Wednesday, February 3, 2010
Monday, January 18, 2010
Paper Art
More amazing stuff! Not exactly origami, but very detailed and beautiful pieces of art. Link here.
Tuesday, December 29, 2009
Case Study
Ok take a look at the couple of pictures here. The first is what I saw once when I went to davidchia.awardspace:
Apparently the site didn't load properly. When I refreshed, this was what I got:
So, this can serve as a case study for how MS word represents its equations from the Equation Editor as a String (a series of characters forming text).
Apparently the site didn't load properly. When I refreshed, this was what I got:
Monday, December 28, 2009
Another Paradox
Pity I'm too lazy to type it out as proper equations... This means that very few will understand what I'm about to type next.
Assuming that differentiation is valid over complex numbers, we have the taylor expansions of sin x, cos x and exp(x). From these we can derive euler's formula (but a more beautiful proof is found in one of the preceding posts):
exp(i theta) = cos theta + i sin theta (*)
(where i = sqrt(-1))
Because the taylor expansion does not restrict the function being expanded to only a real domain (and the definitions of cos, sin and exp also do not say anything about the domain), it is safe to say that theta in (*) can take a complex value. Now, what if theta = i?
exp(-1) = cos i + i sin i (**)
LHS is a real number, so imaginary part is zero
1/e = cos i, 0 = sin i
now if theta = -i:
exp(1) = cos (-i) + i sin (-i) (**)
LHS is a real number
e = cos (-i), 0 = sin (-i)
therefore, cos i is not equivalent to cos (-i).
This contradicts with the property of cos x = cos (-x). This property can be observed from the taylor expansion as well, so there is no conflict with letting x be complex.
Hmm anyone knows a way around this?
Shaun solved it immediately after I showed it to him: I assumed that the imaginary part at the equations marked (**) were i sin i and i sin (-i) repectively, but that was an invalid assumption, since either of them could have a real part as well.
Assuming that differentiation is valid over complex numbers, we have the taylor expansions of sin x, cos x and exp(x). From these we can derive euler's formula (but a more beautiful proof is found in one of the preceding posts):
exp(i theta) = cos theta + i sin theta (*)
(where i = sqrt(-1))
Because the taylor expansion does not restrict the function being expanded to only a real domain (and the definitions of cos, sin and exp also do not say anything about the domain), it is safe to say that theta in (*) can take a complex value. Now, what if theta = i?
exp(-1) = cos i + i sin i (**)
LHS is a real number, so imaginary part is zero
1/e = cos i, 0 = sin i
now if theta = -i:
exp(1) = cos (-i) + i sin (-i) (**)
LHS is a real number
e = cos (-i), 0 = sin (-i)
therefore, cos i is not equivalent to cos (-i).
This contradicts with the property of cos x = cos (-x). This property can be observed from the taylor expansion as well, so there is no conflict with letting x be complex.
Hmm anyone knows a way around this?
Shaun solved it immediately after I showed it to him: I assumed that the imaginary part at the equations marked (**) were i sin i and i sin (-i) repectively, but that was an invalid assumption, since either of them could have a real part as well.
Saturday, December 19, 2009
The Greatest Mystery in Biology
How do we fire nerve signals from the brain according to our will?
Much of life is composed of self-sustaining chemical reactions. These are the real chain reactions, with everything fitting together and one thing leading to the next. However, not everything is automatic; almost all movement in the body is due to muscular contractions and relaxations. Of course, there is an automatic part to them; the myosin heads consume ATP and climb along those fiber things (microfilaments?) whose names I've forgotten, and the cumulative effect forces the muscle to contract. This movement is stimulated by nerve signals. Now, if we trace the "automaticness" to its source, we find that nerve signals are passed from synapse to synapse by automatic chemical reactions, but we don't know how the reaction was started in the first place! What we do know is that when we want to move our finger, a nerve signal is generated and that signal sparks of a chain reaction to make some finger muscles contract or relax. So, what happens when we "want" to move something? This is paramount to finding out the connection between the consciousness and the body, and if only we had the ability to trace every nerve signal to its source, I think we'd see something that can't be explained by science.
How I wish someone would hurry up and find the answer.
I'd say that the two greatest mysteries in science are the Grand Unified Theory (a set of laws describing everything in the material world) and the connection between the consciousness and the material world.
By the way, there are some people who live normal lives with a brain that hasn't developed since they were in the womb. Brain scans show that their brain is just a shrivelled lump of something and they should be incapable of functioning anywhere near like a human should. One of those people became a math honors student, which is saying quite a lot for a shrivelled brain.
Much of life is composed of self-sustaining chemical reactions. These are the real chain reactions, with everything fitting together and one thing leading to the next. However, not everything is automatic; almost all movement in the body is due to muscular contractions and relaxations. Of course, there is an automatic part to them; the myosin heads consume ATP and climb along those fiber things (microfilaments?) whose names I've forgotten, and the cumulative effect forces the muscle to contract. This movement is stimulated by nerve signals. Now, if we trace the "automaticness" to its source, we find that nerve signals are passed from synapse to synapse by automatic chemical reactions, but we don't know how the reaction was started in the first place! What we do know is that when we want to move our finger, a nerve signal is generated and that signal sparks of a chain reaction to make some finger muscles contract or relax. So, what happens when we "want" to move something? This is paramount to finding out the connection between the consciousness and the body, and if only we had the ability to trace every nerve signal to its source, I think we'd see something that can't be explained by science.
How I wish someone would hurry up and find the answer.
I'd say that the two greatest mysteries in science are the Grand Unified Theory (a set of laws describing everything in the material world) and the connection between the consciousness and the material world.
By the way, there are some people who live normal lives with a brain that hasn't developed since they were in the womb. Brain scans show that their brain is just a shrivelled lump of something and they should be incapable of functioning anywhere near like a human should. One of those people became a math honors student, which is saying quite a lot for a shrivelled brain.
Thursday, December 3, 2009
Robert Lang on TED
The post is on my origami blog, the link is under my blog archive (you might want to visit the 5OSME homepage as well).
Sunday, November 15, 2009
Yo 401
Next year I'm gonna be taking math and physics honors, so I'll probably be separated from most of 401. Since everyone is doing class declarations, I might as well do one too. Hehehe 401 rocks cos everyone is nice to me (: I'll do the declarations according to the little groups from seating arrangements.
Jer Sheng Xing Yu Marc Jireh Ryan
Woah having Jer Sheng as a classmate has been quite the experience. In year three he'd been sitting next to me, and he's quite distracting. Luckily Marc is "controlling" him now. Xing Yu Marc Jireh = quiet in class? And Marc has improved lots recently, I was quite impressed during a Vectors lesson when he told Mr. Teo that the direction vector of the intersection of two planes (i.e. a line) is the cross product of the normal vectors of the planes. Ryan's pranks are too much!! Although I can't do anything to stop it, he's just to persistent... Anyway he's really funny and his pro book full of english mistakes is absolutely hilarious.
Jonathan Elston Bradford Stella Siying
Lol poor bradford and siying are NE ambassadors. It must be horrible to present to uninterested ppl T_T JESS always buys from the malay food stall, they're the reason why I dared to start eating from there again after the horrible stomachaches. Jonathan = "gay with a streak of evil" as I described him once. Streak of evil because he suans ppl very well. I really couldn't believe the chong wei thing at first, thought he was joking, but he's shown it so often that I've no chocie but to accept it. Nice person anyway! Elston = quiet guy with spooky laugh, talked with him the most during the Kevin Rudd talk last Friday, ate good food together (: His laugh never fails to make me laugh. Stella Siying = almost sisters! I remembered when I asked them to do this quiz, and the question was like "who's the most important person to you" and they both pointed at the each other at the same time. Stella, you shouldn't let ppl copy your homework whenever they want!!! Bradford's laugh sounds like pretence to me, you try to laugh out of everything! Crazy about Big Bang, I also like some of their songs. You were from a competitive class, but the level of kiasuness is just too high hahaha
Danielle Belinda Boon Siong Kiang Teng
Lol Danielle is so wierd. She has "Danielle moments" where she does really silly things, and she sort of breaks down when someone teases her about the (non-existant) dandruff she has. Scary claws, used for bullying people who tease you. Belinda is really reliable, like almost taking over the work of two mentor reps! Lim, you have some catching up to do (hint hint). Boon Siong is super guai person, I agree with kiang teng on "can't find a single flaw in his character". Nice person, but to me he usually looks emo which makes me nervous cos I'm easily affected by other people's moods. Boon Siong is as reliable as Belinda. Kiang Teng super fun person to be with, talks lots of funny stuff and suans Danielle a lot and gets attacked a lot as a result. That time he spilled curry all over his formal shirt, and had to borrow a girl's formal shirt cos there were no boy's ones left at the general office. After that we kept teasing him about girls' clothing. We all remember that birthday present you got! Never believe what he says easily, cos he scams others quite easily.
Huan Rong Yuhan Xinyi Eugenia Genevieve
Huan Rong = always sleeping! Why so sleepy... Had the interesting chinese debate with you during chinese class, sorry I had to keep asking you to repeat yourself cos you were speaking too fast! Yuhan = nice person who has same music taste as me and watches anime. Some of my favourite music was sent to me by her. She also recommended 07-Ghost and Kuroshitsuji, which are now two of my favourite manga. Cute mushroom emoticons hahaha! Xinyi = super hardworking and always smiling and helpful person. My horrible slackerness makes me cringe in guilt when I see you taking those details and neat notes. Hope you can achieve your goals! After the Kevin Rudd talk, my suspicions were confirmed... Alister, you totally have no chance now so you might as well give up. Eugenia = always laughing sillyly at something. Hehe I remember during one of the class potlucks where you asked me why I'm so smart. I replied, "You dont see me study in school right?" (she nods in agreement) "Well, I don't study at home either." To which she was in total disbelief. Hahaha sorry about the brag just now but I'm such a horrible slacker. Genevieve = quiet and sleeping often. You must have it tough, but we all don't know much about it cos we don't ask about it. Hope your leg recovers soon!
Xing Chen Chong Wei Alister Herng Yi Lim Sung Hyun Juin Bin
Xing Chen = super sleeper + basketball. Heck care in chinese class, I think you should at least try to listen in class even if you can get an A anyway otherwise. However, nice person anyway, eats with me the most. Chong Wei = one of those who talks a bit roughly, I was a bit distant from you so far but recently you acknowledged me as a friend and I'm more comfortable around you now. Last time in Mr. Wu's class, I think you went too far in saying that, and it was pretty stupid seeing as Mr. Wu was already angry at Xing Chen for saying "shut up" already. Ah well, at least everything went ok. You said during a class outing that I looked cool in that buttoned shirt, but up to now I don't know whether I should trust that remark. Alister = Happy and friendly person in a tragic korean (filipino) drama. After I got close to you guys recently I realised that you're actually quite nice to talk to. I remember Mrs. Soong's look on her face when we told her: "You mean... he has already 'confirmed his interests' ?!" And during a class when Mrs. Soong asked ali to look in front, and I said, "but _ is over there." which wa sone of my few successful jokes. As I said before, there's no hope already. Lim = math pro who has a dysfunctional email system. He really misses a lot of stuff and people have trouble contacting him... hope he's doing something about this. Got a band as well, plays drums, I want to hear them perform! Did something after a concert which I think was really really stupid and rash and he agreed with me after that. You need to put more effort in your work! He really cares a lot for the community, he can do things that normal people would never do like approach a beggar and talk to him, I'm really impressed but I don't know if you should be doing such things. He has fun philosophical arguments with me, and during english lessons he occasionally involves mr. valles as well. Juin Bin = another prolific sleeper, bad handwriting, but nice person anyways. You need to stay awake or you'll miss out on the tips and tricks the teachers teach us. You MUST improve you handwriting or you'll miss out on tons of opportunities cos you leave bad impressions. Puts a lot of effort into projects where I worked with him, nice team member. Fellow anime-watcher and manga-reader. Liar Game rules!
Mrs. Soong
Thought of this section last-minute. Our class has lots of problems so it's been hard on you, and we've made you really upset quite a number of times, so thanks for pulling through each time and accepting us! You always try to make classes interesting, and we can tell that you care for the people in 401. Ace flag-seller! You've also started using Georgia, which I think is an excellent choice for small pieces of text but I'd prefer the orgiinal font for lecture notes since that font is easier to read than Georgia. Once again, thanks for these 2 years!
Jer Sheng Xing Yu Marc Jireh Ryan
Woah having Jer Sheng as a classmate has been quite the experience. In year three he'd been sitting next to me, and he's quite distracting. Luckily Marc is "controlling" him now. Xing Yu Marc Jireh = quiet in class? And Marc has improved lots recently, I was quite impressed during a Vectors lesson when he told Mr. Teo that the direction vector of the intersection of two planes (i.e. a line) is the cross product of the normal vectors of the planes. Ryan's pranks are too much!! Although I can't do anything to stop it, he's just to persistent... Anyway he's really funny and his pro book full of english mistakes is absolutely hilarious.
Jonathan Elston Bradford Stella Siying
Lol poor bradford and siying are NE ambassadors. It must be horrible to present to uninterested ppl T_T JESS always buys from the malay food stall, they're the reason why I dared to start eating from there again after the horrible stomachaches. Jonathan = "gay with a streak of evil" as I described him once. Streak of evil because he suans ppl very well. I really couldn't believe the chong wei thing at first, thought he was joking, but he's shown it so often that I've no chocie but to accept it. Nice person anyway! Elston = quiet guy with spooky laugh, talked with him the most during the Kevin Rudd talk last Friday, ate good food together (: His laugh never fails to make me laugh. Stella Siying = almost sisters! I remembered when I asked them to do this quiz, and the question was like "who's the most important person to you" and they both pointed at the each other at the same time. Stella, you shouldn't let ppl copy your homework whenever they want!!! Bradford's laugh sounds like pretence to me, you try to laugh out of everything! Crazy about Big Bang, I also like some of their songs. You were from a competitive class, but the level of kiasuness is just too high hahaha
Danielle Belinda Boon Siong Kiang Teng
Lol Danielle is so wierd. She has "Danielle moments" where she does really silly things, and she sort of breaks down when someone teases her about the (non-existant) dandruff she has. Scary claws, used for bullying people who tease you. Belinda is really reliable, like almost taking over the work of two mentor reps! Lim, you have some catching up to do (hint hint). Boon Siong is super guai person, I agree with kiang teng on "can't find a single flaw in his character". Nice person, but to me he usually looks emo which makes me nervous cos I'm easily affected by other people's moods. Boon Siong is as reliable as Belinda. Kiang Teng super fun person to be with, talks lots of funny stuff and suans Danielle a lot and gets attacked a lot as a result. That time he spilled curry all over his formal shirt, and had to borrow a girl's formal shirt cos there were no boy's ones left at the general office. After that we kept teasing him about girls' clothing. We all remember that birthday present you got! Never believe what he says easily, cos he scams others quite easily.
Huan Rong Yuhan Xinyi Eugenia Genevieve
Huan Rong = always sleeping! Why so sleepy... Had the interesting chinese debate with you during chinese class, sorry I had to keep asking you to repeat yourself cos you were speaking too fast! Yuhan = nice person who has same music taste as me and watches anime. Some of my favourite music was sent to me by her. She also recommended 07-Ghost and Kuroshitsuji, which are now two of my favourite manga. Cute mushroom emoticons hahaha! Xinyi = super hardworking and always smiling and helpful person. My horrible slackerness makes me cringe in guilt when I see you taking those details and neat notes. Hope you can achieve your goals! After the Kevin Rudd talk, my suspicions were confirmed... Alister, you totally have no chance now so you might as well give up. Eugenia = always laughing sillyly at something. Hehe I remember during one of the class potlucks where you asked me why I'm so smart. I replied, "You dont see me study in school right?" (she nods in agreement) "Well, I don't study at home either." To which she was in total disbelief. Hahaha sorry about the brag just now but I'm such a horrible slacker. Genevieve = quiet and sleeping often. You must have it tough, but we all don't know much about it cos we don't ask about it. Hope your leg recovers soon!
Xing Chen Chong Wei Alister Herng Yi Lim Sung Hyun Juin Bin
Xing Chen = super sleeper + basketball. Heck care in chinese class, I think you should at least try to listen in class even if you can get an A anyway otherwise. However, nice person anyway, eats with me the most. Chong Wei = one of those who talks a bit roughly, I was a bit distant from you so far but recently you acknowledged me as a friend and I'm more comfortable around you now. Last time in Mr. Wu's class, I think you went too far in saying that, and it was pretty stupid seeing as Mr. Wu was already angry at Xing Chen for saying "shut up" already. Ah well, at least everything went ok. You said during a class outing that I looked cool in that buttoned shirt, but up to now I don't know whether I should trust that remark. Alister = Happy and friendly person in a tragic korean (filipino) drama. After I got close to you guys recently I realised that you're actually quite nice to talk to. I remember Mrs. Soong's look on her face when we told her: "You mean... he has already 'confirmed his interests' ?!" And during a class when Mrs. Soong asked ali to look in front, and I said, "but _ is over there." which wa sone of my few successful jokes. As I said before, there's no hope already. Lim = math pro who has a dysfunctional email system. He really misses a lot of stuff and people have trouble contacting him... hope he's doing something about this. Got a band as well, plays drums, I want to hear them perform! Did something after a concert which I think was really really stupid and rash and he agreed with me after that. You need to put more effort in your work! He really cares a lot for the community, he can do things that normal people would never do like approach a beggar and talk to him, I'm really impressed but I don't know if you should be doing such things. He has fun philosophical arguments with me, and during english lessons he occasionally involves mr. valles as well. Juin Bin = another prolific sleeper, bad handwriting, but nice person anyways. You need to stay awake or you'll miss out on the tips and tricks the teachers teach us. You MUST improve you handwriting or you'll miss out on tons of opportunities cos you leave bad impressions. Puts a lot of effort into projects where I worked with him, nice team member. Fellow anime-watcher and manga-reader. Liar Game rules!
Mrs. Soong
Thought of this section last-minute. Our class has lots of problems so it's been hard on you, and we've made you really upset quite a number of times, so thanks for pulling through each time and accepting us! You always try to make classes interesting, and we can tell that you care for the people in 401. Ace flag-seller! You've also started using Georgia, which I think is an excellent choice for small pieces of text but I'd prefer the orgiinal font for lecture notes since that font is easier to read than Georgia. Once again, thanks for these 2 years!
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