mireazma

I don't understand "not necessarily the primary". Are you saying I could use other "cell" shapes, i.e. square, and still have equally spaced vertices on the surface of the sphere? I need as few vertices as possible per facet, but since 2 is impossible, I'm left with 3. This is because given a position on the sphere I have to compute its characteristics by interpolation between the neighboring vertices. It seemed at least at the beginning that it's as easy to generate equilateral triangles as it is for squares etc. but considerably more inefficient to interpolate in a square compared to a triangle. EDIT: I would have used a tessellated icosahedron (seen in the link you gave, studiot) but the small triangles resulting by division, projected on the sphere, have a different topology. Otherwise, there would have exist the respective regular polyhedron and I understand the only regular ones are the platonic ones.

Thank you for still sticking with me. Wait, CE = FD < diameter of the sphere which means in our situation -- where they don't intersect -- that the subtending arcs are smaller than a semi great circle. EDIT: thanks a lot for the link, it is a new stub to explore in my quest. I won't give up until I have proof that it's impossible to get what I want.

You have an eye for space geometry. I barely managed to just picture the idea. I'm not sure I have correctly but I don't know what to do further; I added the red points and something is off with eventual additional resulting blue arcs but I can't see what.

Thanks again. I can't get around the fact that the number of points/vectors is fixed (100) and no point resides on the great circle. With these conditions I think I cannot manage to solve it. I can't see a formula but I get the feeling it's doable through derivation [edited] integration over the curvature (maybe I'm talking non-sense but the double integral stands for the double curvature of the sphere -- 1 would by a cylinder) [/edited] and I don't know to derive Yes I missed the derivation and integration classes and from the web all I could learn is what they stand for but can't make calculations of their type, not taking into account the curved geometry. If I could see an example for an arbitrary number of points, i.e. the operations involved, I guess I would know to adapt it to any case, i.e. 100, 1000. So, if I'm not asking too much from anybody who cares to help me with this matter, I'd appreciate such an example very much. It's the only mathematics related part of my software -- the only part that keeps me from developing it.

Thank you both for replying. Daedalus, I'm lost with the double integrals thing; I was hoping for a more intuitive description. I'm just a lousy programmer and an even lousier mathematician and all I can assimilate is that if I know the given triplet (r, θ, φ) for a point I can get its corresponding vector by using the formula you supplied. This solves a half of my problem. The other half is to split the azimuth and inclination angles so to even-space the vectors, where studiot's reasoning comes into play. studiot, I have 2 things to be clarified about in your reasoning: 1. I know curved geometry has some differences compared to euclidian (?) but can a triangle be both right and equilateral at the same time? All appearances say it can and it is in your sketches. 2. Can the circles (horizontal spherical sections) be divided in numbers other than powers of 2 and still mentain equidistant points? Imagine the second picture in your sketch: we section the hemisphere by a horizontal plane at any point. I presume equilateral spherical triangles result independently of the elevation of the sectioning plane (resulting circle). a. If I'm right, if we split any circle in a number other than 4, the horizontal sides will be of course, shorter / longer, so no more "equilaterality". b. If I'm wrong, it follows that there is a unique elevation corresponding to a specific number of arcs of the circle. Although I think (a) is true, how can this be explained: In plane geometry we have a hexagon formed by 6 equilateral triangles (all triangles with a common corner -- center of hex and the additional corners common 2 by 2). If we only take 5 of those triangles and close the resulting gap, the whole shape would pop-up into a pentagonal pyramid. So a curve plane could contain all of the points. This curved "mesh" is a "patch" of a sphere, right? I presumed this patch can be extended by additional equilateral spherical triangles but from this point on it's out of my reach. I'm insisting on non powers of 2 because I have a fixed number of points / vectors, limited by the requirements of the software I'm making (I think it's 1000) I'm sorry I don't have a sketch for a more intuitive reading but I did my best at clarity of the exposition.

I know nothing about curved geometry but maybe I don't have to, after all. I want to evenly distribute an arbitrary number of points on a hemisphere. I visualized for example 1000 vectors starting from the same origin, on one side of a plane containing the origin. If they are normalized they'll resemble a hemisphere. Being evenly spaced means the 'tips" make for corners of equilateral triangles. I guess there are at least 2 possibilities: on the top of the "dome" there's a point OR the center of a triangle. Basically I want to find the vectors. Can it be explained with matrices at most? Quaternions are a little too hard for me and be it only if there's no other choice. Thanks in advance.

Wow, huge leap in time for this thread from 2007 to 2012 (pun intended) Why are you arguing whether time exists or debate its nature without mentioning space? I personally can't differentiate between time and space, as it's the same thing but in 2 colors. I think space is generally easier to comprehend so one should start from describing space first. Time sounds more mystic than space but after all it's another kind of space. It's more difficult to grasp because we only experience (generally) 2 dimensions of time and only one direction. But one can contemplate very easy the other direction of dimension 1. The real difficulty starts from more than 2 time dimensions. What happens if time would "flow" backwards? Well, first, it doesn't even flow. It's just there. It's like you said the road flows as you exist on it. But it's you who position yourself here, then there then back here, so on. You are traveling with respect to the road and not the other way around. I'm saying this because you initiate the movement, not the road. It's just you can't go backwards (or can you?) If we call a 6-th sense the one to perceive time with, say "seeing", we'd say we subjectively "see" time through a slit. Second, to answer, we would merely experience unwinding events. I like (please, indulge my personal preference) to see all events laid on a R3 space. With different observers come different referential systems, so origin is not relevant. The x axis I like to call destiny, y axis -- simultaneity and z axis, alternity. The cause-effect feature of 2 events is in fact the vicinity of 2 points in time. Let's say you are positioned at coordinate xyz. The observable course of events is just a horizontal plane xy at an alternity "height" y, containing that point (you). A line of this plane is a continuous cause-effect chain of an arbitrary infinitesimal events. Note that we call some event "the cause" of an other -- "the effect", depending on the direction of this line. If we take the positive direction of x (the one we experience), we say you knock out a pen on the table and caused it to fall. If you see it backwards, you don't say your finger retracted and the pen followed your finger but instead, the pen rose from the table and pushed your hand, causing it to retract. So a point (event) is the cause/effect of all of its vicinity points. You can draw a line, a curve and yes, even a circle on this plane to determine some causality. You tamper with events, being one of them, and observe the causality only in x positive direction, as discussed. As the causality line approaches x component 0, you make less sense (have a lower "resolution") of the causal feature until the line is perpendicular to x and has only a y component, when you can't relate a cause/effect between events on that line and sense all of them as instantaneity. So instantaneity is a line of causality parallel with simultaneity. The relativistic concept of difference of simultaneity would lie in the rotation about alternity (z) of the reference system but maintaining the xy time plane. Now to the 3rd dimension. If you could go back in time and alter the known course of actions, the only possible way you could alter is to "act" a changed "weave" of causality, different than the one with the known course. This new weave is a higher/lower xy plane, on the alternity axis. But you could only cause so little alteration, that is, to the size of an infinitesimal point in time, to move exactly 1 plane up/down, "after" which you continue on the current new xy plane. The more you do this alteration, the more you get further from the original plane. This is the way I like to look at time.

Related to the op question I can contribute with... some more questions ) like if gravity is not an intrinsic force of a body but is merely a curvature of spacetime, so pure geometry, (my) ignorant logic would dictate that a static body left in mid-air would remain static, as opposed to a moving object that would follow the curved spacetime around the Earth, for example. Where's the moving from, be it accelerated or uniform, after all?

I'm glad someone replied at last and TBH I'm glad it was you, A-wal. Well, I don't mean any discrimination but it's probably cause A-wal is the op. It's just relativity (especially the problems from above) has been drawing half of my mental resources for the past week and it's disturbing the normal college-work-resting cycle Anyway, about the twin paradox no matter how I don't want to allocate time, it's more and more obvious that I have to study more intensively -- I can't see the "picture" and it feels like the answer will come with study and not with contemplating alone. But it's not the case with problem no. 2. I would appreciate very much if I could be told what happens in my example with the 2 ships from my first post. Here it is again, revised: Again, my intention is to correct my view so what does really happen in the given situation, instead of what I wrote it would? Please, let me know.

Thank you for the explanations and the study materials. It'll take some time before I assimilate what's written both in the pdf and the article and I pinned these as the next steps in appropriating relativity. I got to read the time dilation (TD) and length contraction (LC) in the pdf and it raised even more questions I'm trying to put the ideas in order and temper the exponentially growing questions and solve them in order. I know it's not fair to ask you for answers without properly studying physics and mathematics but with planned study in mind, please, bare with me and answer as much as your time and mood allows you Generally I'm trying to provide as much and as clear info in order to help you help me. TD and LC are inseparable in that whenever/wherever one is present, the other is its consequence, right? 1. About the twin paradox, besides the TD/LC occurring between 2 different inertial frames I now know of TD (and LC?) resulting from acceleration. They're not the same thing because the latter effect is irreversible. I mean, it would be necessary that the whole system in which an accelerating ship senses "youth" but not the ship, to accelerate, so to reverse the effect. Here I'm trying to determine whether there's a difference in the way of looking at the two situations, rather than attempting to understand the accelerating TD. For the actual understanding I will study. 2. About the light beam clock, altho I read the "Special Relativity 2" pdf, I can't relate to what happens in the example from the post above, if the light is shot diagonally towards the aft of the flying ship. There are now extra risen questions for which I'll study for answers but it's too much for the moment (like whether TD and LC are mutually exclusive, LC from acceleration and others)

I came here thru Google, cause I was looking for explanations on time dilation. Physics has always intrigued me but my knowledge and understanding is somewhere close to 1, on a 1-10 scale and was even 0.2 like 2 weeks ago (before digging up youtube for vids). The satisfaction and joy in (roughly) understanding relativity and answering questions in different "what if" thought experiments started with the creation of a space game. This is to let you know what my level is. I'm glad I got here and I thank A-wal for this initiative -- it slaps you in the face with clear, concise and substantial explanations, it looks like a synthesis of how to look at things from relativistic perspective. If I were to say it's exactly what I was looking for well, I'd be lying: it's more! With a little systematization, revision in expressing and with an aid of pictures, this deserves to be the reference article on relativity, to which all new comers and amateurs to be referred to. Now the ontopic: Altho I've read it like 2 times I still have a couple of misfit things in my mind and I'd like to set them right. 1. The twin paradox This is what I understand: The more you move thru space, the less you move thru time. So a twin A who travels more, ages less than his twin E, remained on Earth (even if A flies away and then reverses direction, getting back to Earth). I tried to get it by reading A-wal's post multiple times but I think I'm biased by a vid on YouTube to not see it right. This is what I don't understand: motion is relative to one another, so from perspective A, E is also moving away and then closing. How come A is younger? And for the same reciprocal reason, A should see all things around him, not lengthened but shortened and slow. Yet another logical consequence: the time dilation, length contraction or both would not be isotropic, which brings me to: 2. The ceiling floor light beam We have ship A passing by ship E. A measures time by a photon bouncing between the ceiling and floor. But this time, the photon is shot diagonally, towards the back of the ship, so it appears to the stationary E that the photon is bouncing vertically, up and down. So for A the light path is longer and for E it's shorter, which makes time (activities) on A to appear faster seen from E. Following the idea, the direction of motion of an object (or more precisely, region of it) inside A, relative to traveling direction of A, sets the duration of that motion. As you'd suspect, the slowest motion would be in the direction of traveling. Ignoring length contraction, it would be (1 - vA) * tE, where vA is A velocity fraction of c, tE is the time perception of E. In the opposite direction i.e. motions towards back, we'd have (1 + vA) * tE. Any motion in between would be as faster as it approaches 180o but I don't know if I got it right about time anisotropy. I would very much want to understand how these things work. I have other questions as well (like is there theoretically possible to unify gravity time dilation with motion T.D.) but at least I would get these right first. I may not have elsewhere to get clarifications from and this seemed the perfect place. Thank you in advance.