tag:blogger.com,1999:blog-80177550947629556362014-10-04T22:43:01.206-02:30Without Loss of Generality:Musings of a Math NerdMiikehttp://www.blogger.com/profile/17108690155958578476noreply@blogger.comBlogger13125tag:blogger.com,1999:blog-8017755094762955636.post-19114579604256064412007-05-28T15:43:00.000-02:302007-05-28T15:56:04.741-02:30Argno maple makes miike a dull boy<br /><br />that is allMiikehttp://www.blogger.com/profile/17108690155958578476noreply@blogger.com1tag:blogger.com,1999:blog-8017755094762955636.post-38319775221747468542007-05-23T16:19:00.000-02:302007-05-23T16:23:02.043-02:30Invaders from The Fourth DimensionSo no updates for a couple days, but haven't been doing a whole lot of anything new until today.<br /><br />Today I started rewriting my programs... again. Except this time, they're being implemented to work in four-dimensional space. <span style="font-style: italic;">How cool is that?</span> It's not actually as simple as it sounds, though. Namely because, whereas in three dimensions every surface has only one straight line that is perpendicular to it (called the <span style="font-style: italic;">normal</span> vector), in four dimensions those same surfaces have <span style="font-style: italic;">two</span> lines that are perpendicular. Crazy, huh?Miikehttp://www.blogger.com/profile/17108690155958578476noreply@blogger.com0tag:blogger.com,1999:blog-8017755094762955636.post-15068404998977029172007-05-16T16:23:00.000-02:302007-05-16T16:36:57.622-02:30VideoconferencingAll I've been doing for the past two days is continue reading this general relativity book.<br /><br />Except for yesterday afternoon, when I had this meeting with some other NSERC students, one from here, three from Dalhousie in Halifax, and four students University of New Brunswick. But here's the kicker: no-one left their respective schools. The meeting was only to introduce ourselves and to see the system. At our school we had two massive LCD displays side-by-side, with top-quality webcams above, a speaker system below, and microphones scattered about the room. The two screens are hooked up to a computer which runs the software. On the screen we see about a dozen video feeds across the three schools (which all have similar setups, though Dalhousie's was more advanced with five touch-sensitive screens since they developed the system) and we could all talk and interact in real-time. It was very cool. Throughout the summer we're going to be pushing the system to its limits, improving ease-of-use, as well as developing interaction techniques that will be used to eventually teach entire courses over this system. Very high tech, very cool.Miikehttp://www.blogger.com/profile/17108690155958578476noreply@blogger.com0tag:blogger.com,1999:blog-8017755094762955636.post-83434807597904843352007-05-14T16:20:00.000-02:302007-05-14T16:38:51.553-02:30General RelativityToday I started reading about this titular subject, and with a massive scientific irony, is possibly one of the easiest concepts to understand in physics, modern or classical. Ever wonder why the Earth pulls us down? Newton says because two masses attract each other... but why? I mean, we know why two magnets attract each other: they're constantly sending photons back and forth "telling" the other to move closer (or further apart). But there is no such chatter in Newtonian gravity. So how do we know which way is down?<br /><br />Well many people know about "great circles". If you're looking at a map, the path of the plane traveling in a straight line appears an arc of a circle. That's because even though the plane is traveling in a straight line in three dimensions, the <span style="font-style: italic;">projection</span> of the path of the plane into two dimensions is curved. The idea behind general relativity is that objects are always traveling in a straight line in four dimensions (the "spacetime continuum"), but when you project that line into three dimensions (aka, the universe that we observe) you get a curve!<br /><br />Now you're probably wondering how mass is involved. Well, recall what I've been doing for the past couple weeks: calculating curvature. A straight line has no curvature. A slow tilt has small curvature. A sharp turn has high curvature. Well, with gravity and general relativity, mass <span style="font-style: italic;">is</span> the curvature. In free space, nothing around you, you go in a straight line. Near a planet, you curve into it. Near a black hole, you curve into it faster.Miikehttp://www.blogger.com/profile/17108690155958578476noreply@blogger.com0tag:blogger.com,1999:blog-8017755094762955636.post-43717886004952296962007-05-11T16:14:00.001-02:302007-05-14T16:32:35.789-02:30Week in Review: May 7-May 11Okay, so I've missed a few days. Here's what went down. I started calculating the <span style="font-style: italic;">other</span> kind of curvature, which is basically multiplying the same two values that were previously added to find the first kind. And I had to do that in two ways. The first way was proving to be horribly inaccurate (machine error popping up), but the second way (which calculates dead-on) is only valid for 3 dimensions. Once we go to four when only the first way will work. Quite the dilemma.Miikehttp://www.blogger.com/profile/17108690155958578476noreply@blogger.com0tag:blogger.com,1999:blog-8017755094762955636.post-62544751914278878682007-05-08T16:19:00.000-02:302007-05-08T16:26:45.843-02:30Symmetric, bitchToday I rewrote my programs entirely from scratch... again. Except this time there was a useful point. Consider the shapes I was working with. A sphere, torus, parabloid... those are all round, aren't they? You can spin them all you want around the <span style="font-style: italic;">z</span>-axis and it won't change how they look. Any surface like this is called <span style="font-weight: bold;">rotationally symmetric</span> and mathematically that means its curvature is dependent on only one variable instead of two.<br /><br />So now my algorithms calculate the exact same stuff they did before, except they do it in a tiny fraction of the time it used to take, and the error shrinks much faster as well. So woot!Miikehttp://www.blogger.com/profile/17108690155958578476noreply@blogger.com0tag:blogger.com,1999:blog-8017755094762955636.post-29926226448825249282007-05-07T16:11:00.000-02:302007-05-07T16:13:42.185-02:30I am HatedSo originally, the programs I wrote hated me, but now they've come to love me.<br /><br />Then Maple started hating me, but it adapted and now we get along semi-fine.<br /><br />So now OpenOffice Calc @#$%!-ing despises me, crashing, calculating things wrong, just in general pissing me off. And all I need to do is verify my programs are creating the right answers...Miikehttp://www.blogger.com/profile/17108690155958578476noreply@blogger.com0tag:blogger.com,1999:blog-8017755094762955636.post-31040955960129857962007-05-04T16:14:00.000-02:302007-05-04T16:19:17.847-02:30Week in Review: April 30-May 4First week was pretty good. Getting used to this whole 9-5 stuff, and even though I imagine this stuff will ultimately make me immune to ibuprofen from the headaches it's causing, I'm actually enjoying it. I've learned a nice bit about differential geometry, and my programs are starting to look bug-free and does the right job within reasonable error. So what's in store for next week? Stay tuned and find out!Miikehttp://www.blogger.com/profile/17108690155958578476noreply@blogger.com0tag:blogger.com,1999:blog-8017755094762955636.post-87681249413966645502007-05-03T16:05:00.000-02:302007-05-03T16:08:35.328-02:30Numerical Techniques: Part ThreeOkay, so not a whole lot new accomplished today. Merged all the programs into three simpler programs, which makes it a lot easier to test new calculations. Though honestly, I have no idea whether these are coming out right or not :P. But, they do work out perfect for a sphere, and the only changes I make for any other surface are the formulas used to calculate the surfaces, so I'm supposing they work out right.Miikehttp://www.blogger.com/profile/17108690155958578476noreply@blogger.com0tag:blogger.com,1999:blog-8017755094762955636.post-6599040790542225222007-05-02T15:45:00.000-02:302007-05-02T15:50:38.818-02:30Numerical Techniques: Part TwoSo today I expanded on the programs I made yesterday. Now it can handle a sphere, an ellipsoid, or torus, and ultimately finds what is called the "intrinsic curvature" of the object in terms of a particular distortion (currently involving just increasing the radius, essentially, but can be modified for any kind of distorting). I have no idea what this curvature actually represents, but apparently my programs calculate it right! That's always good.Miikehttp://www.blogger.com/profile/17108690155958578476noreply@blogger.com0tag:blogger.com,1999:blog-8017755094762955636.post-72373414623287919182007-05-01T16:17:00.000-02:302007-05-01T16:24:13.701-02:30Numerical TechniquesToday I began implementing some simple numerical differentiation and integration routines, applying them to some techniques found in differential geometry.<br /><br />In the span of nearly three hours, I wrote a series of five programs. These programs must be run sequentially, as each uses the data of the previous program (except, of course, for the first which generates the data).<br /><br />Five programs and three hours and I can successfully calculate the area of a sphere with radius 1.<br /><br />Okay, so maybe I'm demeaning this process a little. My <span style="font-style: italic;">initial</span> application was a unit sphere. But the first program can be modified to generate data for any kind of surface, and the remaining programs can find the area of said surface with little-to-no modification.Miikehttp://www.blogger.com/profile/17108690155958578476noreply@blogger.com0tag:blogger.com,1999:blog-8017755094762955636.post-2624492541672004232007-04-30T15:59:00.000-02:302007-05-01T16:25:53.967-02:30Differential GeometrySo today I read up on this titular subject, and came to the conclusion that it is the possibly cannibis-induced lovechild of set theory and vector calculus. For those unfamiliar with the subjects, they are two very core fields in mathematics. I know many people who are good at one and only one (myself included); I know of none adept at both.<br /><br />The following words have defined mathematical meanings that I was completely unaware of before today:<br /><ul><li>Patch</li><li>Umbilical</li><li>Monkey saddle</li><li>Maximal</li><li>Compact</li><li>Pullback</li><li>Pushout</li><li>Categorical dual<br /></li><li>Bundle</li><li>Fibre product</li><li>Functor</li></ul>And no, I am not exaggerating in any way. Each and every one of these terms is related in some manner to differential geometry, I have discovered them all in my reading.<br /><br /><span style="font-weight: bold;"><span style="font-style: italic;">Update:</span></span><span style="font-style: italic;"> I reworded things a little, as previously this entry sounded far more harshly than I intended.</span>Miikehttp://www.blogger.com/profile/17108690155958578476noreply@blogger.com0tag:blogger.com,1999:blog-8017755094762955636.post-19626565831940754382007-04-30T14:03:00.000-02:302007-04-30T14:05:23.743-02:30WelcomeMy name is Miike Barriault. I am a math student at Memorial University of Newfoundland. Starting April 30, 2007, I began working for one of my profs under a research grant. It is my intention to, every day, post on this blog whatever interesting thing happened to me that day or, if nothing is worth saying, just rant about something that's bugging me (or that I enjoy) about my job.Miikehttp://www.blogger.com/profile/17108690155958578476noreply@blogger.com0