) can be found by dividing up the length evenly. And whenever you play the string, all of those overtones actually are played at once. And in fact, it's precisely the relative strengths of the overtones that give each sound a different quality. Why does flute have such a "pure" sound? Because only the first one or two overtones are very strong. What gives brass instruments their piercing texture? Because of the shape of the instrument (more cone-like than tube-like), it brings out the odd harmonics, while the even harmonics are really weak. In instruments with a "rich" sound like violin or human voice, you can go up a long ways in the overtone series and they'll still be pretty strong. But while the strengths of each overtone change from instrument to instrument, the pitch of each overtone is always the same in most instruments - you can get each overtone by evenly dividing the length.
But, as briefly mentioned in the video, some instruments don't have quite as nicely placed overtones. In particular, 2 or 3-dimensional vibrators have stranger overtones. If not controlled right, usually the overtones will just be messy and result in noise - like most drums. But sometimes the overtones can be controlled to sound not so bad. This 3-d aluminum bar shows this well - the 1st overtone is an octave above the fundamental (so it sounds ok), but the 2nd overtone is just a half step above the 1st! With 1-d vibrators, the overtones almost always sound nice with the original note, because they're so nicely divided - you'd have to go past the 10th overtone before you'd get two overtones a half step away from each other. But 3-d objects aren't limited in that way.
So in conclusion - yes, a note can be out of tune with itself. Case closed.
Answer: YES.
Brief intro to overtones: Everything you hear, you hear because something is moving, causing the air around it to move, which pushes a wave of air into your ear where it is heard. Musical notes come from a specific type of movement - vibrating at a constant speed. When you pluck a string, blow into a tube, hit a metal bar, or anything like that, something vibrates at a regular speed, producing a note. The faster it vibrates, the higher the note you get.
Here's a weird thing though - Only the simplest waves - sine waves (like in http://www.youtube.com/watch?v=R7D1f6U6TpU) - come from just a smooth vibration back and forth. Every real-world instrument actually vibrates in a more complicated way. But the cool thing is that these complicated vibrations can actually be split up into a bunch of simple vibrations - called overtones - that add up to the original!
You can see overtones at work quite easily if you have a stringed instrument like a guitar, or violin, or piano. Touch a string lightly at exactly the halfway point and play that string - you will see that the string still vibrates, but at a higher pitch. This is one of the overtones of the string. The same thing happens if you hold your finger one third of the way down the string, or one fourth, or one fifth. In fact, all the overtones of all instruments that vibrate along a straight line (strings, pipes, etc) can be found by dividing up the length evenly. And whenever you play the string, all of those overtones actually are played at once. And in fact, it's precisely the relative strengths of the overtones that give each sound a different quality. Why does flute have such a "pure" sound? Because only the first one or two overtones are very strong. What gives brass instruments their piercing texture? Because of the shape of the instrument (more cone-like than tube-like), it brings out the odd harmonics, while the even harmonics are really weak. In instruments with a "rich" sound like violin or human voice, you can go up a long ways in the overtone series and they'll still be pretty strong. But while the strengths of each overtone change from instrument to instrument, the pitch of each overtone is always the same in most instruments - you can get each overtone by evenly dividing the length.
But, as briefly mentioned in the video, some instruments don't have quite as nicely placed overtones. In particular, 2 or 3-dimensional vibrators have stranger overtones. If not controlled right, usually the overtones will just be messy and result in noise - like most drums. But sometimes the overtones can be controlled to sound not so bad. This 3-d aluminum bar shows this well - the 1st overtone is an octave above the fundamental (so it sounds ok), but the 2nd overtone is just a half step above the 1st! With 1-d vibrators, the overtones almost always sound nice with the original note, because they're so nicely divided - you'd have to go past the 10th overtone before you'd get two overtones a half step away from each other. But 3-d objects aren't limited in that way.
So in conclusion - yes, a note can be out of tune with itself. Case closed.
Thursday, September 1, 2011
Left-handed Guitar
Wow, I've been getting a lot of traffic on this blog from my last four posts on learning math... hundreds of views in total (though interestingly, less and less for each part... obviously not many people feel like reading through the whole thing :p). It's tempting to devote the blog entirely to that kind of topic now, because I know I'll get a lot of interested readers...
But no, I'm still going to use this blog to do random stuff. I may do another math-or-education-related post in the future, or just a questioning-the-way-we-do-things-in-general post, but the timing for that remains on MY terms. Mwa ha ha I'm so selfish.
So anyways. This was actually from August 17, when we were staying over at someone's house in the maritimes. And they had a left-handed guitar!
(sorry the background hum is kinda loud sometimes... the humidifier was on)
A few thoughts on playing left-handed guitar. First of all, IT'S HARD. Even though I had a solid theoretical base, and I knew chords, notes, fingerings, strum patterns, etc. it still took a long time to actually get my fingers to do what I wanted them to do (I'd actually had about an hour of practice before what you see on video). So for people picking up the instrument for the first time, who don't know the notes or chords, it must be even harder. I hope I can remember this experience for the future so I can empathize better with people learning for the first time.
I think it was actually a lot harder than when I learned right-handed guitar. Which leads me to my second point - having played violin for eight years before picking up guitar made it a lot easier to learn. Not only did I understand music better, but my left hand was used to holding down strings, and my right hand was used to moving back and forth to make the notes. I actually learned bass guitar too, starting in 6th grade - that also helped my left hand hold down strings, and my right hand make the notes. So by the time I got to guitar, my hands were kind of already used to that. But switching directions was hard because neither hand was used to what it was supposed to do.
Thirdly... the next time I get a right-handed guitar, I want to try playing it upside-down. It would be so cool to get good at both ways.
But no, I'm still going to use this blog to do random stuff. I may do another math-or-education-related post in the future, or just a questioning-the-way-we-do-things-in-general post, but the timing for that remains on MY terms. Mwa ha ha I'm so selfish.
So anyways. This was actually from August 17, when we were staying over at someone's house in the maritimes. And they had a left-handed guitar!
(sorry the background hum is kinda loud sometimes... the humidifier was on)
A few thoughts on playing left-handed guitar. First of all, IT'S HARD. Even though I had a solid theoretical base, and I knew chords, notes, fingerings, strum patterns, etc. it still took a long time to actually get my fingers to do what I wanted them to do (I'd actually had about an hour of practice before what you see on video). So for people picking up the instrument for the first time, who don't know the notes or chords, it must be even harder. I hope I can remember this experience for the future so I can empathize better with people learning for the first time.
I think it was actually a lot harder than when I learned right-handed guitar. Which leads me to my second point - having played violin for eight years before picking up guitar made it a lot easier to learn. Not only did I understand music better, but my left hand was used to holding down strings, and my right hand was used to moving back and forth to make the notes. I actually learned bass guitar too, starting in 6th grade - that also helped my left hand hold down strings, and my right hand make the notes. So by the time I got to guitar, my hands were kind of already used to that. But switching directions was hard because neither hand was used to what it was supposed to do.
Thirdly... the next time I get a right-handed guitar, I want to try playing it upside-down. It would be so cool to get good at both ways.
Subscribe to:
Comments (Atom)