Shadow

I don't think this warrants a separate topic and since it's related, I'll ask here. Can motion in 3D be represented as rotation in 4D? I was thinking about the 2D projection of a point on a rotating sphere. It would be moving in 2D. If the answer is yes, can time be understood as 4D rotation, since time is basically something that allows 3D objects to change, ie. move? I apologize if the questions are absolute rubbish, but they have been lying around in my head for a while now, and I'm curious as to what the answers are.

In less technical terms, you basically can't visualize a four dimensional anything the same way you can visualize a 3D cube for example. It's like trying to draw a 3D cube on a piece of paper; just can't be done. What we do instead is called projection. Projection is what you do every time you draw a 3D object, be it a cube, a tetrahedron, or even more complex objects such as a house, on a piece of paper; I could explain further, but I think it'll be much easier to comprehend if you look at the image at the top of the last page ajb links to (stereographic projection). This process can be extended to higher dimensions, which is the closest you can come to "visualizing" a higher dimensional object. But it doesn't actually tell us what it'll look like, it just gets us a little closer. If you only draw the outer edges of a projected 3D cube, you get a sort of weird looking hexagon if my imagination is working correctly, but that doesn't mean a 3D cube will actually look like a hexagon. In the same way, a projection of a tesseract into 2D looks like two 3D cubes projected into 2D, but that doesn't in the slightest mean that it actually looks that way. There are other forms of "visualizing" higher D objects, such as the Shlegel diagrams ajb mentioned or making a net of an object. There are probably more that I'm not aware of, explore the internet and see for yourself. Another thing you can do is conceptualization. Even though you can't picture it in your mind, you can work with an abstract concept obeying the rules of higher dimension geometry, and potentially do lots of cool stuff in you head without actually seeing a single picture. Also, and I probably should have led with this, I'm no expert in this matter. The above was merely an attempt to convey my view of the subject, and may contain untruths. If so, I apologize.

Careful; the mistake I made is the same you just made; [math] 25 \neq 25\%[/math]. Think of the percent sign as a unit, like in physics.

I see what you mean. The multiplycation by 100 in my version was an attempt to convert to percent directly, as in the OP. If I added the percent sign, as in [math]\frac{x}{x+3} * 100 \%[/math], would it then be correct?

Why?

There you go. That's the equation you were looking for.

If the angle with the x-axis is what you want, it's easier to calculate the inverse tan of the slope. But again, be careful; if the angle is larger than 90°, the angle will be negative, and you have to add it, or subtract it's absolute value, to/from 180°.

Very good. Now, we have the general form of the fraction, but that only gives us numbers smaller than one, while we want number in the range <0, 100> (or equivalently, percent). So, instead of getting 0.5, we want 50. Instead of getting 0.625 we want to get 62.5. Try altering the general from so it gives these results.

If you just want someone else to do the thinking for you, practically everyone here can give you the correct answer, or you can use WolframAlpha, which is much quicker and easier to use than Excel. But I think you should try and figure out the answer by yourself if you can, it's good practice. Now, I'm not sure what you're doing in those two divisions; it seem that in the first case, you somehow mistook [math]\frac{3}{3+1}[/math] for [math]\frac{3 + 1}{3}[/math]. There is a very big difference between the two, and it is not arbitrary which way you choose to evaluate the fraction. True, the precedence rule states that division should be performed before addition. But the mistake you made was dividing three by three, when in fact you were supposed to divide three by three plus one. Now, since you can hardly divide a number by two other number and a mathematical operation all in one, you first have to evaluate 3 + 1 = 4; thus, [math]\frac{3}{3+1} = \frac{3}{4} = 0.75[/math]. The person(s) on Yahoo! were correct, but again, you're letting other people do the work for you. Focus on the question I asked previously; can you write down the general form of the fractions [math]\frac{1}{4}, \frac{2}{5}, \frac{3}{6}...[/math]? Here are a couple of hints at how to arrive at the solution: Analyze the fractions you have in front of you. Note that if the numerator is one, the denominator is four, which is one plus three. If the numerator is two, the denominator is five, which is two plus three, if the numerator is three the denominator is six, which is three plus three and so on. Can you find a rule for how the fractions are generated? Ask yourself, if the numerator were four, what would the denominator be? If the numerator were five, what would the denominator be? Finally, ask yourself, if the numerator were some number [math]x[/math], what would the denominator be? Or, and I'm almost telling you the answer here, how much bigger would the denominator be? Hope this helps. PS.: If you can describe the rule using English but not using math, do so, and we'll work from there.

That's like asking "What's the smallest (positive) real number?". There isn't one. Alternatively, you can consider infinitesimals.

I don't think this can be described by a (non-piecewise) equation, unless the last three percentages are 57.1429..., 62.5 and 66.666... . If that is indeed the case (which it should be according to the relationship you gave at the end), it isn't very difficult to find the equation. For starters, can you write down the general form of the fractions 1/4, 2/5, 3/6 etc. ? (for example, the general form of the fractions 1/2, 2/3, 3/4 etc. would be [math]\frac{x}{x+1}[/math]).

You can use the dot product to calculate the angle between two vectors. However, you have to be careful how you choose your vectors if you want the specific angle you were talking about.

The title speaks for itself. I guess the question should be narrowed down to irrational numbers, since the rest are all rationals. I'm leaning towards no, although that's just intuition speaking. Note that I'm not asking if we can find a closed form expression for a given number; just if it always exists.

I never said anything about water. I am asking if he actually fell 22 stories, which, if D H is correct, he didn't. Also, due to surface tension, jumping into water from 22 stories would be like landing on concrete (assuming he's not a world-class cliff diver, and even then), so actually no, it wouldn't cushion his fall.

He landed at ground level? Ie. after falling 22 stories?

Okay, I consider myself owned

If you want a function that gives 0 for x < 0 and x for x> 0: [math]x (\frac{x}{2*|x|} + \frac{1}{2})[/math] Careful though, it's undefined for x=0.

Don't you apply L'Hopital if you get [math]\frac{\infty}{\infty}[/math]? EDIT: Never mind, I missed the point.

That makes a lot more sense. Thanks.

Nobody said you can't divide nothing. What everyone is saying is you can't divide by nothing. You can divide 0 apples between five people [math](\frac{0}{5})[/math]; every person get's exactly 0 apples. Dividing five apples between zero people [math](\frac{5}{0})[/math], now that's impossible. How many apples would each person get? And yes, you were dividing nothing, but apparently thinking you were dividing by nothing all along. Really? I've never heard of that before. Do you know of any resource where I could read up on this? I tried a quick Google search but that didn't turn anything up.

http://you-win-the-internet.com/

Erm...since when is 17 composite? Sorry for the delay in answering Vic, I completely forgot about this topic. I'm pretty tired at the moment, so I'm not all that excited about reading the whole thing again to remember what my argument was, but from what I do remember, your method requires remembering all the primes found thus far, to calculate the gaps (if not, I'm sorry, I really can't remember the method all that well). If that is the case, it is as effective, maybe even less, than the simplest algorithm for primes. If on the other hand you wanted to prove that there was a connection between the distribution of primes and composites (I'm going to go on a whim and assume that's what you meant by "same way as"), and I'm not sure I see how your method proves that, that can be (I think) proven a lot faster. If you take the set of natural number and remove all primes, all that's left are composites (excluding the number 1). Furthermore, primes are the building blocks from which composite numbers are created, so I'd certainly expect there to be a connection between the two. Please note however, I'm not an expert in this matter, far from it. Most of the above is just intuition working, so I apologize if it turns out to be gibberish. Also, I must say that's a pretty interesting way of generating primes. Thumbs up

Okay, that's what I though. In that case, my question is how do you calculate the individual slots? I can see they are prime gaps, but how do you decide how many of them will play a role in the "pattern"? In the pattern for 7, you use 8 prime gaps, in the pattern for 11, you use 48 prime gaps..why? And also, how do you determine the prime gaps in advance of knowing which numbers are prime?

I'm sorry, but I don't understand what "pattern size" and "slot values of patterns" are from your explanation.