gib65

Thanks both, The dot product works for angles from 0 to 180, but beyond that it starts repeating. For example, if (x',y') = (-1,-1) and (x,y) = (0, 0), then theta = 2.356, but if I set (x',y') = (-1,1) then theta = 2.356 - exactly the same. Obviously, I can deduce the angle on the other side by subtracting this result from 2*pi, but it would be best if I had a formula in which I didn't have to do this - the reason being I'm implementing this into a computer program, and I can't just tell it to "eye ball" the angle to tell whether or not it needs to subtract the result from 2*pi. What I need is a formula that gives results that span the whole range from 0 degrees to 360. Fuzzwood, Would your suggestion do this? I'm not sure what you mean by "finding the intersecting point". Do you mean the vertex (which wouldn't necessarily be at the origin)?

I'm not sure if this is in the right sub-forums. If not, please move it. If I were given two points (x,y) and (x',y'), how can I determine the angle they form? Let's assume that (x,y) is the vertex of the angle, and the other line the angle is to be measured from is (x+1,y). If the angle is to be defined by the area swept from (x+1,y), pivoting around (x,y) is a counterclockwise direction, and reaching the line connecting (x,y) and (x',y'), how would one calculate the angle?

Please have a look at this diagram of 5-meo-dmt: Is there any reason why the terminal molecule on the upper right branch says H3C whereas the other molecules on the other branches say CH3? Does it matter in notation whether the H3 comes before or after the C?

Seems counterintuitive after considering my reasoning above, doesn't it? Any way of reconciling this?

We all know that massive objects exert a gravitational influence on other nearby objects. This obviously means that gravity works across space. Should the same not be true of time? I've heard it said once that there's a principle that states that space and time are analogues of each other, meaning that whatever is true of the one is also true of the other. But what does it mean for an object to exert gravity across time? To me, it could mean nothing other than that an object exerts a gravitational force on itself in its future and past states. But what does that mean? To me, it seems it could only mean the same thing that it would mean in terms of space. In terms of space, it means that the object exerting gravity will pull other objects closer to itself, or that it will shorten the distance between them. Therefore, in terms of time, it means the object will shorten the amount of time between its past and future states. And this can only mean that it goes through all its temporal states faster than less massive objects. The more massive the object, the speedier it lives its life in the universe. So what I'd like to know is whether this makes sense theoretically, and whether there is any evidence for it. If the answer to either of these is 'no', why wouldn't it be true (I mean, it does seem to have a certain logic to it, doesn't it)?

Really? And I thought alex was just being humorous.

Personally, I think it's highly cultural, not biological. But if there is a biological component, the "twice the fun" response would have some reasoning behind it when you consider it's the male of the species who is typically more polygomous/permiscuous. So it might be that lesbians - or rather bisexual women - would be more open to a polygomous relationship than heterosexual ones.

I am told that as one falls into a black hole, the gravitational pull eventually becomes so strong that it literally pulls one apart. The difference in force between one's feet and one's head is so powerful that it has just this effect. But then I wonder whatever happens to the electromagnetic force holding the atoms and particles together within one's body. Under normal circumstances (i.e. in the absence of black holes), if a body feels a pull at one end and nothing at the other (or at least a weaker pull), the latter end just gets pulled along (in virtue of the electromagnetic bonds holding it together) by the former end. Why wouldn't the same happen in the case of an astronaught falling into a black hole? Is the difference in gravitational force really that different between head and foot? Is it so different that not even the electromagnetic force can compensate? That's the easy question. But I've got a trickier one: I've also been told that as one falls towards a black hole (approaching the event horizon), time passes by more slowly relative to others who are farther away from the black hole. In fact, as one approaches the event horizon, time for him, relative to those farther away, approaches a full halt (i.e. someone at the event horizon would appear frozen in time). Now, if we carry this over to our astronaught falling into the black hole, what would we say about time for his feet versus time for his head. If his feet are closer to the event horizon than his head, then wouldn't it appear to his head (assuming he's not dead from being torn apart) that his feet were moving much slower? Wouldn't it appear, therefore, as though he were catching up to his feet? Imagine this scenario should his feet be right at the event horizon (standing on it as it were). In that case, his feet should appear to his head to be "frozen in time" and therefore completely stopped. Rather than being pulled apart, he should experience being squished. What's wrong with this picture?

Calories that are hard to burn (like from fats and oils) are 'bad'. Calories that are easy to burn (like carbohydrates) are 'good'. But don't let the labels 'good' and 'bad' fool you; I know that the body needs all sorts of calories (or nutrients in general) and it needs a certain amount of fats and oils along with its carbohydrates. I'm also unsure whether all carbohydrates are equally 'good' for you - are the carbohydrates in a candy bar equally good for you as those in fruits? But I think I'm right in assuming that the kind of calories you take in makes no difference to the formula colries in - calories out = weight gain. So long as you maintain that formula such that the net weight gained is actual weight lost, you will in fact lose weight. Am I right? Is it possible that taking in too many 'bad' calories might result in fatigue or loss of energy? In that case, I can see how it might be harder to maintain the 'calories out' portion of the formula, but that could just be a question of will power or how hard you push yourself.

I've always been taught that the formula for how much weight one gains/loses is very simple: calories in - calories out = weight gained (not in so many words of course). But then I hear of 'good' calories and 'bad' calories - or - calories that are easy to burn vs. those that are hard. If I were on a weight loss routine - reducing calories in through diet and increasing calories out through exercise - why would I need to worry whether the calories I'm taking in are 'good' or 'bad'? If it's as simple as calories in - calories out then does it matter whether the calories in part is mostly 'good' calories or 'bad'? So I guess the question is: if I've got the formula right (calories in - calories out = weight gained), then how does the type of calories ('good' vs. 'bad') affect that formula?

I'm currently taking an online course in mathematics. One of the subjects is about interesting properties that come out of certain number series (or number patterns). Right off the bat, I'll spell out the purpose of this thread so you know where the OP is leading: I'm wondering if we can say that there's something "special" about these number series just because they bear certain interesting properties, or should we say (or prove) that there's always going to be interesting properties of any arbitrary number series no matter what it is? As an example of a number series for which an interesting property emerges, take the following: 1 + 3 = 4 1 + 3 + 5 = 9 1 + 3 + 5 + 7 = 16 1 + 3 + 5 + 7 + 9 = 25 1 + 3 + 5 + 7 + 9 + 11 = 36 The number series in question is simply the odd positive integers. The interesting property that emerges is that the sum of the first n of them is always n^2. I thought that was pretty interesting when I first heard it, but I couldn't figure out why that should be - until the lecturer gave a graphical explanation: Take five squared. It can be represented graphically by a 5 x 5 square. Now do the following: count the number of square cells in the bottom left corner. Obviously, there's 1 of them. Write this down as the first number in the series. Now count the number of cells surounding this first cell. There's one above it, one diagonal to it (above and to the right), and one to its right. That's 3 cells. Write that down as the next number in the series. Do the same for the group of 3 cells. You will find that there are 5 cells surrounding it (we're not counting the 1 cell in the corner since that's already been counted). Write that down as the next number in the series. You should see the pattern here. Since after all the counting is done, you will have counted all the cells in the square, the sum of them should amount to the same as 5^2. Here's another number series with a few interesting properties: The Fibonacci numbers (defined as Fn = Fn-1 + Fn-2, starting with 0 and 1): 0, 1, 1, 2, 3, 5, 8, 13, 21, 34... Did you know that, starting from 0, every third Fibonacci number, and only every third, is even. Furthermore, starting with 0, every fourth, and only every fourth, is divisible by 3. Furthermore, starting with 0, every fifth, and only every fifth, is divisible, not by 4 this time, but 5. The pattern continues: every 6th is divisible by 8, every 7th by 13, every 8th by 21, etc. The pattern turns out to be: Fn is divisible by Fm if and only if n is divisible by m (where F0 = 0). An explanation for the even-ness of every third Fibonacci number was offered (which obviously by now should be interpreted as "divisible by 2"): It's based on a couple simple rules of arithmetic: 1) an odd number added to an even number equals an odd number. 2) Two odd numbers added together equals an even number. Since the Fibonacci sequence is defined by the addition of the last two preceding numbers, the above two rules can determine the parity of the next one. So starting with the first two Fibonacci numbers, the third's parity can be determined, and so on after that: 0 + 1 = 1 (even + odd = odd) 1 + 1 = 2 (odd + odd = even) 1 + 2 = 3 (odd + even = odd) 2 + 3 = 5 (even + odd = odd) 3 + 5 = 8 (odd + odd = even) And so we see that the pattern repeats itself. And that is why every third is even. I'm willing to bet that the divisibility of Fn by Fm can be explained by similar, or more general, rules of arithmetic. Another interesting pattern that falls out of the Fibonacci series is as follows: If GCD(m,n) denotes the greatest common denominator of m and n, then F(GCD(m,n)) = Fm, Fn. This is to be read that the greatest common denominator of the two Fibonacci numbers Fm and Fn is also a Fibonacci number - namely the GCD(m,n)th Fibonacci number. In the course, the lecturer gives this example: GCD(70, 90) = 10 Therefore, GCD(F70, F90) = F10. Unfortunately, he didn't offer an explanation for this one. I find all the above fascinating. It gives one (or me at least) the sense that there's something "special" about the Fibonacci sequence, and to a lesser extent, the odd integers - as though any other arbitrary number sequence wouldn't have nearly as many or interesting properties. But I wonder if this is true. Were I to invent a completely arbitrary number sequence, would I find, if I studied it deeply enough, that just as many and interesting properties emerge from it as well? For example, let's say I defined the Gib numbers as follows: 1, 2, 2, 4, 8, 32, 256, 8192,... That is, starting with 1 and 2, the next number is the product of the previous two: 1 x 2 = 2 2 x 2 = 4 2 x 4 = 8 4 x 8 = 32 8 x 32 = 256 32 x 256 = 8192 Right off the bat, I notice this series can be written: 2^F0, 2^F1, 2^F2, 2^F3, 2^F4, 2^F5, 2^F6, 2^F7,... That is to say, the nth number in the Gib series is 2 raised to the nth Fibonacci number (where n begins at 0). Ok, anything else? I'm not sure. Let me know if you find something. What if I came up with something I bit more complex? Say: 1, 2, 6, 24, 120, 720,... which is to say, starting with 1, the nth number is the (n-1)th times n (where n begins at 1). Will there be just as many "neat" things about this number series as there are in the Fibonacci one or the odd number series? Will there be just as many in any arbitrarily constructed series?

I was thinking about this question today and started wondering whether it even makes sense to talk about the density of a fundamental particle. It makes sense to talk about the density of, say, a nucleus. You would define it as the amount of volume taken up by the protons and neutrons (as opposed to the space between them) over the amount of total volume enclosed by the nucleus as a whole. The density of protons and neutrons could in turn be likewise defined except in terms of quarks and the space between them. But a quark is (supposedly) fundamental. It has no internal parts. So do we say that its density is 1 (occupied volume / total volume = 1), in which case all fundamental particles have a density of 1, or do we put it in terms of mass over volume, in which case some fundamental particles could be said to be more dense than others, or do we say that density simply doesn't apply to fundamental particles?

Oh, I think I understand. So while the light was traveling its 13.7 billion lightyear journey, the light source was on a journey of its own in the other direction (because of expansion), and the total distance between the source and us now is 45 billion lightyears... right?

I'm not getting any numbers. That might be the problem. I'm not thinking of this mathematically, but visually. I'm trying to visualize the size of the visible universe in comparison to one whose circumpherence is 628 billion lightyears. I'm assuming the visible universe has radius 13.7 billion lightyears (it's 13.7 billion years old, and so the farthest light could have traveled is 13.7 billion lightyears, and so the farthest we can see is 13.7 billion lightyears away, and that defines the "visible universe"). If the minimum radius of the universe is roughly 100 billion lightyears as you say, that's a ratio of 13.7:100 or 13.7% the size of the total universe. At this point, I'm just using some layman's reasoning and my visualization to figure that taking a portion of 13.7% of the total universe and trying to measure some curviture therein should give you something, shouldn't it? It ain't like the surface of a penny compared to the surface of the Earth - it's more like the UK compared to the surface of the Earth. The surface area of the UK should have some noticeable curvature. No? This all probably sounds like silly reasoning to an expert like you, Martin, but it's the best reasoning I can figure out

WOW!!! That's an incredibly big universe. But isn't that still not big enough for a margin of error of 2% for measuring the universe to be flat? I can explain my reasoning if necessary. Thanks Martin. I'm used to these thought experiments, so I don't really have much trouble with them or understanding what they signify. But what's new to me is how to imagine a finite flat universe. Whether it's a "three-sphere" or toroidal universe we're living in, those would be closed universes (right?) - and quite finite. But these models could only serve as useful guides for how to think of space travel in a finite flat universe. If the universe is flat, then we can't actually say we're living in a "three-sphere" or on the surface of a 4D torus - we can only say it is as if we were (and only in some respects).

Hmm... I have difficulty dissociating this scenario with the model of a 4D sphere on which our 3D universe is its surface. I guess I could chalk the 4D sphere up to a useful conceptual tool only and that the real nature of space - that is, how it wraps around like you say - to something beyond my ability to comprehend. But is there anything wrong with imagining an infinite amount of matter and energy filling the universe?

Is this more or less what Sisyphus was saying? If it is, then it follows that the amount of matter and energy in the universe is infinite. Is this possible? If it's not infinite, then all matter and energy must be localized around a central point. It doesn't have to be condensed around that point, but if one were to travel far enough, one would eventually enter a region of space where no matter or energy existed (except for the person himself, of course). The universe would essentially have a center. I'm told that no such center exists, and so I must conclude that the amount of matter and energy, in this case, would be infinite. Is this tenable?

See, the problem I have with an infinite space is that it conflicts with my understanding of the Big Bang Theory. According to what I understand, the Big Bang began as a singularity (or something close to one) and this included space itself - that is to say, space itself began to expand with everything else. To my mind, the only model that fits this is the one of a 4D sphere where our 3D universe is its surface. If the universe is flat or open, that means either space has expanded to infinity already (which is untenable) or there is some kind of limit to space - a "wall" so to speak - that is a finite distance away and can supposedly be encountered if one traveled far enough (which seems absurd to me). An alternative is that space is infinite to begin with and the Big Bang occured at some local region in space. Space could still be said to be expanding according to this model, but not all of it starting from a singularity. The problem with this is that, from what I understand, the contents of the early universe (particles, gas, photons, etc.) are said to have filled all space and so they can't be limited to a local region.

Thanks Spyman. This is the part that I'm a bit confused about: I was under the impression that the universe could be thought of as the 3D surface of a 4D sphere and that all planets, stars, and galaxies were like objects on its surface. It expands like a balloon being filled with air. But this means the universe is necessarily finite. Why would it be uncertain whether it is finite or infinite? I started a thread similar to this here. Maybe you can comment on that one too if you have something to say.

Is there such a thing as a factor of intelligence for collecting trivial facts? My wife seems to have something like this. Everytime I learn something interesting and I tell her about it, she replies "Yeah, I knew that". I tried testing her a while back to see if she was just saying that to seem smarter than me: I made up some crap as if it were fact and told her about it. She passed the test - she admitted not knowing about it. So I think there must be some factor of intelligence for picking up on mundane facts. Is there?

One of the predictions of GW is that the polar ice caps will melt and cause sea levels to rise. Have they been rising?

I know that the known universe is roughly 27 billion light years in diameter, but what about the unknown universe - that is, the universe even beyond what is visible? I used to assume it just wasn't known how big it was until I came across a chart the other day depicting the rate of expansion of the early universe. It was like this: Time: size: 10^-40s 1mm 10^-30s 100m 10^-20s Earth 10^-10s solar system 1s 10 Ly Now I'm not sure how reliably we can extrapolate this chart to today's universe. The rate of expansion has been changing after all. From what I understand according to the inflationary theory, the universe started out by accelerating, then decelerated, and is now into an accelerating phase once again. But given that physicists seem to have some kind of grasp on its rate of expansion and size at various points in its history, and that we know the age of the universe, can we estimate with any confidence its size today?

Sorry, that was years ago. I have no idea what article it was. I can ask the guy I'm arguing with for his sources.

I'm in a debate with someone on another forum about whether AD(H)D actually exists. He thinks it doesn't. I argued a point with him that studies have shown that those with AD(H)D are statistically more likely to have smaller right frontal lobes than "normal" people. He countered that with studies that show that ritalin and other amphetamines cause shrinkage in the right frontal lobe (implying that the small frontal lobes in AD(H)D people weren't actually small before medication). I want to know, first, if this is right, and second, if he's right in thinking that there are no neurological differences between AD(H)D people and "normal" people.

Right, and who's to say the fourth dimension isn't time? I sometimes like to think of the radius of the 4D sphere universe as time.